Question 5.3: The sample mean and standard deviation for the cold cranking......
The sample mean and standard deviation for the cold cranking amperages of 100 batter-ies are \overline{X} = 185.5 and s = 5.0. Find an 85% confidence interval for the mean amperage of the batteries.
We must find the point estimate, the standard error, and the critical value. The point estimate is \overline{X} = 185.5. The standard error is \sigma_{\overline{X}}, which we approximate with \sigma_{\overline{X}} ≈ s ∕ \sqrt{n} = 0.5. To find the critical value for an 85% confidence interval, set 1−𝛼 = 0.85 to obtain 𝛼 = 0.15 and 𝛼 ∕ 2 = 0.075. The critical value is z_{.075}, the z-score that cuts off 7.5% of the area in the right-hand tail. From the z table, we find z_{.075} = 1.44. The margin of error is therefore (1.44)(0.5) = 0.72. So the 85% confidence interval is 185.5 ± 0.72 or, equivalently, (184.78, 186.22).