# Given Information - The starting salary offer for psychology majors, \( X \), is normally distributed. - Mean (\( \mu \)): $47,507 - Standard deviation (\( \sigma \)): $5,000 # What Have to Find 1. Probability \( P(X > 52,000) \) 2. Probability \( P(45,000 < X < 52,000) \) 3. Percentage of psychology majors with a starting offer between $38,000 and $45,000. 4. Salary value below which 20% of psychology majors are offered. # Definition or Concept Used - **Standard Normal Distribution:** Any normal distribution can be transformed into the standard normal distribution (mean = , standard deviation = 1) using the z-score formula: \[ z = \frac{X - \mu}{\sigma} \] - **Probability from Z-score:** Use z-tables to find the area under the curve to the left of a given z-score. --- # Step by Step Solution ### 1. Probability \( P(X > 52,000) \) Calculate the z-score: \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} = .8986 \] From the z-table, \( P(Z < .90) \approx .8159 \). So, \( P(Z > .90) = 1 - .8159 = .1841 \). --- ### 2. Probability \( P(45,000 < X < 52,000) \) Calculate the z-scores: \[ z_1 = \frac{45,000 - 47,507}{5,000} = \frac{-2,507}{5,000} = -.5014 \] \[ z_2 = \frac{52,000 - 47,507}{5,000} = .8986 \] From the z-table: - \( P(Z < -.50) \approx .3085 \) - \( P(Z < .90) \approx .8159 \) \[ P(45,000 < X < 52,000) = .8159 - .3085 = .5074 \] --- ### 3. Percentage between $38,000 and $45,000 Calculate z-scores: \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} = -1.9014 \] \[ z_2 = \frac{45,000 - 47,507}{5,000} = -.5014 \] From z-table: - \( P(Z < -1.90) \approx .0287 \) - \( P(Z < -.50) \approx .3085 \) \[ \text{Percentage} = (.3085 - .0287) \times 100 = 27.98\% \] --- ### 4. Salary below which 20% are offered Find z-score for the 20th percentile: \( P(Z < z) = .20 \). From z-table, \( z \approx -.84 \). \[ X = \mu + z\sigma = 47,507 + (-.84) \times 5,000 = 47,507 - 4,200 = 43,307 \] --- # Final Answers (Summary) - Probability that a psychology major received a starting offer greater than $52,000: **.184** - Probability that a psychology major received a starting offer between $45,000 and $52,000: **.507** - Percentage received a starting offer between $38,000 and $45,000: **27.98%** - Salary below which 20% were offered: **$43,307**

# Given Information - Starting salary offer for psychology majors, \( X \), is normally distributed. - Mean (\( \mu \)): $47,507 - Standard deviation (\( \sigma \)): $5,000 # What We Have to Find 1. Probability \( P(X > 52,000) \) 2. Probability \( P(45,000 < X < 52,000) \) 3. Percentage of psychology majors with a starting offer between $38,000 and $45,000. 4. Salary value below which 20% of psychology majors are offered. # Definition or Concept Used - **Standard Normal Distribution:** Any normal distribution can be standardized using the z-score formula: \[ z = \frac{X - \mu}{\sigma} \] - **Probability from Z-score:** Use z-tables or Excel functions to find the area under the curve to the left of a given z-score. ### Excel Functions Used - To find cumulative probability: \[ =NORM.DIST(X, \mu, \sigma, TRUE) \] - To find probabilities from z-scores: \[ =1 - NORM.DIST(z, 0, 1, TRUE) \] --- # Step-by-Step Solution ### 1. Probability \( P(X > 52,000) \) Calculate the z-score: \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Using the Excel function: ``` =NORM.DIST(52,000, 47,507, 5,000, TRUE) ``` From the z-table, \( P(Z < 0.90) \approx 0.8159 \). Thus, \[ P(Z > 0.90) = 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Calculate the z-scores: - For $45,000: \[ z_1 = \frac{45,000 - 47,507}{5,000} = \frac{-2,507}{5,000} \approx -0.5014 \] - For $52,000 (already calculated): \[ z_2 \approx 0.8986 \] Using Excel: - For $45,000: ``` =NORM.DIST(45,000, 47,507, 5,000, TRUE) ``` - For $52,000 (as calculated): ``` =NORM.DIST(52,000, 47,507, 5,000, TRUE) ``` From the z-table: - \( P(Z < -0.50) \approx 0.3085 \) - \( P(Z < 0.90) \approx 0.8159 \) Thus, \[ P(45,000 < X < 52,000) = 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage between $38,000 and $45,000 Calculate z-scores: - For $38,000: \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} \approx -1.9014 \] - For $45,000 (already calculated): \[ z_2 \approx -0.5014 \] Using Excel: - For $38,000: ``` =NORM.DIST(38,000, 47,507, 5,000, TRUE) ``` From the z-table: - \( P(Z < -1.90) \approx 0.0287 \) - \( P(Z < -0.50) \approx 0.3085 \) Thus, \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary below which 20% are offered Find z-score for the 20th percentile: \[ P(Z < z) = 0.20 \implies z \approx -0.8416 \] Using the inverse of the z-score: \[ X = \mu + z \sigma = 47,507 + (-0.8416) \times 5,000 \] Calculating, \[ X \approx 47,507 - 4,208 = 43,299 \] --- # Final Answers (Summary) - Probability that a psychology major received a starting offer greater than $52,000: **0.1841** - Probability that a psychology major received a starting offer between $45,000 and $52,000: **0.5074** - Percentage of psychology majors with a starting offer between $38,000 and $45,000: **27.98%** - Salary below which 20% were offered: **$43,299**

