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 batteries are \bar{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 \bar{X} = 185.5. The standard error is \sigma_{\bar{X}}, which we approximate with \sigma_{\bar{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).