Question

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.

Short answer

(a) The sample proportion is less than $60\%$i.e. $45\%$.

(b) It's also feasible that the sample proportion is lower than $60\%$because the population proportion is likewise lower than $60\%$.

(c) One basic random sampling of $100$students revealed that $73$of the $100$had finished their assigned homework the previous week.

(d) Yes, it is surprising to get the sample proportion.

(e) Yes, there is a piece of convincing evidence that less than sixty percent of all students at the school completed their assigned homework last week.

Step-by-step solution

Part (a) Step 1: Given information

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

Part (a) Step 2: Explanation

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.

Part (b) Step 1: Given information

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

Part (b) Step 2: Explanation

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 $60\%$.

Part (c) Step 1: Given information

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

Part (c) Step 2: Explanation

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 $0.73$).

Part (d) Step 1: Given information

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

Part (d) Step 2: Explanation

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 $p=0.45$.

Part (e) Step 1: Given information

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

Part (e) Step 2: Explanation

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.