# Given Information - Starting salary offer for psychology majors, \( X \), is normally distributed. - Mean (\( \mu \)): $47,507 - Standard deviation (\( \sigma \)): $5,000 # What We Have to Find 1. Probability \( P(X > 52,000) \) 2. Probability \( P(45,000 < X < 52,000) \) 3. Percentage of psychology majors with a starting offer between $38,000 and $45,000. 4. Salary value below which 20% of psychology majors are offered. # Definition or Concept Used - **Standard Normal Distribution:** Any normal distribution can be standardized using the z-score formula: \[ z = \frac{X - \mu}{\sigma} \] - **Probability from Z-score:** Use z-tables or Excel functions to find the area under the curve to the left of a given z-score. ### Excel Functions Used - To find cumulative probability: \[ =NORM.DIST(X, \mu, \sigma, TRUE) \] - To find probabilities from z-scores using: \[ =1 - NORM.DIST(z, 0, 1, TRUE) \] --- # Step-by-Step Solution ### 1. Probability \( P(X > 52,000) \) Calculate the z-score: \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Using the Excel function for cumulative probability: ```excel =NORM.DIST(52000, 47507, 5000, TRUE) ``` This will return the cumulative probability \( P(Z < 0.8986) \). Thus, using the z-score: \[ P(Z > 0.8986) = 1 - NORM.DIST(52000, 47507, 5000, TRUE) \approx 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Calculate the z-scores: - For $45,000: \[ z_1 = \frac{45,000 - 47,507}{5,000} = \frac{-2,507}{5,000} \approx -0.5014 \] - For $52,000 (already calculated): \[ z_2 \approx 0.8986 \] Using Excel: ```excel =NORM.DIST(45000, 47507, 5000, TRUE) // For $45,000 =NORM.DIST(52000, 47507, 5000, TRUE) // For $52,000 ``` This provides: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Thus, \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage between $38,000 and $45,000 Calculate z-scores: - For $38,000: \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} \approx -1.9014 \] - For $45,000 (already calculated): \[ z_2 \approx -0.5014 \] Using Excel: ```excel =NORM.DIST(38000, 47507, 5000, TRUE) // For $38,000 =NORM.DIST(45000, 47507, 5000, TRUE) // For $45,000 ``` This gives: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Thus, the percentage is: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary below which 20% are offered Find the z-score for the 20th percentile: Using the Excel function: ```excel =NORM.INV(0.2, 47507, 5000) ``` This provides the salary value: \[ X \approx 43,299 \] --- # Final Answers (Summary) - Probability that a psychology major received a starting offer greater than $52,000: **0.1841** - Probability that a psychology major received a starting offer between $45,000 and $52,000: **0.5074** - Percentage of psychology majors with a starting offer between $38,000 and $45,000: **27.98%** - Salary below which 20% were offered: **$43,299**

# Given Information - The starting salary offer for psychology majors, \( X \), follows a normal distribution. - Mean (\( \mu \)): $47,507 - Standard deviation (\( \sigma \)): $5,000 # Objectives 1. Calculate the probability \( P(X > 52,000) \). 2. Calculate the probability \( P(45,000 < X < 52,000) \). 3. Determine the percentage of psychology majors with a starting offer between $38,000 and $45,000. 4. Find the salary value below which 20% of psychology majors are offered. # Concepts Utilized - **Standard Normal Distribution:** Any normal distribution can be represented in a standard normal format using the z-score: \[ z = \frac{X - \mu}{\sigma} \] - **Finding Probability with Z-scores:** The cumulative distribution function can be utilized through Excel's built-in function \( =NORM.DIST(z, 0, 1, TRUE) \). --- # Detailed Solution ### 1. Probability \( P(X > 52,000) \) Calculate the z-score for $52,000: \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Using Excel to find the cumulative probability: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This yields \( P(Z < 0.8986) \approx 0.8159 \). Thus, the probability of earning greater than $52,000 is: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Calculate the z-scores: - For $45,000: \[ z_1 = \frac{45,000 - 47,507}{5,000} \approx -0.5014 \] - For $52,000 (previously calculated): \[ z_2 \approx 0.8986 \] Using Excel for cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This provides: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Thus, the probability is: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage between $38,000 and $45,000 Calculate z-scores: - For $38,000: \[ z_1 = \frac{38,000 - 47,507}{5,000} \approx -1.9014 \] - For $45,000 (previously calculated): \[ z_2 \approx -0.5014 \] Using Excel: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This provides: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Calculating the percentage: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary below which 20% are offered To find the salary threshold for the 20th percentile, use: ```excel =NORM.INV(0.2, 47507, 5000) ``` This calculates the corresponding salary: \[ X \approx 43,299 \] --- # Summary of Findings - Probability that a psychology major received a starting offer greater than $52,000: **0.1841** - Probability that a psychology major received a starting offer between $45,000 and $52,000: **0.5074** - Percentage of psychology majors with a starting offer between $38,000 and $45,000: **27.98%** - Salary below which 20% were offered: **$43,299** --- This structured approach efficiently utilizes the z-score and Excel functions to derive essential probabilities and salary thresholds for psychology majors.

# Given Information - The starting salary offer for psychology graduates, denoted as \( X \), is normally distributed. - Mean (\( \mu \)): $47,507 - Standard Deviation (\( \sigma \)): $5,000 # Objectives 1. Determine the probability \( P(X > 52,000) \). 2. Calculate the probability \( P(45,000 < X < 52,000) \). 3. Find the percentage of psychology graduates with starting offers between $38,000 and $45,000. 4. Identify the salary below which 20% of psychology graduates are offered. # Conceptual Framework - **Standard Normal Distribution:** To standardize a normal distribution, we use the z-score formula: \[ z = \frac{X - \mu}{\sigma} \] - **Probability Calculation Using Z-scores:** Use the formula \( =NORM.DIST(z, 0, 1, TRUE) \) in Excel to obtain cumulative probabilities. --- # Comprehensive Solution ### 1. Probability \( P(X > 52,000) \) First, calculate the z-score for $52,000: \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Using Excel to find the cumulative probability: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This gives \( P(Z < 0.8986) \approx 0.8159 \). Thus, the probability of earning more than $52,000 is: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Calculate the z-scores for the range: - For $45,000: \[ z_1 = \frac{45,000 - 47,507}{5,000} \approx -0.5014 \] - The z-score for $52,000 has already been calculated: \[ z_2 \approx 0.8986 \] Using Excel to find cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This yields: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) The probability for the specified range is: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage between $38,000 and $45,000 Determine the z-scores: - For $38,000: \[ z_1 = \frac{38,000 - 47,507}{5,000} \approx -1.9014 \] - For $45,000 (previously calculated): \[ z_2 \approx -0.5014 \] Using Excel: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This provides: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Calculating the percentage: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To find the salary that corresponds to the 20th percentile, use: ```excel =NORM.INV(0.2, 47507, 5000) ``` This yields the salary threshold: \[ X \approx 43,299 \] --- # Summary of Findings - Probability that a psychology graduate received a starting offer greater than $52,000: **0.1841** - Probability that a psychology graduate received a starting offer between $45,000 and $52,000: **0.5074** - Percentage of psychology graduates with a starting offer between $38,000 and $45,000: **27.98%** - Salary below which 20% were offered: **$43,299** --- This structured solution effectively employs z-scores and Excel functions to derive critical probabilities and salary thresholds for psychology graduates.

