Question

6.37 More than one confidence interval. As we prepare to take a sample and compute a 95% confidence interval, we know that the probability that the interval we compute will cover the parameter is 0.95. That’s the meaning of 95% confidence. If we plan to use several such intervals, however, our confidence that all of them will give correct results is less than 95%. Suppose that we plan to take independent samples each month for five months and report a 95% confidence interval for each set of data. (a) What is the probability that all five intervals will cover the true means? This probability (expressed as a percent) is our overall confidence level for the five simultaneous statements. (b) Suppose we instead considered individual 99% confidence intervals. Now, what is the overall confidence level for the five simultaneous statements? (c) Based on the results of parts (a) and (b), how could you keep

Answer 1

A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. The probability that all five intervals will cover the true means is approximately 77%. The overall confidence level decreases as the number of intervals increases and the confidence level for each interval decreases.

Step-by-step explanation:

A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. In this case, the confidence level is 95%, which means there is a 95% probability that the true means of the population will be within the interval. For each set of data collected over five months, the probability that all five intervals will cover the true means is calculated by multiplying the individual probabilities together. Therefore, the probability that all five intervals will cover the true means is 95% multiplied by 95% multiplied by 95% multiplied by 95% multiplied by 95%, which is approximately 77%.

If we consider individual 99% confidence intervals instead, the overall confidence level for the five simultaneous statements is calculated by multiplying 99% by 99% by 99% by 99% by 99%, which is approximately 95%.

Based on the results of these calculations, we can see that the overall confidence level decreases as the number of intervals increases and the confidence level for each interval decreases. To keep a high overall confidence level, it is important to adjust the confidence level for each individual interval.