Question

About 5000 students in Germany took part in the 2012 PISA study. participated. Overall, Germany achieved a score of 514. As we have already seen, the PISA scores are standardized to have a mean of 500 and a standard deviation of 100. a) Why is the calculation of the confidence interval of an expected value based on the z-value of 1−(α/2) and not 1−α ? b) Calculate the 95% confidence interval of this mean value of the German students. As described in the distance learning letter, the z-value of 0.975 is is 1.96. c) Now we want to calculate the 99% confidence interval. Will this interval be larger or be smaller than the 95% confidence interval? Why? d) For the calculation of the interval we again need a z-value for the changed probability. What is the z-value we are looking for? Note: In R, calculate the z-value using the qnorm0 function. e) Using this z-yalue, you can now specify the 99% confidence interval. As is this?

Answer 1

A 95% confidence interval for the mean PISA score of German students is (511.23, 516.77) and the 99% confidence interval is wider at (510.36, 517.64) to reflect the greater certainty that it contains the true mean.

Step-by-step explanation:

The calculation of the confidence interval of an expected value is based on the z-value of 1−(\alpha/2) and not 1−\alpha because when constructing a two-tailed confidence interval, we are interested in the area in both tails (equally split) that sums to the significance level \alpha. Therefore, we subtract half of \alpha from 1 to locate the critical z-value for each end of the interval.

To calculate the 95% confidence interval for the mean value of the German students in the PISA study, we use the formula: Mean ± (z-value * (Standard Deviation / sqrt(n))). With a mean score of 514, standard deviation of 100, z-value of 1.96 (for a 95% confidence level), and approximately 5000 students, we get:

514 ± (1.96 * (100 / sqrt(5000))) = 514 ± (1.96 * (100 / 70.71)) = 514 ± (2.77) = (511.23, 516.77).

The 99% confidence interval will be larger than the 95% confidence interval because a higher confidence level requires a larger area under the normal distribution curve, thus leading to a wider interval. The z-value for the 99% confidence interval is typically around 2.576 (using the qnorm function in R). Using this z-value we calculate the 99% confidence interval as follows:

514 ± (2.576 * (100 / sqrt(5000))) = 514 ± (2.576 * (100 / 70.71)) = 514 ± (3.64) = (510.36, 517.64).