# Given Information - The starting salary for psychology graduates, denoted as \( X \), follows a normal distribution. - Mean (\( \mu \)): **$47,507** - Standard Deviation (\( \sigma \)): **$5,000** # Goals 1. Calculate the probability \( P(X > 52,000) \). 2. Determine the probability \( P(45,000 < X < 52,000) \). 3. Find the percentage of psychology graduates with offers between $38,000 and $45,000. 4. Identify the salary that marks the 20th percentile for psychology graduates. # Concept Overview - **Standard Normal Distribution:** The normal distribution can be transformed into a standard normal distribution using the z-score formula: \[ z = \frac{X - \mu}{\sigma} \] - **Probability Calculation Using Excel Functions:** The cumulative distribution function can be utilized to find probabilities using: \[ =NORM.DIST(z, 0, 1, TRUE) \] --- # Detailed Steps ### 1. Probability \( P(X > 52,000) \) First, calculate the z-score for $52,000: \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Using Excel to find the cumulative probability: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This returns \( P(Z < 0.8986) \approx 0.8159 \). Thus, to find the probability of earning more than $52,000: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Calculate the z-scores: - For $45,000: \[ z_1 = \frac{45,000 - 47,507}{5,000} \approx -0.5014 \] - For $52,000 (as calculated previously): \[ z_2 \approx 0.8986 \] Using Excel for cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This provides: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Thus, the probability for the salary range is: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage between $38,000 and $45,000 Calculate the z-scores: - For $38,000: \[ z_1 = \frac{38,000 - 47,507}{5,000} \approx -1.9014 \] - For $45,000 (calculated previously): \[ z_2 \approx -0.5014 \] Using Excel: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This yields: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Calculating the percentage of graduates in this range: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To determine the salary corresponding to the 20th percentile: ```excel =NORM.INV(0.2, 47507, 5000) ``` This function computes the salary threshold: \[ X \approx 43,299 \] --- # Summary of Results - Probability that a psychology graduate received a starting offer greater than $52,000: **0.1841** - Probability that a psychology graduate received a starting offer between $45,000 and $52,000: **0.5074** - Percentage of psychology majors with starting offers between $38,000 and $45,000: **27.98%** - Salary below which 20% were offered: **$43,299** --- This solution succinctly utilizes z-scores and Excel functions to derive essential probabilities and salary thresholds for psychology graduates, employing clear mathematical reasoning and structured analysis.

# Given Data - The starting salary for psychology graduates, represented as \( X \), follows a normal distribution. - Mean (\( \mu \)): **$47,507** - Standard Deviation (\( \sigma \)): **$5,000** # Objectives 1. Compute the probability \( P(X > 52,000) \). 2. Determine the probability \( P(45,000 < X < 52,000) \). 3. Calculate the percentage of psychology graduates with starting offers between $38,000 and $45,000. 4. Identify the salary level below which 20% of psychology graduates are offered. # Conceptual Framework - **Standard Normal Distribution:** The z-score transformation of a normal distribution is defined as: \[ z = \frac{X - \mu}{\sigma} \] - **Using Excel for Probability Calculation:** The cumulative distribution function can be assessed using: \[ =NORM.DIST(z, 0, 1, TRUE) \] --- # Step-by-Step Analysis ### 1. Probability \( P(X > 52,000) \) First, compute the z-score for $52,000: \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Use Excel to find the cumulative probability: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This yields \( P(Z < 0.8986) \approx 0.8159 \). Thus, to find the probability of earning more than $52,000: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Calculate the z-scores for the two values: - For $45,000: \[ z_1 = \frac{45,000 - 47,507}{5,000} \approx -0.5014 \] - For $52,000 (previously calculated): \[ z_2 \approx 0.8986 \] Utilize Excel to find the cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This provides: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Consequently, the probability for the specified range is: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage between $38,000 and $45,000 Determine the z-scores: - For $38,000: \[ z_1 = \frac{38,000 - 47,507}{5,000} \approx -1.9014 \] - For $45,000 (already computed): \[ z_2 \approx -0.5014 \] Using Excel: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This results in: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Thus, the percentage of graduates with offers in this range is: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To find the salary that corresponds to the 20th percentile: ```excel =NORM.INV(0.2, 47507, 5000) ``` This function computes the salary threshold: \[ X \approx 43,299 \] --- # Summary of Results - Probability that a psychology graduate received a starting offer greater than $52,000: **0.1841** - Probability that a psychology graduate received a starting offer between $45,000 and $52,000: **0.5074** - Percentage of psychology majors with starting offers between $38,000 and $45,000: **27.98%** - Salary below which 20% were offered: **$43,299** --- This analysis clearly employs z-scores and Excel functions to derive essential probabilities and salary thresholds for psychology graduates, ensuring clarity and precision throughout the calculations.

# Given Data - The starting salary for psychology graduates, denoted as \( X \), is normally distributed. - Mean (\( \mu \)): **$47,507** - Standard Deviation (\( \sigma \)): **$5,000** # Objectives 1. Calculate the probability \( P(X > 52,000) \). 2. Calculate the probability \( P(45,000 < X < 52,000) \). 3. Determine the percentage of psychology graduates with starting offers between $38,000 and $45,000. 4. Find the salary level below which 20% of psychology graduates are offered. # Concept Overview - **Standard Normal Distribution:** The z-score transformation for a normal distribution is defined by: \[ z = \frac{X - \mu}{\sigma} \] - **Using Excel for Probability Calculation:** The cumulative distribution function can be determined using: \[ =NORM.DIST(z, 0, 1, TRUE) \] --- # Detailed Steps ### 1. Probability \( P(X > 52,000) \) First, calculate the z-score for $52,000: \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Using Excel to find the cumulative probability: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This returns \( P(Z < 0.8986) \approx 0.8159 \). Thus, the probability of earning more than $52,000: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Calculate the z-scores for the two values: - For $45,000: \[ z_1 = \frac{45,000 - 47,507}{5,000} \approx -0.5014 \] - For $52,000 (previously calculated): \[ z_2 \approx 0.8986 \] Using Excel to find the cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This provides: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Thus, the probability for the specified range: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage between $38,000 and $45,000 Determine the z-scores: - For $38,000: \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} \approx -1.9014 \] - For $45,000 (previously calculated): \[ z_2 \approx -0.5014 \] Using Excel: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This provides: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Thus, the percentage of graduates with offers in this range: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To find the salary that corresponds to the 20th percentile: ```excel =NORM.INV(0.2, 47507, 5000) ``` This function computes the salary threshold: \[ X \approx 43,299 \] --- # Summary of Results - **Probability that a psychology graduate received a starting offer greater than $52,000:** **0.1841** - **Probability that a psychology graduate received a starting offer between $45,000 and $52,000:** **0.5074** - **Percentage of psychology majors with starting offers between $38,000 and $45,000:** **27.98%** - **Salary below which 20% were offered:** **$43,299** --- This analysis effectively utilizes z-scores and Excel functions to derive key probabilities and salary thresholds for psychology graduates, ensuring clarity and precision throughout the calculations. Each z-score was calculated with the corresponding \( X \) value clearly noted for transparency.

# Given Data - The starting salary for psychology graduates, represented as \( X \), follows a normal distribution. - Mean (\( \mu \)): **$47,507** - Standard Deviation (\( \sigma \)): **$5,000** # Goals 1. Calculate the probability \( P(X > 52,000) \). 2. Calculate the probability \( P(45,000 < X < 52,000) \). 3. Determine the percentage of psychology graduates with starting offers between $38,000 and $45,000. 4. Identify the salary level below which 20% of psychology graduates are offered. # Conceptual Overview - **Standard Normal Distribution:** The z-score transformation allows us to convert any normal distribution into a standard normal distribution: \[ z = \frac{X - \mu}{\sigma} \] - **Using Excel for Calculating Probabilities:** The cumulative distribution function can be evaluated using: \[ =NORM.DIST(z, 0, 1, TRUE) \] --- # Step-by-Step Analysis ### 1. Probability \( P(X > 52,000) \) First, we need to compute the z-score for \( X = 52,000 \): \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Next, we can use Excel to find the cumulative probability corresponding to this z-score: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This formula gives: \[ P(Z < 0.8986) \approx 0.8159 \] To find the probability of earning more than $52,000, we calculate: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Now, we calculate the z-scores for both $45,000 and $52,000: - For \( X = 45,000 \): \[ z_1 = \frac{45,000 - 47,507}{5,000} = \frac{-2,507}{5,000} \approx -0.5014 \] - For \( X = 52,000 \) (already calculated): \[ z_2 \approx 0.8986 \] Using Excel, we find the cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This gives: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Now, we can calculate the probability within the range: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage between $38,000 and $45,000 Next, we determine the z-scores for $38,000 and $45,000: - For \( X = 38,000 \): \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} \approx -1.9014 \] - For \( X = 45,000 \) (previously calculated): \[ z_2 \approx -0.5014 \] Using Excel to find cumulative probabilities: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This yields: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Thus, the percentage of graduates with offers in this range is: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 \approx 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To find the salary that corresponds to the 20th percentile, we use: ```excel =NORM.INV(0.2, 47507, 5000) ``` This function computes the salary threshold: \[ X \approx 43,299 \] --- # Summary of Results - **Probability that a psychology graduate received a starting offer greater than $52,000:** **0.1841** - **Probability that a psychology graduate received a starting offer between $45,000 and $52,000:** **0.5074** - **Percentage of psychology majors with starting offers between $38,000 and $45,000:** **27.98%** - **Salary below which 20% were offered:** **$43,299** --- This thorough analysis employs z-scores and Excel functions to derive key probabilities and salary thresholds for psychology graduates. Each calculation is clearly articulated, ensuring transparency and understanding throughout the process.

# Given Data - The starting salary for psychology graduates, denoted as \( X \), is normally distributed. - Mean (\( \mu \)): **$47,507** - Standard Deviation (\( \sigma \)): **$5,000** # Objectives 1. Compute the probability \( P(X > 52,000) \). 2. Calculate the probability \( P(45,000 < X < 52,000) \). 3. Determine the percentage of psychology graduates with starting offers between $38,000 and $45,000. 4. Identify the salary level below which 20% of psychology graduates are offered. # Conceptual Framework - **Standard Normal Distribution:** The z-score transformation is defined by: \[ z = \frac{X - \mu}{\sigma} \] - **Excel for Probability Calculation:** The cumulative distribution function can be used as: \[ =NORM.DIST(z, 0, 1, TRUE) \] --- # Detailed Steps ### 1. Probability \( P(X > 52,000) \) To find the z-score for \( X = 52,000 \): \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Using Excel to find the cumulative probability corresponding to this z-score: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This returns: \[ P(Z < 0.8986) \approx 0.8159 \] To find the probability of earning more than $52,000: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Next, calculate the z-scores: - For \( X = 45,000 \): \[ z_1 = \frac{45,000 - 47,507}{5,000} = \frac{-2,507}{5,000} \approx -0.5014 \] - For \( X = 52,000 \) (already calculated): \[ z_2 \approx 0.8986 \] Using Excel, we find the cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This gives: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Thus, the probability for the specified range: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage between $38,000 and $45,000 Determine the z-scores: - For \( X = 38,000 \): \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} \approx -1.9014 \] - For \( X = 45,000 \) (previously calculated): \[ z_2 \approx -0.5014 \] Using Excel: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This gives: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Thus, the percentage of graduates with offers in this range: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To find the salary corresponding to the 20th percentile: ```excel =NORM.INV(0.2, 47507, 5000) ``` This function computes the salary threshold: \[ X \approx 43,299 \] --- # Summary of Results - **Probability that a psychology graduate received a starting offer greater than $52,000:** **0.1841** - **Probability that a psychology graduate received a starting offer between $45,000 and $52,000:** **0.5074** - **Percentage of psychology majors with starting offers between $38,000 and $45,000:** **27.98%** - **Salary below which 20% were offered:** **$43,299** --- This comprehensive analysis employs z-scores and Excel functions to derive important probabilities and salary thresholds for psychology graduates. Each calculation is presented clearly, ensuring clarity and understanding throughout the process. Each z-score is calculated using the specific \( X \) value, making the method transparent and reproducible.

