# Given Information ### Question 1 - The weight of a package delivered by a courier company is normally distributed. - Mean (\(\mu\)) = 2.5 kg - Standard deviation (\(\sigma\)) = .6 kg #### (a) - 14% of the packages weigh heavier than \( w \) kg. Find the value of \( w \). #### (b) - Shipping fee = \$30 basic charge + \$12 per kg of the package. - Find the following for shipping fee: 1. Mean 2. Standard deviation 3. Variance 4. Median 5. 86th percentile --- ### Question 2 - Expenditure weights and prices of items for years 2019, 202, 2021 are given. | Item | Weight | 2019 | 202 | 2021 | |-----------|--------|------|------|------| | Clothes | 10% | 15 | 12 | 15 | | Food | 40% | 12 | 15 | 18 | | Living | 35% | 40 | 44 | 48 | | Commute | 15% | 10 | 12 | 10 | #### (a) - Compile the CPIs for 202 and 2021 using 2019 as base year. #### (b) - Calculate the percentage change in prices from 202 to 2021. #### (c) - Which items have experienced price rise from 202 to 2021? --- # Definitions and Concepts Used - **Normal Distribution**: The probability that a value \( X \) is greater than \( x \) can be found using the Z-score formula: \( Z = \frac{x - \mu}{\sigma} \). - **Percentile**: The value below which a given percentage of data falls. - **Linear Transformation**: For \( Y = a + bX \), new mean = \( a + b\mu \), new std. dev. = \( |b|\sigma \), new variance = \( b^2\sigma^2 \), new median = \( a + b(\text{median of } X) \). - **Consumer Price Index (CPI)**: \( \text{CPI} = \frac{\sum (\text{weight} \times \text{price index})}{\sum \text{weights}} \), where \( \text{price index} = \frac{\text{price in given year}}{\text{price in base year}} \times 100 \). --- # Solution ## Question 1 ### (a) Find the value of \( w \): Given: \( P(X > w) = .14 \). So, \( P(X < w) = .86 \). Find the Z-score corresponding to .86 cumulative probability (from standard normal tables): \( Z \approx 1.08 \). Using \( Z = \frac{w - 2.5}{.6} \): \[ 1.08 = \frac{w - 2.5}{.6} \] \[ w - 2.5 = 1.08 \times .6 = .648 \] \[ w = 2.5 + .648 = 3.148 \] --- ### (b) Shipping Fee Statistics Let \( X \) be the package weight (\( N(2.5, .6^2) \)), fee \( Y = 30 + 12X \): - **Mean**: \( E[Y] = 30 + 12 \times 2.5 = 60 \) - **Std. Dev.**: \( SD[Y] = 12 \times .6 = 7.2 \) - **Variance**: \( Var[Y] = (12^2) \times (.6^2) = 144 \times .36 = 51.84 \) - **Median**: Median of \( X \) is 2.5, so median of \( Y \) is \( 30 + 12 \times 2.5 = 60 \) - **86th percentile**: \( X_{86\%} = 3.148 \) (from part a), so \( Y_{86\%} = 30 + 12 \times 3.148 = 67.776 \) --- ## Question 2 ### (a) Compile the CPIs for 202 and 2021 using 2019 as the base year Calculate price relatives for each year: - Clothes: 202: \( \frac{12}{15} \times 100 = 80 \), 2021: \( \frac{15}{15} \times 100 = 100 \) - Food: 202: \( \frac{15}{12} \times 100 = 125 \), 2021: \( \frac{18}{12} \times 100 = 150 \) - Living: 202: \( \frac{44}{40} \times 100 = 110 \), 2021: \( \frac{48}{40} \times 100 = 120 \) - Commute: 202: \( \frac{12}{10} \times 100 = 120 \), 2021: \( \frac{10}{10} \times 100 = 100 \) CPI = \( \sum (\text{weight} \times \text{price index}) \): #### CPI for 202: \[ (.10 \times 80) + (.40 \times 125) + (.35 \times 110) + (.15 \times 120) = 8 + 50 + 38.5 + 18 = 114.5 \] #### CPI for 