Question
The managers of a brokerage firm are interested in finding out if the number of new clients a broker brings into the firm affects the sales generated by the broker. They sample 12 brokers and determine the number of new clients they have enrolled in the last year and their sales amounts in thousands of dollars. These data are presented in the table that follows.
Broker Clients Sales
1 27 52
2 11 37
3 42 64
4 33 55
5 15 29
6 15 34
7 25 58
8 36 59
9 28 44
10 30 48
11 17 31
12 22 38
Suppose the managers of the brokerage firm want to construct a 99% confidence interval estimate for the mean sales made by brokers who have brought into the firm 24 new clients. The confidence interval is from ________ to ________.
39.19 to 49.89
40.29 to 48.62
38.24 to 50.26
37.56 to 52.25
Broker Clients Sales
1 27 52
2 11 37
3 42 64
4 33 55
5 15 29
6 15 34
7 25 58
8 36 59
9 28 44
10 30 48
11 17 31
12 22 38
Suppose the managers of the brokerage firm want to construct a 99% confidence interval estimate for the mean sales made by brokers who have brought into the firm 24 new clients. The confidence interval is from ________ to ________.
39.19 to 49.89
40.29 to 48.62
38.24 to 50.26
37.56 to 52.25
Answers
GPT-4o mini
To construct a 99% confidence interval for the mean sales made by brokers who have brought in 24 new clients, we need to follow several steps:
1. **Identify the relevant data points**: We need to find the sales amounts of brokers with 24 or fewer new clients since we want to estimate the mean sales for brokers who have brought in 24 or fewer new clients.
2. **Filter the data**:
- Broker 1: 27 clients, Sales: 52
- Broker 2: 11 clients, Sales: 37
- Broker 3: 42 clients, Sales: 64
- Broker 4: 33 clients, Sales: 55
- Broker 5: 15 clients, Sales: 29
- Broker 6: 15 clients, Sales: 34
- Broker 7: 25 clients, Sales: 58
- Broker 8: 36 clients, Sales: 59
- Broker 9: 28 clients, Sales: 44
- Broker 10: 30 clients, Sales: 48
- Broker 11: 17 clients, Sales: 31
- Broker 12: 22 clients, Sales: 38
Brokers with 24 or fewer new clients are Brokers 2, 5, 6, 11, and 12.
3. **Calculate the sales for these brokers**:
- Broker 2: 37
- Broker 5: 29
- Broker 6: 34
- Broker 11: 31
- Broker 12: 38
4. **Calculate the sample mean (\( \bar{x} \)) and sample standard deviation (s)** for these sales:
- Sample Mean:
\[
\bar{x} = \frac{37 + 29 + 34 + 31 + 38}{5} = \frac{169}{5} = 33.8
\]
- Sample Standard Deviation:
\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
\]
Breaking the calculations down:
- \(x_1 = 37\), \(x_2 = 29\), \(x_3 = 34\), \(x_4 = 31\), \(x_5 = 38\).
- Deviations from mean:
- \(37 - 33.8 = 3.2\)
- \(29 - 33.8 = -4.8\)
- \(34 - 33.8 = 0.2\)
- \(31 - 33.8 = -2.8\)
- \(38 - 33.8 = 4.2\)
- Square of deviations:
- \(3.2^2 = 10.24\)
- \((-4.8)^2 = 23.04\)
- \(0.2^2 = 0.04\)
- \((-2.8)^2 = 7.84\)
- \(4.2^2 = 17.64\)
Sum of squares = \(10.24 + 23.04 + 0.04 + 7.84 + 17.64 = 58.8\)
Now plug it in:
\[
s = \sqrt{\frac{58.8}{4}} = \sqrt{14.7} \approx 3.84
\]
5. **Construct the 99% confidence interval**:
Using the t-distribution because the sample size is small (\(n = 5\)):
- Degrees of freedom = \(n - 1 = 5 - 1 = 4\)
- The critical value \(t\) for a 99% confidence interval and 4 degrees of freedom is approximately \(3.747\) (you can find this in a t-table).