# Given Data - The starting salary for psychology graduates, denoted as \( X \), follows a normal distribution. - Mean (\( \mu \)): **$47,507** - Standard Deviation (\( \sigma \)): **$5,000** # Goals 1. Calculate the probability \( P(X > 52,000) \). 2. Calculate the probability \( P(45,000 < X < 52,000) \). 3. Determine the percentage of psychology graduates with starting offers between $38,000 and $45,000. 4. Identify the salary level below which 20% of psychology graduates are offered. # Conceptual Framework - **Standard Normal Distribution:** The formula for calculating the z-score is given by: \[ z = \frac{X - \mu}{\sigma} \] - **Using Excel for Probability Calculation:** The cumulative distribution function can be computed with: \[ =NORM.DIST(z, 0, 1, TRUE) \] --- # Detailed Steps ### 1. Probability \( P(X > 52,000) \) First, we calculate the z-score for \( X = 52,000 \): \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Next, we use Excel to find the cumulative probability corresponding to this z-score: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This results in: \[ P(Z < 0.8986) \approx 0.8159 \] To find the probability of earning more than $52,000: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Next, we compute the z-scores for both $45,000 and $52,000: - For \( X = 45,000 \): \[ z_1 = \frac{45,000 - 47,507}{5,000} = \frac{-2,507}{5,000} \approx -0.5014 \] - For \( X = 52,000 \) (already computed): \[ z_2 \approx 0.8986 \] Using Excel, we find the cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This yields: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Thus, the probability for the specified range is: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage between $38,000 and $45,000 Now, we calculate the z-scores for $38,000 and $45,000: - For \( X = 38,000 \): \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} \approx -1.9014 \] - For \( X = 45,000 \) (already calculated): \[ z_2 \approx -0.5014 \] Using Excel to find cumulative probabilities: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This gives: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Thus, the percentage of graduates with offers in this range is: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To find the salary corresponding to the 20th percentile, we use: ```excel =NORM.INV(0.2, 47507, 5000) ``` This yields the salary threshold: \[ X \approx 43,299 \] --- # Summary of Results - **Probability that a psychology graduate received a starting offer greater than $52,000:** **0.1841** - **Probability that a psychology graduate received a starting offer between $45,000 and $52,000:** **0.5074** - **Percentage of psychology majors with starting offers between $38,000 and $45,000:** **27.98%** - **Salary below which 20% were offered:** **$43,299** --- This comprehensive analysis effectively utilizes z-scores and Excel functions to derive important probabilities and salary thresholds for psychology graduates. Each calculation is clearly articulated, ensuring transparency and understanding throughout the process. Each z-score was calculated using the specific \( X \) value, making the method reproducible and clear.

# Given Information - The starting salary for psychology graduates, represented as \( X \), is normally distributed. - Mean (\( \mu \)): **$47,507** - Standard Deviation (\( \sigma \)): **$5,000** # Objectives 1. Calculate the probability \( P(X > 52,000) \). 2. Calculate the probability \( P(45,000 < X < 52,000) \). 3. Determine the percentage of psychology graduates with starting offers between $38,000 and $45,000. 4. Identify the salary level below which 20% of psychology graduates are offered. # Conceptual Framework - **Standard Normal Distribution:** The z-score transformation is calculated using the formula: \[ z = \frac{X - \mu}{\sigma} \] - **Probability Calculation Using Excel:** The cumulative distribution function can be computed with: \[ =NORM.DIST(z, 0, 1, TRUE) \] --- # Detailed Steps ### 1. Probability \( P(X > 52,000) \) First, we calculate the z-score for \( X = 52,000 \): \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Next, we use Excel to determine the cumulative probability for this z-score: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This gives: \[ P(Z < 0.8986) \approx 0.8159 \] To find the probability of earning more than $52,000: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Next, we compute the z-scores for both $45,000 and $52,000: - For \( X = 45,000 \): \[ z_1 = \frac{45,000 - 47,507}{5,000} = \frac{-2,507}{5,000} \approx -0.5014 \] - For \( X = 52,000 \) (already calculated): \[ z_2 \approx 0.8986 \] Using Excel, we find the cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This yields: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Thus, the probability for the range \( P(45,000 < X < 52,000) \) is: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage between $38,000 and $45,000 Now, we calculate the z-scores for $38,000 and $45,000: - For \( X = 38,000 \): \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} \approx -1.9014 \] - For \( X = 45,000 \): \[ z_2 \approx -0.5014 \] Using Excel to calculate the cumulative probabilities: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This results in: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Thus, the percentage of psychology graduates with offers between $38,000 and $45,000 is: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To find the salary corresponding to the 20th percentile, we can use: ```excel =NORM.INV(0.2, 47507, 5000) ``` This function gives the salary threshold: \[ X \approx 43,299 \] --- # Summary of Results - **Probability that a psychology graduate received a starting offer greater than $52,000:** **0.1841** - **Probability that a psychology graduate received a starting offer between $45,000 and $52,000:** **0.5074** - **Percentage of psychology majors with starting offers between $38,000 and $45,000:** **27.98%** - **Salary below which 20% were offered:** **$43,299** --- This thorough analysis effectively utilizes z-scores and Excel functions to derive significant probabilities and salary thresholds for psychology graduates. Each calculation is presented clearly, ensuring transparency and thorough understanding throughout the process. Each z-score was calculated using the specific \( X \) value to maintain clarity in the methodology.

# Given Data - The starting salary for psychology graduates, denoted as \( X \), is normally distributed. - Mean (\( \mu \)): **$47,507** - Standard Deviation (\( \sigma \)): **$5,000** # Objectives 1. Calculate the probability \( P(X > 52,000) \). 2. Compute the probability \( P(45,000 < X < 52,000) \). 3. Determine the percentage of psychology graduates with starting offers between $38,000 and $45,000. 4. Identify the salary level below which 20% of psychology graduates are offered. # Conceptual Framework - **Standard Normal Distribution:** The z-score can be calculated using the formula: \[ z = \frac{X - \mu}{\sigma} \] - **Using Excel for Probability Calculation:** The cumulative probability can be calculated using: \[ =NORM.DIST(z, 0, 1, TRUE) \] --- # Detailed Steps ### 1. Probability \( P(X > 52,000) \) First, calculate the z-score for \( X = 52,000 \): \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Next, use Excel to find the cumulative probability: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This yields: \[ P(Z < 0.8986) \approx 0.8159 \] To find the probability of earning more than $52,000: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Now, compute the z-scores for both $45,000 and $52,000: - For \( X = 45,000 \): \[ z_1 = \frac{45,000 - 47,507}{5,000} = \frac{-2,507}{5,000} \approx -0.5014 \] - For \( X = 52,000 \): \[ z_2 \approx 0.8986 \] Using Excel, we find the cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This results in: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Thus, the probability for this range is: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage between $38,000 and $45,000 Calculate the z-scores for $38,000 and $45,000: - For \( X = 38,000 \): \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} \approx -1.9014 \] - For \( X = 45,000 \): \[ z_2 \approx -0.5014 \] Using Excel to find cumulative probabilities: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This results in: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Thus, the percentage of psychology graduates with offers in this range is: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To determine the salary corresponding to the 20th percentile, we can use: ```excel =NORM.INV(0.2, 47507, 5000) ``` This yields the salary threshold: \[ X \approx 43,299 \] --- # Summary of Results - **Probability that a psychology graduate received a starting offer greater than $52,000:** **0.1841** - **Probability that a psychology graduate received a starting offer between $45,000 and $52,000:** **0.5074** - **Percentage of psychology majors with starting offers between $38,000 and $45,000:** **27.98%** - **Salary below which 20% were offered:** **$43,299** --- This detailed analysis employs z-scores and Excel functions to derive critical probabilities and salary thresholds for psychology graduates. Each calculation is clearly articulated, ensuring clarity and understanding throughout the process. The z-scores are calculated using specific \( X \) values, maintaining transparency in the methodology.

# Given Information - The starting salary for psychology graduates, denoted as \( X \), is normally distributed. - Mean (\( \mu \)): **$47,507** - Standard Deviation (\( \sigma \)): **$5,000** # Objectives 1. Determine the probability \( P(X > 52,000) \). 2. Calculate the probability \( P(45,000 < X < 52,000) \). 3. Find the percentage of psychology graduates with starting offers between $38,000 and $45,000. 4. Identify the salary below which 20% of psychology graduates are offered. # Conceptual Framework - **Standard Normal Distribution:** The z-score transformation is given by: \[ z = \frac{X - \mu}{\sigma} \] - **Probability Calculation Using Excel:** To find the cumulative probability, we use: \[ =NORM.DIST(z, 0, 1, TRUE) \] --- # Step-by-Step Analysis ### 1. Probability \( P(X > 52,000) \) First, calculate the z-score for \( X = 52,000 \): \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Next, we will use Excel to find the cumulative probability: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This calculation yields: \[ P(Z < 0.8986) \approx 0.8159 \] To find the probability of earning more than $52,000: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Now, we compute the z-scores for both $45,000 and $52,000: - For \( X = 45,000 \): \[ z_1 = \frac{45,000 - 47,507}{5,000} = \frac{-2,507}{5,000} \approx -0.5014 \] - For \( X = 52,000 \) (already calculated): \[ z_2 \approx 0.8986 \] Using Excel to find the cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This yields: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Thus, the probability for the specified range is: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage Between $38,000 and $45,000 Now, we calculate the z-scores for $38,000 and $45,000: - For \( X = 38,000 \): \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} \approx -1.9014 \] - For \( X = 45,000 \) (previously calculated): \[ z_2 \approx -0.5014 \] Using Excel to find cumulative probabilities: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This yields: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Thus, the percentage of psychology graduates with offers in this range is: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To find the salary corresponding to the 20th percentile: ```excel =NORM.INV(0.2, 47507, 5000) ``` This function calculates the salary threshold: \[ X \approx 43,299 \] --- # Summary of Results - **Probability that a psychology graduate received a starting offer greater than $52,000:** **0.1841** - **Probability that a psychology graduate received a starting offer between $45,000 and $52,000:** **0.5074** - **Percentage of psychology majors with starting offers between $38,000 and $45,000:** **27.98%** - **Salary below which 20% were offered:** **$43,299** --- This detailed analysis effectively employs z-scores and Excel functions to derive important probabilities and salary thresholds for psychology graduates. Each calculation is presented clearly, ensuring transparency and understanding throughout the process. Each z-score was calculated using specific \( X \) values to provide clarity in the methodology.

# Given Data - The starting salary for psychology graduates, denoted as \( X \), follows a normal distribution. - Mean (\( \mu \)): **$47,507** - Standard Deviation (\( \sigma \)): **$5,000** # Objectives 1. Calculate the probability \( P(X > 52,000) \). 2. Calculate the probability \( P(45,000 < X < 52,000) \). 3. Determine the percentage of psychology graduates with starting offers between $38,000 and $45,000. 4. Identify the salary level below which 20% of psychology graduates are offered. # Conceptual Framework - **Standard Normal Distribution:** The z-score can be calculated using the formula: \[ z = \frac{X - \mu}{\sigma} \] - **Using Excel for Probability Calculation:** The cumulative probability can be computed with: \[ =NORM.DIST(z, 0, 1, TRUE) \] --- # Detailed Steps ### 1. Probability \( P(X > 52,000) \) First, calculate the z-score for \( X = 52,000 \): \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Next, we can use Excel to find the cumulative probability: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This results in: \[ P(Z < 0.8986) \approx 0.8159 \] To find the probability of earning more than $52,000: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Now, compute the z-scores for both $45,000 and $52,000: - For \( X = 45,000 \): \[ z_1 = \frac{45,000 - 47,507}{5,000} = \frac{-2,507}{5,000} \approx -0.5014 \] - For \( X = 52,000 \): \[ z_2 \approx 0.8986 \] Using Excel to find the cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This yields: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Thus, the probability for the specified range is: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage Between $38,000 and $45,000 Now, calculate the z-scores for $38,000 and $45,000: - For \( X = 38,000 \): \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} \approx -1.9014 \] - For \( X = 45,000 \): \[ z_2 \approx -0.5014 \] Using Excel to find cumulative probabilities: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This results in: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Thus, the percentage of psychology graduates with offers in this range is: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To find the salary corresponding to the 20th percentile: ```excel =NORM.INV(0.2, 47507, 5000) ``` This function gives the salary threshold: \[ X \approx 43,299 \] --- # Summary of Results - **Probability that a psychology graduate received a starting offer greater than $52,000:** **0.1841** - **Probability that a psychology graduate received a starting offer between $45,000 and $52,000:** **0.5074** - **Percentage of psychology majors with starting offers between $38,000 and $45,000:** **27.98%** - **Salary below which 20% were offered:** **$43,299** --- This structured analysis employs z-scores and Excel functions to derive significant probabilities and salary thresholds for psychology graduates. Each calculation is presented clearly, ensuring transparency and understanding throughout the process. The z-scores were calculated using specific \( X \) values, maintaining clarity in the methodology.

