A large school district knows that $75 \%$ of students in previous years rode the bus to school. Administrators wondered if that figure was still accurate, so they took a random sample of $n=80$ students and found that $\hat{p}=65 \%$ of those sampled rode the bus to school.

To see how likely a sample like this was to happen by random chance alone, the school district performed a simulation. They simulated 120 samples of $n=80$ students from a large population where $75 \%$ of the students rode the bus to school. They recorded the proportion of students who rode the bus in each sample. Here are the sample proportions from their 120 samples:

They want to test $H_{0}: p=75 \%$ vs. $H_{\mathrm{a}}: p
eq 75 \%$ where $p$ is the true proportion of students in this district that ride the bus to school.

Based on these simulated results, what is the approximate $p$-value of the test?
Note: The sample result was $\hat{p}=65 \%$.
Choose 1 answer:
(A) $p$-value $\approx 0.07$
(B) $p$-value $\approx 0.058$
(C) $p$-value $\approx 0.04$
(D) $p$-value $\approx 0.0 \overline{3}$

Question

A large school district knows that $75 \%$ of students in previous years rode the bus to school. Administrators wondered if that figure was still accurate, so they took a random sample of $n=80$ students and found that $\hat{p}=65 \%$ of those sampled rode the bus to school.

To see how likely a sample like this was to happen by random chance alone, the school district performed a simulation. They simulated 120 samples of $n=80$ students from a large population where $75 \%$ of the students rode the bus to school. They recorded the proportion of students who rode the bus in each sample. Here are the sample proportions from their 120 samples:

They want to test $H_{0}: p=75 \%$ vs. $H_{\mathrm{a}}: p 
eq 75 \%$ where $p$ is the true proportion of students in this district that ride the bus to school.

Based on these simulated results, what is the approximate $p$-value of the test?
Note: The sample result was $\hat{p}=65 \%$.
Choose 1 answer:
(A) $p$-value $\approx 0.07$
(B) $p$-value $\approx 0.058$
(C) $p$-value $\approx 0.04$
(D) $p$-value $\approx 0.0 \overline{3}$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the observed sample proportion and the null hypothesis proportion. ‖ ‖ step 2 ⋮ Determine the number of simulated proportions that are as extreme or more extreme than the observed sample proportion. ‖ ‖ step 3 ⋮ Calculate the approximate p-value by dividing the number of extreme simulated proportions by the total number of simulations. ‖ ∻Answer∻ ⚹ The approximate p-value of the test is not provided in the question, but it would be calculated using the steps above. ⚹ ∻Key Concept∻ ⚹p-value in hypothesis testing⚹ ∻Explanation∻ ⚹The p-value represents the probability of obtaining a sample proportion as extreme as the observed one (or more extreme) under the assumption that the null hypothesis is true. It is calculated by comparing the observed sample proportion to the distribution of sample proportions from the simulation.⚹◊What is the formula for calculating the $p$-value in hypothesis testing when the sample proportion is known?⍭ Generate me a similar question◊

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