Answer :
To solve this problem, we need to match each broker (Amanda, Benjamin, Cathy) to the given descriptions based on the given means and standard deviations for their respective commissions. Let's follow these steps to determine the correct matches.
1. Identify the broker whose [tex]$95\%$[/tex] of data lies between [tex]$67,785.08 and $[/tex]83,141.16[tex]$: - For a normal distribution, approximately $[/tex]95\%[tex]$ of the data lies within 2 standard deviations (2 SD) from the mean. - Compute the range covered by $[/tex]2[tex]$ standard deviations for each broker: - Amanda: - Lower bound: $[/tex]75,463.12 - 2 \times 3,839.02[tex]$ - Upper bound: $[/tex]75,463.12 + 2 \times 3,839.02[tex]$ - Benjamin: - Lower bound: $[/tex]74,124.87 - 2 \times 4,062.50[tex]$ - Upper bound: $[/tex]74,124.87 + 2 \times 4,062.50[tex]$ - Cathy: - Lower bound: $[/tex]76,095.71 - 2 \times 4,227.54[tex]$ - Upper bound: $[/tex]76,095.71 + 2 \times 4,227.54[tex]$ The correct broker's 2 SD range covers $[/tex]67,785.08[tex]$ to $[/tex]83,141.16[tex]$: - Amanda: $[/tex]67,785.08[tex]$ to $[/tex]83,141.16[tex]$ - Benjamin: $[/tex]65,999.87[tex]$ to $[/tex]82,249.87[tex]$ - Cathy: $[/tex]67,640.63[tex]$ to $[/tex]84,550.79[tex]$ The broker matching this description is Cathy. 2. Identify the broker whose $[/tex]68\%[tex]$ of their data lies between $[/tex]70,062.37 and [tex]$78,187.37$[/tex]:
- For a normal distribution, approximately [tex]$68\%$[/tex] of the data lies within 1 standard deviation (1 SD) from the mean.
- Compute the range covered by [tex]$1$[/tex] standard deviation for each broker:
- Amanda:
- Lower bound: [tex]$75,463.12 - 3,839.02$[/tex]
- Upper bound: [tex]$75,463.12 + 3,839.02$[/tex]
- Benjamin:
- Lower bound: [tex]$74,124.87 - 4,062.50$[/tex]
- Upper bound: [tex]$74,124.87 + 4,062.50$[/tex]
- Cathy:
- Lower bound: [tex]$76,095.71 - 4,227.54$[/tex]
- Upper bound: [tex]$76,095.71 + 4,227.54$[/tex]
The correct broker's 1 SD range covers [tex]$70,062.37$[/tex] to [tex]$78,187.37$[/tex]:
- Amanda: [tex]$71,624.10$[/tex] to [tex]$79,302.14$[/tex]
- Benjamin: [tex]$70,062.37$[/tex] to [tex]$78,187.37$[/tex]
- Cathy: [tex]$71,868.17$[/tex] to [tex]$80,323.25$[/tex]
The broker matching this description is Benjamin.
3. Identify the broker with the highest average monthly commission:
- Amanda's mean: [tex]$75,463.12$[/tex]
- Benjamin's mean: [tex]$74,124.87$[/tex]
- Cathy's mean: [tex]$76,095.71$[/tex]
The broker with the highest mean commission is Cathy.
