7 fill in the blank 7 points\nsuppose we wanted to test whether dress codes in schools has an effect on…

7 fill in the blank 7 points\nsuppose we wanted to test whether dress codes in schools has an effect on student self-esteem using a standardized self-esteem test (the rosenburg self-esteem test, or rse score). we know the standard deviation for this standardized self-esteem test is 12. if we took two new samples from each group and found that:\n$\\bar{x}_1 = 124$ and $n_1 = 25$\n$\\bar{x}_2 = 118$ and $n_2 = 25$\nassuming an alpha-level of 0.01, state the null and the alternative hypotheses and determine whether the results are significant.\nis this a directional or a non-directional hypothesis?\nnon-directional

7 fill in the blank 7 points\nsuppose we wanted to test whether dress codes in schools has an effect on student self-esteem using a standardized self-esteem test (the rosenburg self-esteem test, or rse score). we know the standard deviation for this standardized self-esteem test is 12. if we took two new samples from each group and found that:\n$\\bar{x}_1 = 124$ and $n_1 = 25$\n$\\bar{x}_2 = 118$ and $n_2 = 25$\nassuming an alpha-level of 0.01, state the null and the alternative hypotheses and determine whether the results are significant.\nis this a directional or a non-directional hypothesis?\nnon-directional

Answer

Explanation:

Step1: State the null and alternative hypotheses.

$H_0: \mu_1 = \mu_2$ and $H_a: \mu_1 \neq \mu_2$

Step2: Calculate the standard error of the difference.

$\sigma_{\bar{x}_1-\bar{x}_2} = \sqrt{\frac{\sigma^2}{n_1} + \frac{\sigma^2}{n_2}} = \sqrt{\frac{12^2}{25} + \frac{12^2}{25}} = \sqrt{5.76 + 5.76} \approx 3.394$

Step3: Calculate the z-test statistic.

$z = \frac{\bar{x}_1 - \bar{x}2}{\sigma{\bar{x}_1-\bar{x}_2}} = \frac{124 - 118}{3.394} \approx 1.768$

Step4: Determine significance at alpha = 0.01.

For a two-tailed test at $\alpha = 0.01$, the critical $z$-value is $\pm 2.576$. Since $|1.768| < 2.576$, we fail to reject the null hypothesis.

Answer:

Null hypothesis: $H_0: \mu_1 = \mu_2$; Alternative hypothesis: $H_a: \mu_1 \neq \mu_2$. The results are not statistically significant because the calculated $z$-score of $1.768$ is less than the critical value of $2.576$. The hypothesis is non-directional.