Question

the prices of a random sample of 23 new motorcycles have a sample standard deviation of $3710. assume the sample is from a normally distributed population. construct a confidence interval for the population variance \\( \sigma ^ { 2 } \\) and the population standard deviation \\( \sigma \\). use a 95% level of confidence. interpret the results.

c. with 5% confidence, you can say that the population variance is between and
d. with 95% confidence, you can say that the population variance is between 8232853 and 27573365
what is the confidence interval for the population standard deviation \\( \sigma \\)?
\\( ( 2869,5251 ) \\) (round to the nearest integer as needed.)
interpret the results. select the correct choice below and fill in the answer box(es) to complete your choice. (round to the nearest integer as needed.)
a. with 5% confidence, you can say that the population standard deviation is between \\( \\$ \\) and \\( \\$ \\).
b. with 95% confidence, you can say that the population standard deviation is between \\( \\$ \\) and \\( \\$ \\).
c. with 5% confidence, you can say that the population standard deviation is less than \\( \\$ \\).
d. with 95% confidence, you can say that the population standard deviation is greater than \\( \\$ \\)

Explanation:

Step1: Recall the relationship between variance and standard deviation

The confidence interval for the population standard deviation \(\sigma\) is constructed from the confidence interval for the population variance \(\sigma^{2}\). If the confidence interval for \(\sigma^{2}\) is \((L, U)\) (where \(L = 8232853\) and \(U=27573365\)), then for \(\sigma\), we use the formula \(\sqrt{L}<\sigma <\sqrt{U}\).

Step2: Calculate the square - roots

Calculate \(\sqrt{8232853}\) and \(\sqrt{27573365}\).
\(\sqrt{8232853}\approx2869\) (using a calculator, \(\sqrt{8232853}=\sqrt{8232853}\approx2869\))
\(\sqrt{27573365}\approx5251\) (using a calculator, \(\sqrt{27573365}=\sqrt{27573365}\approx5251\))

Answer:

B. With 95% confidence, you can say that the population standard deviation is between \(\$2869\) and \(\$5251\).