https://file.aiquickdraw.com/mathbot/file/1727664920344d0nns3fy.jpg

It looks like you've uploaded an image showing a problem related to hypothesis testing, specifically a Z-test for a population proportion.

Here's a breakdown of the problem:

- **Claim**: 35% of students will find a match the first time using a school-sponsored roommate-matching website.
- **Sample size (n)**: 170 students.
- **Sample result (x)**: 51 students found a match on their first time.
- **Hypothesis test**: To test if the actual proportion is different from the claimed 35%.

You are asked to compute the **Z-test statistic** for this hypothesis test.

### Formula for Z-test statistic:
\[
Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}
\]
Where:
- \(\hat{p} = \frac{x}{n}\) is the sample proportion.
- \(p_0 = 0.35\) is the hypothesized population proportion.
- \(n = 170\) is the sample size.
- \(x = 51\) is the number of successes (students who found a match).

Let's calculate this step by step.

1. Compute the sample proportion:
\[
\hat{p} = \frac{51}{170}
\]

2. Use the hypothesized proportion \(p_0 = 0.35\) and the sample size \(n = 170\) to compute the standard error:
\[
\text{SE} = \sqrt{\frac{0.35 \times (1 - 0.35)}{170}}
\]

3. Calculate the Z-statistic.

I'll do the math now.The calculated Z-test statistic is approximately \(-1.367\).

This value suggests how many standard deviations the sample proportion is away from the hypothesized proportion of 35%.

Would you like further details on interpreting this result or have any additional questions?

Here are 5 related questions you could explore:
1. How do we interpret the Z-test statistic in terms of rejecting or failing to reject the null hypothesis?
2. What is the significance level typically used in hypothesis testing, and how does it affect the conclusion?
3. How would the result change if the sample size were larger or smaller?
4. Can this Z-test be applied to small samples, and if not, what test would be more appropriate?
5. How does the choice of a one-tailed vs. two-tailed test affect the result?

**Tip**: Remember to compare the Z-statistic with the critical value from the Z-table to make a decision about the null hypothesis.

At your college, there is a school-sponsored website that matches people looking for roommates. The school claims that 35% of students will find a match their first time using the site. You are a writer for the school newspaper and are suspicious of the claim. To test it, you decide to perform a hypothesis test. To do so, you choose a random sample of 170 students who visited the site looking for a roommate. Of the students surveyed, 51 said they found a match their first time using the site. Find the value of the test statistic for your Z-test. Round your answer to three or more decimal places.

Learn how to perform a Z-test for a population proportion using the roommate matching example. This problem includes calculating the Z-test statistic for a sample of 170 students, with 51 successes, to test a school claim that 35% find a match. Step-by-step solution with detailed explanation for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation