At a nearby college, there is a school-sponsored website that matches people looking for roommates. According to the school's reports, 38% of students will find a match their first time using the site. A writer for the school newspaper tests this claim by choosing a random sample of 165 students who visited the site looking for a roommate. Of the students surveyed, 48 said they found a match their first time using the site. Complete the parts below to perform a hypothesis test to see if there is enough evidence, at the 0.05 level of significance, to reject the claim that the proportion, p, of all students who will find a match their first time using the site is 38%. (a) State the null hypothesis Ho and the alternative hypothesis H, that you would use for the test.
At a nearby college, there is a school-sponsored website that matches people looking for roommates. According to the school's reports, 38% of students will find a match their first time using the site. A writer for the school newspaper tests this claim by choosing a random sample of 165 students who visited the site looking for a roommate. Of the students surveyed, 48 said they found a match their first time using the site. Complete the parts below to perform a hypothesis test to see if there is enough evidence, at the 0.05 level of significance, to reject the claim that the proportion, p, of all students who will find a match their first time using the site is 38%. (a) State the null hypothesis Ho and the alternative hypothesis H, that you would use for the test.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![# Introduction to Hypothesis Tests for a Population Proportion
At a nearby college, there is a school-sponsored website that matches people looking for roommates. According to the school's reports, 38% of students will find a match their first time using the site. A writer for the school newspaper tests this claim by choosing a random sample of 165 students who visited the site looking for a roommate. Of the students surveyed, 48 said they found a match their first time using the site.
Complete the parts below to perform a hypothesis test to see if there is enough evidence, at the 0.05 level of significance, to reject the claim that the proportion, \( p \), of all students who will find a match their first time using the site is 38%.
### (a) State the Null Hypothesis \( H_0 \) and the Alternative Hypothesis \( H_1 \) that you would use for the test.
- **Null Hypothesis \( H_0 \):** \( p = 0.38 \)
- **Alternative Hypothesis \( H_1 \):** \( p \neq 0.38 \)
### (b) For your hypothesis test, you will use a Z-test. Find the values of \( np \) and \( n(1-p) \) to confirm that a Z-test can be used.
(One standard is that \( np \geq 10 \) and \( n(1-p) \geq 10 \) under the assumption that the null hypothesis is true.)
- \( np = 165 \times 0.38 = 62.7 \)
- \( n(1-p) = 165 \times (1 - 0.38) = 102.3 \)
### (c) Perform a Z-test and find the \( p \)-value.
Here is some information to help you with your Z-test:
- The value of the test statistic is given by
\[
\hat{p} = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}
\]
- The \( p \)-value is two times the area under the curve to the left of the value of the test statistic.
### Diagram
There is a diagram showing the "Standard Normal Distribution." This diagram illustrates the normal curve with sections marked under the curve to represent areas related to the test statistic and corresponding \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53adddb0-ebb2-4e76-a872-01ab126d1a60%2F61f6afe8-755f-4958-9f9b-74c3c1acf6f2%2Fhxy56lk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Introduction to Hypothesis Tests for a Population Proportion
At a nearby college, there is a school-sponsored website that matches people looking for roommates. According to the school's reports, 38% of students will find a match their first time using the site. A writer for the school newspaper tests this claim by choosing a random sample of 165 students who visited the site looking for a roommate. Of the students surveyed, 48 said they found a match their first time using the site.
Complete the parts below to perform a hypothesis test to see if there is enough evidence, at the 0.05 level of significance, to reject the claim that the proportion, \( p \), of all students who will find a match their first time using the site is 38%.
### (a) State the Null Hypothesis \( H_0 \) and the Alternative Hypothesis \( H_1 \) that you would use for the test.
- **Null Hypothesis \( H_0 \):** \( p = 0.38 \)
- **Alternative Hypothesis \( H_1 \):** \( p \neq 0.38 \)
### (b) For your hypothesis test, you will use a Z-test. Find the values of \( np \) and \( n(1-p) \) to confirm that a Z-test can be used.
