Question

a private and a public university are located in the same city. for the private university, 1046 alumni were surveyed and 653 said that they attended at least one class reunion. for the public university, 791 out of 1327 sampled alumni claimed they have attended at least one class reunion. is the difference in the sample proportions statistically significant at the 1% level?

Answer 1

Using the z-distribution, it is found that since the absolute value of the test statistic is less than the critical value, the difference in the sample proportions is not statistically significant at the 1% level.

At the null hypothesis, we test if the proportions are the same, that is, their subtraction is 0, hence:

$H_0: p_1 - p_2 = 0$

At the alternative hypothesis, it is tested if they are different, that is, their subtraction is not 0, hence:

$H_1: p_1 - p_2 \neq 0$

The proportions and their respective standard errors are given by:

$p_1 = \frac{653}{1046} = 0.6343, s_1 = \sqrt{\frac{0.6343(0.3657)}{1046}} = 0.0149$

$p_2 = \frac{791}{1327} = 0.5961, s_2 = \sqrt{\frac{0.5961(0.4039)}{1327}} = 0.0135$

For the distribution of the difference, the mean and the standard error are given by:

$\overline{p} = p_1 - p_2 = 0.6343 - 0.5961 = 0.0382$

$s = \sqrt{s_1^2 + s_2^2} = \sqrt{0.0149^2 + 0.0135^2} = 0.0201$

The test statistic is:

$z = \frac{\overline{p} - p}{s}$

In which $p = 0$ is the value tested at the null hypothesis.

Hence:

$z = \frac{\overline{p} - p}{s}$

$z = \frac{0.0382 - 0}{0.0201}$

$z = 1.9$

The critical value for a two-tailed test, as we are testing if two values are different, with a significance level of 0.01, is of $|z^{\ast}| = 2.576$

Since the absolute value of the test statistic is less than the critical value, the difference in the sample proportions is not statistically significant at the 1% level.

A similar problem is given at https://brainly.com/question/25728144