Question

In class we had 41 95% confidence intervals that we believe to be calculated correctly. The confidence intervals were collected by taking a sample of 60 data points from the population data.

41 of the confidence intervals appear to be calculated correctly.

Of these 4 of them do not have the population mean inside the confidence interval.

Based on a 95% confidence, we expected 2 to not contain the population mean.

Did this happen by chance alone?

To find your answer, complete the 7 steps of hypothesis testing and compute Beta.

Answer 1

As we are testing here whether the true confidence interval confidence level here is 95%, therefore this is a test for proportion being equal to 0.95.

The sample proportion here is computed as:
p = (41 - 4)/41 = 0.9024

As we are testing here whether the confidence level is 0.95, therefore the test statistic here is computed as:

P-P 0.9024 – 0.95 = -1.3973 P(1-P) 0.95*0.05 41 n

As this is a one tailed test, we have the p-value from standard normal tables here as:
p = P( Z < -1.3973) = 2*0.0812 = 0.1624

As the p-value here is 0.0812 > 0.05, therefore this is not an unusual event for 5% level of significance here, therefore we cannot reject the null hypothesis here. Therefore this can happen by chance alone. At 5% level of significance we dont have sufficient evidence here that the confidence level is not equal to 95% here.