As part of an annual review of its accounts, a discount brokerage selected a random sample of 26 customers and reviewed the value of their accounts. The mean was $32,100, with a sample standard deviation of $8,550. (Use t Distribution Table & z Distribution Table.)What is a 95% confidence interval to estimate the mean account value of the population of customers? (Round your answers to the nearest dollar amount.)
Question
As part of an annual review of its accounts, a discount brokerage selected a random sample of 26 customers and reviewed the value of their accounts. The mean was 8,550. (Use t Distribution Table & z Distribution Table.)What is a 95% confidence interval to estimate the mean account value of the population of customers? (Round your answers to the nearest dollar amount.)
Solution
To calculate the 95% confidence interval for the mean account value of the population of customers, we will use the t-distribution because the sample size is less than 30. Here are the steps:
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Identify the sample mean () and sample standard deviation (s):
- Sample mean () = $32,100
- Sample standard deviation (s) = $8,550
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Determine the sample size (n):
- Sample size (n) = 26
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Find the degrees of freedom (df):
- Degrees of freedom (df) = n - 1 = 26 - 1 = 25
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Determine the t-value for a 95% confidence interval:
- For a 95% confidence interval and 25 degrees of freedom, we look up the t-value in the t-distribution table. The t-value for 25 degrees of freedom at a 95% confidence level is approximately 2.060.
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Calculate the standard error of the mean (SE):
- Standard error (SE) = s / √n = 1,676.92
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Calculate the margin of error (ME):
- Margin of error (ME) = t-value * SE = 2.060 * 3,454.46
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Determine the confidence interval:
- Lower limit = - ME = 3,454.46 ≈ $28,645.54
- Upper limit = + ME = 3,454.46 ≈ $35,554.46
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Round the confidence interval to the nearest dollar:
- Lower limit ≈ $28,646
- Upper limit ≈ $35,554
Therefore, the 95% confidence interval to estimate the mean account value of the population of customers is approximately 35,554.
Similar Questions
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