Question

1. you intend to estimate a population mean with a confidence interval. you believe the population to have a normal distribution. your sample size is 10. find the critical value that corresponds to a confidence level of 85%. (report answer accurate to three decimal places with appropriate rounding.) ( t_{a / 2}=pm )

Explanation:

Step1: Calculate the significance level

The confidence level \(C = 0.85\). The significance level \(\alpha=1 - C=1 - 0.85 = 0.15\).

Step2: Calculate \(\frac{\alpha}{2}\)

\(\frac{\alpha}{2}=\frac{0.15}{2}=0.075\).

Step3: Calculate the degrees of freedom

The sample size \(n = 10\). The degrees of freedom \(df=n - 1=10 - 1 = 9\).

Step4: Find the critical value

Using a \(t\) - distribution table or a calculator (e.g., in R: qt(1 - 0.075, 9) or in Excel: T.INV.2T(0.15,9)), we find the critical value.

Answer:

\(\pm1.574\)