Let's go through the problem step-by-step:
a) Determine the null and alternative hypotheses.
- The null hypothesis ([tex]\(H_0\)[/tex]) represents the status quo or the current accepted value. In this case, it states that 71% of college students work.
[tex]\[
H_0: p = 0.71
\][/tex]
- The alternative hypothesis ([tex]\(H_a\)[/tex]) represents the claim being tested. The researcher thinks the percentage of college students who work has changed. So, the alternative hypothesis is:
[tex]\[
H_a: p \neq 0.71
\][/tex]
b) Determine the test statistic. Round to two decimals.
- To find the test statistic, we use the formula for the z-score in the context of proportions:
[tex]\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}
\][/tex]
where:
[tex]\(\hat{p}\)[/tex] = sample proportion
[tex]\(p_0\)[/tex] = population proportion
[tex]\(n\)[/tex] = sample size
- Given:
[tex]\[
\hat{p} = \frac{91}{110} = 0.8273
\][/tex]
[tex]\[
p_0 = 0.71
\][/tex]
[tex]\[
n = 110
\][/tex]
- Calculating the standard error:
[tex]\[
SE = \sqrt{\frac{0.71 \times (1 - 0.71)}{110}} \approx 0.0425
\][/tex]
- Calculating the z-score:
[tex]\[
z = \frac{0.8273 - 0.71}{0.0425} \approx 2.76
\][/tex]
c) Find the p-value. Round to 4 decimals.
- To find the p-value associated with the calculated z-score for a two-tailed test:
[tex]\[
p\text{-value} = 2 \times P(Z > 2.76)
\][/tex]
- Using standard normal distribution tables or computational tools, we find:
[tex]\[
p\text{-value} \approx 0.0057
\][/tex]
d) Make a decision.
- The significance level ([tex]\(\alpha\)[/tex]) is 0.01. We compare the p-value to the significance level:
[tex]\[
0.0057 < 0.01
\][/tex]
- Since the p-value is less than the significance level, we reject the null hypothesis.
e) Write the conclusion.
- Given that we rejected the null hypothesis, we conclude that:
[tex]\[
\text{There is sufficient evidence to support the claim that the percentage of college students who work is different than 71\%.}
\][/tex]
Summary:
a) [tex]\(H_0: p = 0.71\)[/tex], [tex]\(H_a: p \neq 0.71\)[/tex]
b) [tex]\(z = 2.76\)[/tex]
c) [tex]\(p\text{-value} = 0.0057\)[/tex]
d) Reject the null hypothesis
e) There is sufficient evidence to support the claim that the percentage of college students who work is different than 71\%.
