More homework Some skeptical Ap® Statistics students want to investigate the newspaper's claim in Exercise $11$, so they choose an SRS of $100$students from the school to interview. In their sample, $45$students completed their homework last week. Does this provide convincing evidence that less than $60\%$of all students at the school completed their assigned homework last week?

a. What is the evidence that less than $60\%$of all students completed their assigned homework last week?

b. Provide two explanations for the evidence described in part (a).

We used technology to simulate choosing $250$SRSs of size $n={100}^{n=100}$from a population of $2000$students where $60\%$completed their assigned homework last week. The dotplot shows $p^{\hat{p}}$the sample proportion of students who completed their assigned homework last week for each of the $250$simulated samples.

c. There is one dot on the graph at $0.73$. Explain what this value represents.

d. Would it be surprising to get a sample proportion of $p=0.{45}^{\hat{p}=0.45}$or smaller in an SRS of size $100$when $p=0.{60}^{p=0.60}$? Justify your answer.

e. Based on your previous answers, is there convincing evidence that less than $60\%$of all students at the school completed their assigned homework last week? Explain your reasoning.

We need to find out the evidence for all the students completing their assigned work last week.

We know that

The sample proportion is a division of the number of successes and sample size i.e. $\hat{p}=\frac{x}{n}=\frac{45}{100}=0.45=45\%$

We should notice that the sample proportion of $45\%$is lower than the advertised $60\%$, implying that less than$60\%$ of all students at the school completed their assigned homework last week.

We need to find out the explanations for the evidence described in part (a).

Because the population proportion is $60\%$and we obtained a sample with a sample proportion of only $45\%$by chance, the sample proportion could be less than $60\%$.

However, it's also feasible that the sample proportion is lower than $60\%$because the population proportion is likewise lower than $60\%$.

We need to find out the representation of value $0.73$.

Each dot in the dotplot indicates a sample proportion $p$for a simple random sample (SRS) of $100$students, where the sample proportion is the proportion of students in the sample who completed their assigned homework the previous week.

The dot at $0.73$indicates a basic random sample of $100$pupils, with $73$of them having done their assigned homework the previous week (which then results in a sample proportion of $0.73$).

We need to find whether it is surprising to get smaller SRS or not.

We can see that there are no dots above $0.45$and no dots to the left of $0.45$in the following dotplot. As a result, obtaining a sample with a sample percentage of $p=0.45$is highly implausible when the population proportion is actually $p=0.60$, and we would be shocked if we obtained a sample with a sample proportion of $p=0.45$.

We need to find the convincing answer for part (d).

We can see that there are no dots above $0.45$and no dots to the left of $0.45$in the following dotplot. This means that the sample proportion is very unlikely to be $p=0.45$when the population proportion is actually $p=0.60$, and we would be shocked if we got a sample with sample proportion $p=0.45$.

This suggests that less than $60\%$of all pupils at the school completed their given homework last week, according to credible information.