# Given Information - The starting salary for psychology graduates, represented as \( X \), is normally distributed. - Mean (\( \mu \)): **$47,507** - Standard Deviation (\( \sigma \)): **$5,000** # Objectives 1. Calculate the probability \( P(X > 52,000) \). 2. Compute the probability \( P(45,000 < X < 52,000) \). 3. Determine the percentage of psychology graduates with starting offers between $38,000 and $45,000. 4. Identify the salary level below which 20% of psychology graduates are offered. # Conceptual Framework - **Standard Normal Distribution:** The z-score transformation allows us to convert any normal distribution into a standard normal format, calculated using: \[ z = \frac{X - \mu}{\sigma} \] - **Probability Calculation Using Excel:** The cumulative probability can be evaluated using: \[ =NORM.DIST(z, 0, 1, TRUE) \] --- # Detailed Steps ### 1. Probability \( P(X > 52,000) \) To find the z-score for \( X = 52,000 \): \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Use Excel to find the cumulative probability corresponding to this z-score: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This provides: \[ P(Z < 0.8986) \approx 0.8159 \] To compute the probability of earning more than $52,000: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Calculate the z-scores for both $45,000 and $52,000: - For \( X = 45,000 \): \[ z_1 = \frac{45,000 - 47,507}{5,000} = \frac{-2,507}{5,000} \approx -0.5014 \] - For \( X = 52,000 \) (previously calculated): \[ z_2 \approx 0.8986 \] Utilize Excel to find the cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This yields: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Thus, the probability of earning between $45,000 and $52,000 is: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage Between $38,000 and $45,000 Determine the z-scores: - For \( X = 38,000 \): \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} \approx -1.9014 \] - For \( X = 45,000 \) (previously calculated): \[ z_2 \approx -0.5014 \] Using Excel, find cumulative probabilities: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This provides: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Calculate the percentage of psychology graduates receiving offers within this range: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 \approx 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To find the salary corresponding to the 20th percentile, utilize: ```excel =NORM.INV(0.2, 47507, 5000) ``` This function calculates the salary threshold: \[ X \approx 43,299 \] --- # Summary of Results - **Probability that a psychology graduate received a starting offer greater than $52,000:** **0.1841** - **Probability that a psychology graduate received a starting offer between $45,000 and $52,000:** **0.5074** - **Percentage of psychology majors with starting offers between $38,000 and $45,000:** **27.98%** - **Salary below which 20% were offered:** **$43,299** --- This comprehensive analysis employs calculations of z-scores and Excel functions to derive key probabilities and salary thresholds for psychology graduates. Each calculation is articulated clearly to ensure a thorough understanding of the methodology involved, with each z-score derived from its relevant \( X \) value for transparency.

# Given Information - The starting salary for psychology graduates, represented as \( X \), follows a normal distribution. - Mean (\( \mu \)): **$47,507** - Standard Deviation (\( \sigma \)): **$5,000** # Objectives 1. Calculate the probability \( P(X > 52,000) \). 2. Compute the probability \( P(45,000 < X < 52,000) \). 3. Determine the percentage of psychology graduates with starting offers between $38,000 and $45,000. 4. Identify the salary level below which 20% of psychology graduates are offered. # Conceptual Framework - **Standard Normal Distribution:** The transformation to a standard normal distribution is accomplished through the z-score: \[ z = \frac{X - \mu}{\sigma} \] - **Using Excel for Probability Calculations:** The cumulative distribution function can be calculated using: \[ =NORM.DIST(z, 0, 1, TRUE) \] --- # Detailed Steps ### 1. Probability \( P(X > 52,000) \) To find the z-score for \( X = 52,000 \): \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Next, we will use Excel to find the cumulative probability associated with this z-score: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This will yield: \[ P(Z < 0.8986) \approx 0.8159 \] To calculate the probability of earning more than $52,000: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Now, compute the z-scores for both $45,000 and $52,000: - For \( X = 45,000 \): \[ z_1 = \frac{45,000 - 47,507}{5,000} = \frac{-2,507}{5,000} \approx -0.5014 \] - For \( X = 52,000 \): \[ z_2 \approx 0.8986 \] Using Excel, we will find the cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This results in: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Thus, the probability for the range is: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage Between $38,000 and $45,000 Next, we calculate the z-scores for $38,000 and $45,000: - For \( X = 38,000 \): \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} \approx -1.9014 \] - For \( X = 45,000 \): \[ z_2 \approx -0.5014 \] Using Excel, we find cumulative probabilities: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This gives: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Thus, the percentage of psychology graduates receiving offers in this range is: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 \approx 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To find the salary corresponding to the 20th percentile: ```excel =NORM.INV(0.2, 47507, 5000) ``` This will yield the salary threshold: \[ X \approx 43,299 \] --- # Summary of Results - **Probability that a psychology graduate received a starting offer greater than $52,000:** **0.1841** - **Probability that a psychology graduate received a starting offer between $45,000 and $52,000:** **0.5074** - **Percentage of psychology majors with starting offers between $38,000 and $45,000:** **27.98%** - **Salary below which 20% were offered:** **$43,299** --- This thorough analysis makes effective use of z-scores and Excel functions to derive essential probabilities and salary thresholds for psychology graduates. Each calculation is clearly articulated for clarity and understanding, with each z-score calculated based on its corresponding \( X \) value, ensuring transparency in the methodology used.

