Hello, my problem is I have a random sample of only 15 sales average is $115 with a known population standard deviation of $25.00. Obtain a 92% confidence interval and explain what your answer means. I am not sure what to do because the sample size is only 15. Thanks for your help.

mean = $115

Sd = $25
n = 15
1-a = .92

a = 0.08
Za/2 = z.04 = 1.75

115 -+ 1.75 * 25/sqrt15))

115-+ 11.296

(103.704, 126.296)

Thank you, I got the same answer, but I thought that because the sample size was less than 30, there was something I missed.

To calculate a confidence interval for the population mean with a known standard deviation when the sample size is small (less than 30), we can use the t-distribution. Here's how you can calculate the 92% confidence interval for your problem:

Step 1: Determine the critical value
The critical value corresponds to the desired confidence level and the degrees of freedom, which is equal to the sample size minus one (n - 1). For a 92% confidence level with 14 degrees of freedom, you can refer to a t-distribution table or use statistical software to find the critical value. In this case, the critical value is approximately 1.768.

Step 2: Calculate the margin of error
The margin of error represents the maximum expected distance between the sample mean and the true population mean. It is calculated by multiplying the critical value with the standard error.

Standard error (SE) = (population standard deviation) / sqrt(sample size)
SE = $25 / sqrt(15) ≈ $6.45

Margin of error (ME) = critical value * standard error
ME = 1.768 * $6.45 ≈ $11.42

Step 3: Calculate the confidence interval
To find the lower and upper bounds of the confidence interval, subtract and add the margin of error to the sample mean, respectively.

Lower bound = sample mean - margin of error
Lower bound = $115 - $11.42 ≈ $103.58

Upper bound = sample mean + margin of error
Upper bound = $115 + $11.42 ≈ $126.42

The 92% confidence interval for your sales data is approximately $103.58 to $126.42. This means that if you were to repeatedly take random samples of the same size and calculate confidence intervals, approximately 92% of those intervals would contain the true population mean.

In practical terms, this means you can be 92% confident that the true average sales amount falls within this range.