Thus, the matches based on the given descriptions are:
- [tex]$95\%$[/tex] of data between [tex]$67,785.08$[/tex] and [tex]$83,141.16$[/tex]: Cathy
- [tex]$68\%$[/tex] of their data between [tex]$70,062.37$[/tex] and [tex]$78,187.37$[/tex]: Benjamin
- Highest average monthly commission: Cathy
The resulting mapped responses are:
- Amanda
- Cathy
- Benjamin
- [tex]$95\%$[/tex] of their data lie between [tex]$\$[/tex] 67,785.08[tex]$ and $[/tex]\[tex]$ 83,141.16$[/tex] (2 SD from the mean) [tex]$\longleftrightarrow$[/tex] [tex]$\boxed{\text{Cathy}}$[/tex]
- [tex]$68\%$[/tex] of their data lie between [tex]$\$[/tex] 70,062.37[tex]$ and $[/tex]\[tex]$ 78,187.37$[/tex] (1 SD from the mean) [tex]$\boxed{\text{Benjamin}}$[/tex]
- Had the highest average monthly commission during the past year [tex]$\boxed{\text{Cathy}}$[/tex]
1. Identify the broker whose [tex]$95\%$[/tex] of data lies between [tex]$67,785.08 and $[/tex]83,141.16[tex]$: - For a normal distribution, approximately $[/tex]95\%[tex]$ of the data lies within 2 standard deviations (2 SD) from the mean. - Compute the range covered by $[/tex]2[tex]$ standard deviations for each broker: - Amanda: - Lower bound: $[/tex]75,463.12 - 2 \times 3,839.02[tex]$ - Upper bound: $[/tex]75,463.12 + 2 \times 3,839.02[tex]$ - Benjamin: - Lower bound: $[/tex]74,124.87 - 2 \times 4,062.50[tex]$ - Upper bound: $[/tex]74,124.87 + 2 \times 4,062.50[tex]$ - Cathy: - Lower bound: $[/tex]76,095.71 - 2 \times 4,227.54[tex]$ - Upper bound: $[/tex]76,095.71 + 2 \times 4,227.54[tex]$ The correct broker's 2 SD range covers $[/tex]67,785.08[tex]$ to $[/tex]83,141.16[tex]$: - Amanda: $[/tex]67,785.08[tex]$ to $[/tex]83,141.16[tex]$ - Benjamin: $[/tex]65,999.87[tex]$ to $[/tex]82,249.87[tex]$ - Cathy: $[/tex]67,640.63[tex]$ to $[/tex]84,550.79[tex]$ The broker matching this description is Cathy. 2. Identify the broker whose $[/tex]68\%[tex]$ of their data lies between $[/tex]70,062.37 and [tex]$78,187.37$[/tex]:
- For a normal distribution, approximately [tex]$68\%$[/tex] of the data lies within 1 standard deviation (1 SD) from the mean.
- Compute the range covered by [tex]$1$[/tex] standard deviation for each broker:
- Amanda:
- Lower bound: [tex]$75,463.12 - 3,839.02$[/tex]
- Upper bound: [tex]$75,463.12 + 3,839.02$[/tex]
- Benjamin:
- Lower bound: [tex]$74,124.87 - 4,062.50$[/tex]
- Upper bound: [tex]$74,124.87 + 4,062.50$[/tex]
- Cathy:
- Lower bound: [tex]$76,095.71 - 4,227.54$[/tex]
- Upper bound: [tex]$76,095.71 + 4,227.54$[/tex]
The correct broker's 1 SD range covers [tex]$70,062.37$[/tex] to [tex]$78,187.37$[/tex]:
- Amanda: [tex]$71,624.10$[/tex] to [tex]$79,302.14$[/tex]
- Benjamin: [tex]$70,062.37$[/tex] to [tex]$78,187.37$[/tex]
- Cathy: [tex]$71,868.17$[/tex] to [tex]$80,323.25$[/tex]
The broker matching this description is Benjamin.
3. Identify the broker with the highest average monthly commission:
- Amanda's mean: [tex]$75,463.12$[/tex]
- Benjamin's mean: [tex]$74,124.87$[/tex]
- Cathy's mean: [tex]$76,095.71$[/tex]
The broker with the highest mean commission is Cathy.
Thus, the matches based on the given descriptions are:
- [tex]$95\%$[/tex] of data between [tex]$67,785.08$[/tex] and [tex]$83,141.16$[/tex]: Cathy
- [tex]$68\%$[/tex] of their data between [tex]$70,062.37$[/tex] and [tex]$78,187.37$[/tex]: Benjamin
- Highest average monthly commission: Cathy
The resulting mapped responses are:
- Amanda
- Cathy
- Benjamin
- [tex]$95\%$[/tex] of their data lie between [tex]$\$[/tex] 67,785.08[tex]$ and $[/tex]\[tex]$ 83,141.16$[/tex] (2 SD from the mean) [tex]$\longleftrightarrow$[/tex] [tex]$\boxed{\text{Cathy}}$[/tex]
- [tex]$68\%$[/tex] of their data lie between [tex]$\$[/tex] 70,062.37[tex]$ and $[/tex]\[tex]$ 78,187.37$[/tex] (1 SD from the mean) [tex]$\boxed{\text{Benjamin}}$[/tex]
- Had the highest average monthly commission during the past year [tex]$\boxed{\text{Cathy}}$[/tex]