# Given Information - The starting salary for psychology graduates, denoted as \( X \), follows a normal distribution. - Mean (\( \mu \)): **$47,507** - Standard Deviation (\( \sigma \)): **$5,000** # Objectives 1. Calculate the probability \( P(X > 52,000) \). 2. Compute the probability \( P(45,000 < X < 52,000) \). 3. Determine the percentage of psychology graduates with starting offers between $38,000 and $45,000. 4. Identify the salary level below which 20% of psychology graduates are offered. # Conceptual Framework - **Standard Normal Distribution:** The z-score transformation is given by: \[ z = \frac{X - \mu}{\sigma} \] - **Using Excel for Probability Calculation:** The cumulative probability can be computed using: \[ =NORM.DIST(z, 0, 1, TRUE) \] --- # Detailed Steps ### 1. Probability \( P(X > 52,000) \) To find the z-score for \( X = 52,000 \): \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Use Excel to calculate the cumulative probability associated with this z-score: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This calculation yields: \[ P(Z < 0.8986) \approx 0.8159 \] To find the probability of earning more than $52,000: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Now, compute the z-scores for both $45,000 and $52,000: - For \( X = 45,000 \): \[ z_1 = \frac{45,000 - 47,507}{5,000} = \frac{-2,507}{5,000} \approx -0.5014 \] - For \( X = 52,000 \): \[ z_2 \approx 0.8986 \] Using Excel, we find the cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This results in: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Thus, the probability for the specified range is: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage Between $38,000 and $45,000 Next, we calculate the z-scores for $38,000 and $45,000: - For \( X = 38,000 \): \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} \approx -1.9014 \] - For \( X = 45,000 \): \[ z_2 \approx -0.5014 \] Using Excel, we find cumulative probabilities: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This provides: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Thus, the percentage of psychology graduates receiving offers in this range is: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To find the salary corresponding to the 20th percentile: ```excel =NORM.INV(0.2, 47507, 5000) ``` This function yields the salary threshold: \[ X \approx 43,299 \] --- # Summary of Results - **Probability that a psychology graduate received a starting offer greater than $52,000:** **0.1841** - **Probability that a psychology graduate received a starting offer between $45,000 and $52,000:** **0.5074** - **Percentage of psychology majors with starting offers between $38,000 and $45,000:** **27.98%** - **Salary below which 20% were offered:** **$43,299** --- This thorough analysis utilizes z-scores and Excel functions to derive key probabilities and salary thresholds for psychology graduates. Each calculation is presented clearly, ensuring transparency and understanding throughout the process. Each z-score is derived from specific \( X \) values, maintaining clarity in the methodology.

# Given Information - The starting salary for psychology graduates, denoted as \( X \), is normally distributed. - Mean (\( \mu \)): **$47,507** - Standard Deviation (\( \sigma \)): **$5,000** # Objectives 1. Calculate the probability \( P(X > 52,000) \). 2. Compute the probability \( P(45,000 < X < 52,000) \). 3. Determine the percentage of psychology graduates with starting offers between $38,000 and $45,000. 4. Identify the salary level below which 20% of psychology graduates are offered. # Conceptual Framework - **Standard Normal Distribution:** The z-score transformation is calculated using the formula: \[ z = \frac{X - \mu}{\sigma} \] - **Using Excel for Probability Calculation:** The cumulative probability can be calculated using: \[ =NORM.DIST(z, 0, 1, TRUE) \] --- # Detailed Steps ### 1. Probability \( P(X > 52,000) \) To calculate the z-score for \( X = 52,000 \): \[ z = \frac{52,000 - 47,507}{5,000} = \frac{4,493}{5,000} \approx 0.8986 \] Now, we will use Excel to find the cumulative probability: ```excel =NORM.DIST(0.8986, 0, 1, TRUE) ``` This gives: \[ P(Z < 0.8986) \approx 0.8159 \] To find the probability of earning more than $52,000: \[ P(X > 52,000) = 1 - P(Z < 0.8986) \approx 1 - 0.8159 = 0.1841 \] --- ### 2. Probability \( P(45,000 < X < 52,000) \) Next, calculate the z-scores for both $45,000 and $52,000: - For \( X = 45,000 \): \[ z_1 = \frac{45,000 - 47,507}{5,000} = \frac{-2,507}{5,000} \approx -0.5014 \] - For \( X = 52,000 \): \[ z_2 \approx 0.8986 \] Using Excel, we can find the cumulative probabilities: ```excel =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 =NORM.DIST(0.8986, 0, 1, TRUE) // For $52,000 ``` This results in: - \( P(Z < -0.5014) \approx 0.3085 \) - \( P(Z < 0.8986) \approx 0.8159 \) Thus, the probability for the range \( P(45,000 < X < 52,000) \) is: \[ P(45,000 < X < 52,000) = P(Z < 0.8986) - P(Z < -0.5014) \approx 0.8159 - 0.3085 = 0.5074 \] --- ### 3. Percentage Between $38,000 and $45,000 Now, we calculate the z-scores for $38,000 and $45,000: - For \( X = 38,000 \): \[ z_1 = \frac{38,000 - 47,507}{5,000} = \frac{-9,507}{5,000} \approx -1.9014 \] - For \( X = 45,000 \): \[ z_2 \approx -0.5014 \] Using Excel, we can find the cumulative probabilities: ```excel =NORM.DIST(-1.9014, 0, 1, TRUE) // For $38,000 =NORM.DIST(-0.5014, 0, 1, TRUE) // For $45,000 ``` This yields: - \( P(Z < -1.9014) \approx 0.0287 \) - \( P(Z < -0.5014) \approx 0.3085 \) Thus, the percentage of psychology graduates with offers in this range is: \[ \text{Percentage} = (0.3085 - 0.0287) \times 100 = 27.98\% \] --- ### 4. Salary Below Which 20% Are Offered To find the salary that corresponds to the 20th percentile, we can use: ```excel =NORM.INV(0.2, 47507, 5000) ``` This function calculates the salary threshold: \[ X \approx 43,299 \] --- # Summary of Results - **Probability that a psychology graduate received a starting offer greater than $52,000:** **0.1841** - **Probability that a psychology graduate received a starting offer between $45,000 and $52,000:** **0.5074** - **Percentage of psychology majors with starting offers between $38,000 and $45,000:** **27.98%** - **Salary below which 20% were offered:** **$43,299** --- This analysis effectively employs z-scores and Excel functions to derive key probabilities and salary thresholds for psychology graduates. Each calculation is clearly explained to ensure understanding, with specific \( X \) values used in the z-score calculations for transparency in the methodology.