Answer to Question #335141 in Statistics and Probability for Gaby

Question

About 24% of the population claim blue as their favorite color. Suppose a random sample of 56 students were surveyed, and 12 of them said that blue is their favorite color. Test the hypothesis that the color preference of all college students is different from that of the population. Use a 5% significance level.

(a) State the null and alternative hypotheses.

(b) Calculate the test statistic.

(c) Find or estimate the p-value for the test.

(d) State and interpret your conclusion.

Answer 1

The following null and alternative hypotheses for the population proportion needs to be tested:

$H_0:p=0.24$

$H_a:p\not=0.24$

This corresponds to a two-tailed test, for which a z-test for one population proportion will be used.

The z-statistic is computed as follows:

$$z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{\dfrac{12}{56}-0.24}{\sqrt{\dfrac{0.24(1-0.24)}{56}}}\approx-0.45056$$

The p-value is $p =2P(Z<-0.45056)=2(0.32615335)$

$= 0.652307.$

Based on the information provided, the significance level is $\alpha = 0.05,$ and the critical value for a two-tailed test is $z_c = 1.96.$

The rejection region for this two-tailed test is $R = \{z: |z|> 1.96\}.$

Since it is observed that $|z| = 0.45056\le1.96= z_c ,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is $p =0.652307,$ and since $p = 0.652307 \ge 0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population proportion $p$ is different than 0.24, at the $\alpha = 0.05$ significance level.