(One standard is that \( np \geq 10 \) and \( n(1-p) \geq 10 \) under the assumption that the null hypothesis is true.)
- \( np = 165 \times 0.38 = 62.7 \)
- \( n(1-p) = 165 \times (1 - 0.38) = 102.3 \)
### (c) Perform a Z-test and find the \( p \)-value.
Here is some information to help you with your Z-test:
- The value of the test statistic is given by
\[
\hat{p} = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}
\]
- The \( p \)-value is two times the area under the curve to the left of the value of the test statistic.
### Diagram
There is a diagram showing the "Standard Normal Distribution." This diagram illustrates the normal curve with sections marked under the curve to represent areas related to the test statistic and corresponding \(
![**Standard Normal Distribution**
**Step 1:** Select one-tailed or two-tailed.
- O One-tailed
- O Two-tailed
**Step 2:** Enter the test statistic.
*(Round to 3 decimal places.)*
[Text Box]
**Step 3:** Shade the area represented by the *p*-value.
[Button with Diamond Icon]
**Step 4:** Enter the *p*-value.
*(Round to 3 decimal places.)*
[Text Box]
---
**Graph Description:**
The graph is a standard normal distribution curve, symmetrical around the vertical axis at 0. It spans from approximately -3 to +3 on the horizontal axis, with the highest point at 0. The vertical axis represents probability density, with markings roughly at 0.1, 0.2, and 0.3. The curve demonstrates a bell shape typical of normal distributions.
---
**(d)** Based on your answer to part (c), choose what can be concluded, at the 0.05 level of significance, about the claim made in the school's reports.
- O Since the *p*-value is less than (or equal to) the level of significance, the null hypothesis is rejected. So, there is enough evidence to reject the claim that 38% of students will find a match their first time using the site.
- O Since the *p*-value is less than (or equal to) the level of significance, the null hypothesis is not rejected. So, there is not enough evidence to reject the claim that 38% of students will find a match their first time using the site.
- O Since the *p*-value is greater than the level of significance, the null hypothesis is rejected. So, there is enough evidence to reject the claim that 38% of students will find a match their first time using the site.
- O Since the *p*-value is greater than the level of significance, the null hypothesis is not rejected. So, there is not enough evidence to reject the claim that 38% of students will find a match their first time using the site.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53adddb0-ebb2-4e76-a872-01ab126d1a60%2F61f6afe8-755f-4958-9f9b-74c3c1acf6f2%2Fa5eyhx8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Standard Normal Distribution**
**Step 1:** Select one-tailed or two-tailed.
- O One-tailed
- O Two-tailed
**Step 2:** Enter the test statistic.
*(Round to 3 decimal places.)*
[Text Box]
**Step 3:** Shade the area represented by the *p*-value.
[Button with Diamond Icon]
**Step 4:** Enter the *p*-value.
*(Round to 3 decimal places.)*
[Text Box]
---
**Graph Description:**
The graph is a standard normal distribution curve, symmetrical around the vertical axis at 0. It spans from approximately -3 to +3 on the horizontal axis, with the highest point at 0. The vertical axis represents probability density, with markings roughly at 0.1, 0.2, and 0.3. The curve demonstrates a bell shape typical of normal distributions.
---
**(d)** Based on your answer to part (c), choose what can be concluded, at the 0.05 level of significance, about the claim made in the school's reports.
- O Since the *p*-value is less than (or equal to) the level of significance, the null hypothesis is rejected. So, there is enough evidence to reject the claim that 38% of students will find a match their first time using the site.
- O Since the *p*-value is less than (or equal to) the level of significance, the null hypothesis is not rejected. So, there is not enough evidence to reject the claim that 38% of students will find a match their first time using the site.
- O Since the *p*-value is greater than the level of significance, the null hypothesis is rejected. So, there is enough evidence to reject the claim that 38% of students will find a match their first time using the site.
- O Since the *p*-value is greater than the level of significance, the null hypothesis is not rejected. So, there is not enough evidence to reject the claim that 38% of students will find a match their first time using the site.
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