2021: \[ (.10 \times 100) + (.40 \times 150) + (.35 \times 120) + (.15 \times 100) = 10 + 60 + 42 + 15 = 127 \] --- ### (b) Calculate the percentage change in prices from 202 to 2021 \[ \text{Percentage change} = \frac{127 - 114.5}{114.5} \times 100 \approx 10.91\% \] --- ### (c) Which items have experienced price rise from 202 to 2021? Compare 202 and 2021 prices: - Clothes: 12 → 15 (Rise) - Food: 15 → 18 (Rise) - Living: 44 → 48 (Rise) - Commute: 12 → 10 (Fall) So, Clothes, Food, and Living have experienced price rise. --- # Summary - The value of \( w \) such that 14% of packages are heavier is **3.148 kg**. - Shipping fee statistics: - Mean: **\$60** - Standard deviation: **\$7.2** - Variance: **\$51.84** - Median: **\$60** - 86th percentile: **\$67.78** - CPI for 202: **114.5** - CPI for 2021: **127** - Percentage change in prices from 202 to 2021: **10.91%** - Items with price rise from 202 to 2021: **Clothes, Food, Living**.

# Given Information ### Question 1 - The weight of a package delivered by a courier company is normally distributed. - Mean (\(\mu\)) = 2.5 kg - Standard deviation (\(\sigma\)) = 0.6 kg #### (a) - 14% of the packages weigh heavier than \( w \) kg. Find the value of \( w \). #### (b) - Shipping fee = \$30 basic charge + \$12 per kg of the package. - Find the following for the shipping fee: 1. Mean 2. Standard deviation 3. Variance 4. Median 5. 86th percentile --- ### Question 2 - Expenditure weights and prices of items for years 2019, 2020, and 2021 are given. | Item | Weight | 2019 | 2020 | 2021 | |-----------|--------|------|------|------| | Clothes | 10% | 15 | 12 | 15 | | Food | 40% | 12 | 15 | 18 | | Living | 35% | 40 | 44 | 48 | | Commute | 15% | 10 | 12 | 10 | #### (a) - Compile the CPIs for 2020 and 2021 using 2019 as the base year. #### (b) - Calculate the percentage change in prices from 2020 to 2021. #### (c) - Which items have experienced a price rise from 2020 to 2021? --- # Definitions and Concepts Used - **Normal Distribution**: The probability that a value \( X \) is greater than \( w \) can be found using the Z-score formula: \( Z = \frac{w - \mu}{\sigma} \). - **Percentile**: The value below which a given percentage of data falls. - **Linear Transformation**: For \( Y = a + bX \), the new mean is \( a + b\mu \), the new standard deviation is \( |b|\sigma \), and the new variance is \( b^2\sigma^2 \). - **Consumer Price Index (CPI)**: \[ \text{CPI} = \frac{\sum (\text{weight} \times \text{price index})}{\sum \text{weights}}, \] where \( \text{price index} = \frac{\text{price in given year}}{\text{price in base year}} \times 100 \). --- # Solution ## Question 1 ### (a) Find the value of \( w \) Given: \( P(X > w) = 0.14 \), so \( P(X < w) = 0.86 \). Find the Z-score corresponding to 0.86 cumulative probability (using Excel function `=NORM.S.INV(0.86)`): \[ Z \approx 1.08 \] Using the Z-score formula: \[ Z = \frac{w - 2.5}{0.6} \] \[ 1.08 = \frac{w - 2.5}{0.6} \] Solving for \( w \): \[ w - 2.5 = 1.08 \times 0.6 = 0.648 \] \[ w = 2.5 + 0.648 = 3.148 \] --- ### (b) Shipping Fee Statistics Let \( X \) be the package weight (\( N(2.5, 0.6^2) \)), and the shipping fee \( Y = 30 + 12X \): - **Mean**: \[ E[Y] = 30 + 12 \times 2.5 = 60 \] - **Standard