A software application (app) lets users enter questions to receive answers in the form of images, texts, or videos. Research indicates that 22% of students in Country W use the app to help them with their homework at least once per week. Karen is a student in Country W at a high school that has more than 2,000 students. She believes the proportion of all students at her school who use the app to help them with their homework at least once per week is greater than the proportion for her country. To investigate her belief, she took a simple random sample of 130 students from her school and found that 38 of the sampled students use the app to help them with their homework at least once per week.
Is there convincing statistical evidence, at a 0.05 significance level, to support Karen’s belief? Justify your answer with the appropriate inference procedure.
Answer:
You will need a full explanation to justify answers for quizzes and exams (see solution).
Solution.
We want to perform a test of $$ \begin{align*} H_0:p&=0.22\\ H_a:p&>0.22\\ \end{align*} $$ where $p$ is the actual proportion of students at Karen's school who use the app to help with homework at least once per week. We'll use an $\alpha=0.05$ significance level.
Since Karen took an SRS of 130 students out of a population that is at least $10\times$ the size of $130$ ($2000>130\times 10$) (10% Condition), the SRS is well-approximated by sampling with replacement. Next, assuming $H_0:p=0.22$ is true, observe that $$ \begin{align*} np&=130(0.22)=28.6>10\\[0.3em] n(1-p)&=130(0.78)=101.4>10 \end{align*} $$ so the Large Counts Assumption is also satisfied. Therefore the sampling distribution of $\hat p$ is approximately normal, and we can proceed with a one-sample $z$-test for the population proportion $p$.
We compute:
$$ z=\dfrac{\hat p - p}{\sqrt{\dfrac{p(1-p)}{n}}}\approx\dfrac{0.292308 - 0.22}{\sqrt{\dfrac{0.22(0.78)}{130}}}\approx1.9902 $$This gives a $p$-value of $p=0.023284$ which is less than $\alpha=0.05$. Therefore we reject the null hypothesis and accept the alternative hypothesis.
Conclusion: There is sufficient evidence to suggest that the proportion of all students at Karen's school who use the app to help them with their homework at least once per week is greater than the proportion for her country.
TODO.
Answer:Solution.
TODO