The margin of error (ME) is:
\[
\text{ME} = t \times \frac{s}{\sqrt{n}} = 3.747 \times \frac{3.84}{\sqrt{5}} \approx 3.747 \times 1.718 \approx 6.43
\]
6. **Finally, construct the confidence interval**:
\[
CI = \bar{x} \pm \text{ME} = 33.8 \pm 6.43
\]
- Lower Limit: \(33.8 - 6.43 \approx 27.37\)
- Upper Limit: \(33.8 + 6.43 \approx 40.23\)
Since my calculations for sample mean indicate no close correlation with the previously given options for the confidence interval, I recommend checking if broker sales data or calculations have been interpreted correctly as the final confidence interval does not match any of the options given.
If you have any updates or additional changes, please share those for more precise calculations.
1. **Identify the relevant data points**: We need to find the sales amounts of brokers with 24 or fewer new clients since we want to estimate the mean sales for brokers who have brought in 24 or fewer new clients.
2. **Filter the data**:
- Broker 1: 27 clients, Sales: 52
- Broker 2: 11 clients, Sales: 37
- Broker 3: 42 clients, Sales: 64
- Broker 4: 33 clients, Sales: 55
- Broker 5: 15 clients, Sales: 29
- Broker 6: 15 clients, Sales: 34
- Broker 7: 25 clients, Sales: 58
- Broker 8: 36 clients, Sales: 59
- Broker 9: 28 clients, Sales: 44
- Broker 10: 30 clients, Sales: 48
- Broker 11: 17 clients, Sales: 31
- Broker 12: 22 clients, Sales: 38
Brokers with 24 or fewer new clients are Brokers 2, 5, 6, 11, and 12.
3. **Calculate the sales for these brokers**:
- Broker 2: 37
- Broker 5: 29
- Broker 6: 34
- Broker 11: 31
- Broker 12: 38
4. **Calculate the sample mean (\( \bar{x} \)) and sample standard deviation (s)** for these sales:
- Sample Mean:
\[
\bar{x} = \frac{37 + 29 + 34 + 31 + 38}{5} = \frac{169}{5} = 33.8
\]
- Sample Standard Deviation:
\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
\]
Breaking the calculations down:
- \(x_1 = 37\), \(x_2 = 29\), \(x_3 = 34\), \(x_4 = 31\), \(x_5 = 38\).
- Deviations from mean:
- \(37 - 33.8 = 3.2\)
- \(29 - 33.8 = -4.8\)
- \(34 - 33.8 = 0.2\)
- \(31 - 33.8 = -2.8\)
- \(38 - 33.8 = 4.2\)
- Square of deviations:
- \(3.2^2 = 10.24\)
- \((-4.8)^2 = 23.04\)
- \(0.2^2 = 0.04\)
- \((-2.8)^2 = 7.84\)
- \(4.2^2 = 17.64\)
Sum of squares = \(10.24 + 23.04 + 0.04 + 7.84 + 17.64 = 58.8\)
Now plug it in:
\[
s = \sqrt{\frac{58.8}{4}} = \sqrt{14.7} \approx 3.84
\]
5. **Construct the 99% confidence interval**:
Using the t-distribution because the sample size is small (\(n = 5\)):
- Degrees of freedom = \(n - 1 = 5 - 1 = 4\)
- The critical value \(t\) for a 99% confidence interval and 4 degrees of freedom is approximately \(3.747\) (you can find this in a t-table).
The margin of error (ME) is:
\[
\text{ME} = t \times \frac{s}{\sqrt{n}} = 3.747 \times \frac{3.84}{\sqrt{5}} \approx 3.747 \times 1.718 \approx 6.43
\]
6. **Finally, construct the confidence interval**:
\[
CI = \bar{x} \pm \text{ME} = 33.8 \pm 6.43
\]
- Lower Limit: \(33.8 - 6.43 \approx 27.37\)
- Upper Limit: \(33.8 + 6.43 \approx 40.23\)
Since my calculations for sample mean indicate no close correlation with the previously given options for the confidence interval, I recommend checking if broker sales data or calculations have been interpreted correctly as the final confidence interval does not match any of the options given.
If you have any updates or additional changes, please share those for more precise calculations.