Deviation**: \[ SD[Y] = 12 \times 0.6 = 7.2 \] - **Variance**: \[ Var[Y] = (12^2) \times (0.6^2) = 144 \times 0.36 = 51.84 \] - **Median**: The median of \( X \) is 2.5, so the median of \( Y \) is: \[ 30 + 12 \times 2.5 = 60 \] - **86th Percentile**: Using the result from part (a): \[ X_{86\%} = 3.148 \Rightarrow Y_{86\%} = 30 + 12 \times 3.148 = 67.776 \] --- ## Question 2 ### (a) Compile CPIs for 2020 and 2021 using 2019 as the base year Calculate price relatives for each year: - **Clothes**: - 2020: \( \frac{12}{15} \times 100 = 80 \) - 2021: \( \frac{15}{15} \times 100 = 100 \) - **Food**: - 2020: \( \frac{15}{12} \times 100 = 125 \) - 2021: \( \frac{18}{12} \times 100 = 150 \) - **Living**: - 2020: \( \frac{44}{40} \times 100 = 110 \) - 2021: \( \frac{48}{40} \times 100 = 120 \) - **Commute**: - 2020: \( \frac{12}{10} \times 100 = 120 \) - 2021: \( \frac{10}{10} \times 100 = 100 \) CPI calculations: #### CPI for 2020: \[ \text{CPI}_{2020} = (0.10 \times 80) + (0.40 \times 125) + (0.35 \times 110) + (0.15 \times 120) = 8 + 50 + 38.5 + 18 = 114.5 \] #### CPI for 2021: \[ \text{CPI}_{2021} = (0.10 \times 100) + (0.40 \times 150) + (0.35 \times 120) + (0.15 \times 100) = 10 + 60 + 42 + 15 = 127 \] --- ### (b) Calculate the percentage change in prices from 2020 to 2021 \[ \text{Percentage change} = \frac{127 - 114.5}{114.5} \times 100 \approx 10.91\% \] --- ### (c) Items with price rise from 2020 to 2021 Compare 2020 and 2021 prices: - **Clothes**: 12 → 15 (Rise) - **Food**: 15 → 18 (Rise) - **Living**: 44 → 48 (Rise) - **Commute**: 12 → 10 (Fall) Items that experienced price rise: **Clothes, Food, Living**. --- # Summary - The value of \( w \) such that 14% of packages are heavier is **3.148 kg**. - Shipping fee statistics: - Mean: **\$60** - Standard deviation: **\$7.2** - Variance: **\$51.84** - Median: **\$60** - 86th percentile: **\$67.78** - CPI for 2020: **114.5** - CPI for 2021: **127** - Percentage change in prices from 2020 to 2021: **10.91%** - Items with price rise from 2020 to 2021: **Clothes, Food, Living**.

# Given Information ### Question 1 - The weight of a package delivered by a courier company is normally distributed. - Mean (\(\mu\)) = 2.5 kg - Standard deviation (\(\sigma\)) = 0.6 kg #### (a) - 14% of the packages weigh heavier than \( w \) kg. Find the value of \( w \). #### (b) - Shipping fee = \$30 basic charge + \$12 per kg of the package. - Find the following for the shipping fee: 1. Mean 2. Standard deviation 3. Variance 4. Median 5. 86th percentile --- ### Question 2 - Expenditure weights and prices of items for years 2019, 2020, and 2021 are given. | Item | Weight | 2019 | 2020 | 2021 | |-----------|--------|------|------|------| | Clothes | 10% | 15 | 12 | 18 | | Food | 40% | 12 | 15 | 18 | | Living | 35% | 40 | 44 | 48 | | Commute | 15% | 10 | 10 | 10 | #### (a) - Compile the CPIs for 2020 and 2021 using 2019 as the base year. #### (b) - Calculate the percentage change in prices from 2020 to 2021. #### (c) - Which items have experienced a price rise from 2020 to 2021? --- # Definitions and Concepts Used - **Normal Distribution**: The probability that a value \( X \) is greater than \( w \) can be found using the Z-score formula: \( Z = \frac{w - \mu}{\sigma} \). - **Percentile**: The value below which a given percentage of data falls. - **Linear Transformation**: For \( Y = a + bX \), the new mean is \( a + b\mu \), the new standard deviation is \( |b|\sigma \), and the new variance is \( b^2\sigma^2 \). - **Consumer Price Index (CPI)**: \[ \text{CPI} = \sum w \times \left(\frac{P_t}{P_0}\right) \times 100 \] where \( P_t \) is the price in year \( t \) and \( P_0 \) is the price in the base year. --- # Solution ## Question 1 ### (a) Find the value of \( w \) Given: \( P(X > w) = 0.14 \), so \( P(X < w) = 0.86 \). Find the Z-score corresponding to 0.86 cumulative probability (using Excel function `=NORM.S.INV(0.86)`): \[ Z \approx 1.08 \] Using the Z-score formula: \[ Z = \frac{w - 2.5}{0.6} \] \[ 1.08 = \frac{w - 2.5}{0.6} \] Solving for \( w \): \[ w - 2.5 = 1.08 \times 0.6 = 0.648 \] \[ w = 2.5 + 0.648 = 3.148 \] --- ### (b) Shipping Fee Statistics Let \( X \) be the package weight (\( N(2.5, 0.6^2) \)), and the shipping fee \( Y = 30 + 12X \): - **Mean**: \[ E[Y] = 30 + 12 \times 2.5 = 60 \] - **Standard Deviation**: \[ SD[Y] = 12 \times 0.6 = 7.2 \] - **Variance**: \[ Var[Y] = (12^2) \times (0.6^2) = 144 \times 0.36 = 51.84 \] - **Median**: The median of \( X \) is 2.5, so the median of \( Y \) is: \[ 30 + 12 \times 2.5 = 60 \] - **86th Percentile**: Using the result from part (a): \[ X_{86\%} = 3.148 \Rightarrow Y_{86\%} = 30 + 12 \times 3.148 = 67.776 \] --- ## Question 2 ### (a) Compile CPIs for 2020 and 2021 using 2019 as the base year **Price relatives for 2020**: - **Clothes**: \( \frac{12}{15} = 0.8 \) - **Food**: \( \frac{15}{12} = 1.25 \) - **Living**: \( \frac{44}{40} = 1.1 \) - **Commute**: \( \frac{10}{10} = 1 \) Calculating the weighted sum for CPI 2020: \[ \text{CPI}_{2020} = (0.10 \times 0.8) + (0.40 \times 1.25) + (0.35 \times 1.1) + (0.15 \times 1) \] \[ = 0.08 + 0.5 + 0.385 + 0.15 = 1.115 \] \[ \text{CPI}_{2020} = 1.115 \times 100 = 111.5 \] --- **Price relatives for 2021**: - **Clothes**: \( \frac{18}{15} = 1.2 \) - **Food**: \( \frac{18}{12} = 1.5 \) - **Living**: \( \frac{48}{40} = 1.2 \) - **Commute**: \( \frac{10}{10} = 1 \) Calculating the weighted sum for CPI 2021: \[ \text{CPI}_{2021} = (0.10 \times 1.2) + (0.40 \times 1.5) + (0.35 \times 1.2) + (0.15 \times 1) \] \[ = 0.12 + 0.6 + 0.42 + 0.15 = 1.32 \] \[ \text{CPI}_{2021} = 1.32 \times 100 = 132 \] --- ### (b) Calculate the percentage change in prices from 2020 to 2021 \[ \text{Percentage change} = \frac{132 - 111.5}{111.5} \times 100 = \frac{20.5}{111.5} \times 100 \approx 18.41\% \] --- ### (c) Items with price rise from 2020 to 2021 Compare 2020 and 2021 prices: - **Clothes**: 12 → 18 (Increase) - **Food**: 15 → 18 (Increase) - **Living**: 44 → 48 (Increase) - **Commute**: 10 → 10 (No change) Items that experienced price rise: **Clothes, Food, Living**. --- # Summary - The value of \( w \) such that 14% of packages are heavier is **3.148 kg**. - Shipping fee statistics: - Mean: **\$60** - Standard deviation: **\$7.2** - Variance: **\$51.84** - Median: **\$60** - 86th percentile: **\$67.78** - CPI for 2020: **111.5** - CPI for 2021: **132** - Percentage change in prices from 2020 to 2021: **18.41%** - Items with price rise from 2020 to 2021: **Clothes, Food, Living**.

# Given Information ### Question 1 - The weight of a package delivered by a courier company is normally distributed. - Mean (\(\mu\)) = 2.5 kg - Standard deviation (\(\sigma\)) = 0.6 kg #### (a) - 14% of the packages weigh heavier than \( w \) kg. Find the value of \( w \). #### (b) - Shipping fee = \$30 basic charge + \$12 per kg of the package. - Find the following for the shipping fee: 1. Mean 2. Standard deviation 3. Variance 4. Median 5. 86th percentile --- ### Question 2 - Expenditure weights and prices of items for years 2019, 2020, and 2021 are given. | Item | Weight | 2019 | 2020 | 2021 | |-----------|--------|------|------|------| | Clothes | 10% | 12 | 15 | 18 | | Food | 30% | 40 | 45 | 48 | | Living | 45% | 115 | 114 | 120 | | Commute | 15% | 10 | 10 | 10 | #### (a) - Compile the CPIs for 2020 and 2021 using 2019 as the base year. #### (b) - Calculate the percentage change in prices from 2020 to 2021. #### (c) - Which items have experienced a price rise from 2020 to 2021? --- # Definitions and Concepts Used - **Normal Distribution**: The probability that a value \( X \) is greater than \( w \) can be found using the Z-score formula: \( Z = \frac{w - \mu}{\sigma} \). - **Percentile**: The value below which a given percentage of data falls. - **Linear Transformation**: For \( Y = a + bX \), the new mean is \( a + b\mu \), the new standard deviation is \( |b|\sigma \), and the new variance is \( b^2\sigma^2 \). - **Consumer Price Index (CPI)**: \[ \text{CPI} = \sum w \times \left(\frac{P_t}{P_0}\right) \times 100 \] where \( P_t \) is the price in year \( t \) and \( P_0 \) is the price in the base year. --- # Solution ## Question 1 ### (a) Find the value of \( w \) Given: \( P(X > w) = 0.14 \), so \( P(X < w) = 0.86 \). Find the Z-score corresponding to 0.86 cumulative probability (using Excel function `=NORM.S.INV(0.86)`): \[ Z \approx 1.08 \] Using the Z-score formula: \[ Z = \frac{w - 2.5}{0.6} \] \[ 1.08 = \frac{w - 2.5}{0.6} \] Solving for \( w \): \[ w - 2.5 = 1.08 \times 0.6 = 0.648 \] \[ w = 2.5 + 0.648 = 3.148 \] --- ### (b) Shipping Fee Statistics Let \( X \) be the package weight (\( N(2.5, 0.6^2) \)), and the shipping fee \( Y = 30 + 12X \): - **Mean**: \[ E[Y] = 30 + 12 \times 2.5 = 60 \] - **Standard Deviation**: \[ SD[Y] = 12 \times 0.6 = 7.2 \] - **Variance**: \[ Var[Y] = (12^2) \times (0.6^2) = 144 \times 0.36 = 51.84 \] - **Median**: The median of \( X \) is 2.5, so the median of \( Y \) is: \[ 30 + 12 \times 2.5 = 60 \] - **86th Percentile**: Using the result from part (a): \[ X_{86\%} = 3.148 \Rightarrow Y_{86\%} = 30 + 12 \times 3.148 = 67.776 \] --- ## Question 2 ### (a) Compile CPIs for 2020 and 2021 using 2019 as the base year **Price relatives for 2020**: - **Clothes**: \( \frac{15}{12} = 1.25 \) - **Food**: \( \frac{45}{40} = 1.125 \) - **Living**: \( \frac{114}{115} \approx 0.9913 \) - **Commute**: \( \frac{10}{10} = 1 \) Calculating the weighted sum for CPI 2020: \[ \text{CPI}_{2020} = (0.10 \times 1.25) + (0.30 \times 1.125) + (0.45 \times 0.9913) + (0.15 \times 1) \] \[ = 0.125 + 0.3375 + 0.4461 + 0.15 = 1.0586 \] \[ \text{CPI}_{2020} = 1.0586 \times 100 = 105.86 \] --- **Price relatives for 2021**: - **Clothes**: \( \frac{18}{12} = 1.5 \) - **Food**: \( \frac{48}{40} = 1.2 \) - **Living**: \( \frac{120}{115} \approx 1.0435 \) - **Commute**: \( \frac{10}{10} = 1 \) Calculating the weighted sum for CPI 2021: \[ \text{CPI}_{2021} = (0.10 \times 1.5) + (0.30 \times 1.2) + (0.45 \times 1.0435) + (0.15 \times 1) \] \[ = 0.15 + 0.36 + 0.4696 + 0.15 = 1.1296 \] \[ \text{CPI}_{2021} = 1.1296 \times 100 = 112.96 \] --- ### (b) Calculate the percentage change in prices from 2020 to 2021 \[ \text{Percentage change} = \frac{112.96 - 105.86}{105.86} \times 100 = \frac{7.10}{105.86} \times 100 \approx 6.71\% \] --- ### (c) Items with price rise from 2020 to 2021 Compare 2020 and 2021 prices: - **Clothes**: 15 → 18 (Increase) - **Food**: 45 → 48 (Increase) - **Living**: 114 → 120 (Increase) - **Commute**: 10 → 10 (No change) Items that experienced price rise: **Clothes, Food, Living**. --- # Summary - The value of \( w \) such that 14% of packages are heavier is **3.148 kg**. - Shipping fee statistics: - Mean: **\$60** - Standard deviation: **\$7.2** - Variance: **\$51.84** - Median: **\$60** - 86th percentile: **\$67.78** - CPI for 2020: **105.86** - CPI for 2021: **112.96** - Percentage change in prices from 2020 to 2021: **6.71%** - Items with price rise from 2020 to 2021: **Clothes, Food, Living**.

# Given Information ### Question 1 - The weight of a package delivered by a courier company is normally distributed. - Mean (\(\mu\)) = 2.5 kg - Standard deviation (\(\sigma\)) = 0.6 kg #### (a) - 14% of the packages weigh heavier than \( w \) kg. Find the value of \( w \). #### (b) - Shipping fee = \$30 basic charge + \$12 per kg of the package. - Find the following for the shipping fee: 1. Mean 2. Standard deviation 3. Variance 4. Median 5. 86th percentile --- ### Question 2 - Expenditure weights and prices of items for years 2019, 2020, and 2021 are given. | Item | Weight | 2019 | 2020 | 2021 | |-----------|--------|------|------|------| | Clothes | 10% | 12 | 15 | 18 | | Food | 30% | 40 | 45 | 48 | | Living | 45% | 115 | 114 | 120 | | Commute | 15% | 10 | 10 | 10 | #### (a) - Compile the CPIs for 2020 and 2021 using 2019 as the base year. #### (b) - Calculate the percentage change in prices from 2020 to 2021. #### (c) - Which items have experienced a price rise from 2020 to 2021? --- # Definitions and Concepts Used - **Normal Distribution**: The probability that a value \( X \) is greater than \( w \) can be found using the Z-score formula: \( Z = \frac{w - \mu}{\sigma} \). - **Percentile**: The value below which a given percentage of data falls. - **Linear Transformation**: For \( Y = a + bX \), the new mean is \( a + b\mu \), the new standard deviation is \( |b|\sigma \), and the new variance is \( b^2\sigma^2 \). - **Consumer Price Index (CPI)**: \[ \text{CPI} = \sum w \times \left(\frac{P_t}{P_0}\right) \times 100 \] where \( P_t \) is the price in year \( t \) and \( P_0 \) is the price in the base year. --- # Solution ## Question 1 ### (a) Find the value of \( w \) Given: \( P(X > w) = 0.14 \), so \( P(X < w) = 0.86 \). Find the Z-score corresponding to 0.86 cumulative probability (using Excel function `=NORM.S.INV(0.86)`): \[ Z \approx 1.08 \] Using the Z-score formula: \[ Z = \frac{w - 2.5}{0.6} \] \[ 1.08 = \frac{w - 2.5}{0.6} \] Solving for \( w \): \[ w - 2.5 = 1.08 \times 0.6 = 0.648 \] \[ w = 2.5 + 0.648 = 3.148 \] --- ### (b) Shipping Fee Statistics Let \( X \) be the package weight (\( N(2.5, 0.6^2) \)), and the shipping fee \( Y = 30 + 12X \): - **Mean**: \[ E[Y] = 30 + 12 \times 2.5 = 60 \] - **Standard Deviation**: \[ SD[Y] = 12 \times 0.6 = 7.2 \] - **Variance**: \[ Var[Y] = (12^2) \times (0.6^2) = 144 \times 0.36 = 51.84 \] - **Median**: The median of \( X \) is 2.5, so the median of \( Y \) is: \[ 30 + 12 \times 2.5 = 60 \] - **86th Percentile**: Using the result from part (a): \[ X_{86\%} = 3.148 \Rightarrow Y_{86\%} = 30 + 12 \times 3.148 = 67.776 \] --- ## Question 2 ### (a) Compile CPIs for 2020 and 2021 using 2019 as the base year **Price relatives for 2020**: - **Clothes**: \( \frac{15}{12} = 1.25 \) - **Food**: \( \frac{45}{40} = 1.125 \) - **Living**: \( \frac{114}{115} \approx 0.9913 \) - **Commute**: \( \frac{10}{10} = 1 \) Calculating the weighted sum for CPI 2020: \[ \text{CPI}_{2020} = (0.10 \times 1.25) + (0.30 \times 1.125) + (0.45 \times 0.9913) + (0.15 \times 1) \] \[ = 0.125 + 0.3375 + 0.4461 + 0.15 = 1.0586 \] \[ \text{CPI}_{2020} = 1.0586 \times 100 = 105.86 \] --- **Price relatives for 2021**: - **Clothes**: \( \frac{18}{12} = 1.5 \) - **Food**: \( \frac{48}{40} = 1.2 \) - **Living**: \( \frac{120}{115} \approx 1.0435 \) - **Commute**: \( \frac{10}{10} = 1 \) Calculating the weighted sum for CPI 2021: \[ \text{CPI}_{2021} = (0.10 \times 1.5) + (0.30 \times 1.2) + (0.45 \times 1.0435) + (0.15 \times 1) \] \[ = 0.15 + 0.36 + 0.4696 + 0.15 = 1.1296 \] \[ \text{CPI}_{2021} = 1.1296 \times 100 = 112.96 \] --- ### (b) Calculate the percentage change in prices from 2020 to 2021 \[ \text{Percentage change} = \frac{112.96 - 105.86}{105.86} \times 100 = \frac{7.10}{105.86} \times 100 \approx 6.71\% \] --- ### (c) Items with price rise from 2020 to 2021 Compare 2020 and 2021 prices: - **Clothes**: 15 → 18 (Increase) - **Food**: 45 → 48 (Increase) - **Living**: 114 → 120 (Increase) - **Commute**: 10 → 10 (No change) Items that experienced price rise: **Clothes, Food, Living**. --- # Summary - The value of \( w \) such that 14% of packages are heavier is **3.148 kg**. - Shipping fee statistics: - Mean: **\$60** - Standard deviation: **\$7.2** - Variance: **\$51.84** - Median: **\$60** - 86th percentile: **\$67.78** - CPI for 2020: **105.86** - CPI for 2021: **112.96** - Percentage change in prices from 2020 to 2021: **6.71%** - Items with price rise from 2020 to 2021: **Clothes, Food, Living**.