Using the site. A writer for the school newspaper tests this claim by choosing a random sample of 170 students who visited the site looking for a roommate. Of the students surveyed, 56 said they found a match their first time using the site. Foommates. According to the school's reports, 38% of students will Complete the parts below to perform a hypothesis test to see if there is enough evidence, at the 0.10 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. Ho: D 0 Р ô OO H₁: 0 E 020 0=0 *O X 5 ? (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≥10 and n (1-p)≥10 under the assumption that the null hypothesis is true.) Heren is the sample size and p is the population proportion you are testing. np=0 n (1-p)= (c) Perform a Z-test and find the p-value. Here is some information to help you with your Z-test.
Using the site. A writer for the school newspaper tests this claim by choosing a random sample of 170 students who visited the site looking for a roommate. Of the students surveyed, 56 said they found a match their first time using the site. Foommates. According to the school's reports, 38% of students will Complete the parts below to perform a hypothesis test to see if there is enough evidence, at the 0.10 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. Ho: D 0 Р ô OO H₁: 0 E 020 0=0 *O X 5 ? (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≥10 and n (1-p)≥10 under the assumption that the null hypothesis is true.) Heren is the sample size and p is the population proportion you are testing. np=0 n (1-p)= (c) Perform a Z-test and find the p-value. Here is some information to help you with your Z-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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![**Hypothesis Testing for Proportions**
When conducting research, one often needs to perform hypothesis testing to draw conclusions about data. This exercise will guide you through the steps of conducting a hypothesis test to check whether there is enough evidence to reject the claim about a proportion of students finding a match using a roommate site.
**Step-by-Step Guide:**
1. **Setting Hypotheses:**
- **Classify Null Hypothesis \( H_0 \)** and **Alternative Hypothesis \( H_1 \)**
\[
H_0: \quad \text{p = 0.38}
\]
\[
H_1: \quad \text{p > 0.38}
\]
2. **Confirming Z-test Conditions:**
- Calculate \( np \) and \( n(1 - p) \) under the assumption that the null hypothesis is true. Here, \( n \) is the sample size and \( p \) is the population proportion you are testing.
\[
np = 170 \times 0.38 = 64.6
\]
\[
n(1 - p) = 170 \times (1 - 0.38) = 105.4
\]
Both \( np \) and \( n(1 - p) \) must be greater than or equal to 10 for the Z-test to be valid.
3. **Performing the Z-test:**
- Calculate the test statistic which is given by:
\[
Z = \frac{\hat{p} - p}{\sqrt{\frac{p(1 - p)}{n}}}
\]
Where \(\hat{p}\) is the sample proportion.
4. **Finding the p-value:**
- The p-value represents the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct.
- Typically, the p-value is derived from a standard normal distribution graph and is two times the area under the curve to the left of the Z value.
**Graphs and Diagrams Explanation:**
- There is a visual representation to help select one-tailed or two-tailed tests, and decide whether the critical area lies in one or two tails of the normal distribution.
- The bell-shaped curve shown represents a standard normal distribution S](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd1b6d45-a7eb-422c-8cfa-b963b51ff783%2F2ec93499-fda6-481e-9901-44b59d06826f%2Frsalcm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Hypothesis Testing for Proportions**
When conducting research, one often needs to perform hypothesis testing to draw conclusions about data. This exercise will guide you through the steps of conducting a hypothesis test to check whether there is enough evidence to reject the claim about a proportion of students finding a match using a roommate site.
**Step-by-Step Guide:**
1. **Setting Hypotheses:**
- **Classify Null Hypothesis \( H_0 \)** and **Alternative Hypothesis \( H_1 \)**
\[
H_0: \quad \text{p = 0.38}
\]
\[
H_1: \quad \text{p > 0.38}
\]
2. **Confirming Z-test Conditions:**
- Calculate \( np \) and \( n(1 - p) \) under the assumption that the null hypothesis is true. Here, \( n \) is the sample size and \( p \) is the population proportion you are testing.
\[
np = 170 \times 0.38 = 64.6
\]
\[
n(1 - p) = 170 \times (1 - 0.38) = 105.4
\]
Both \( np \) and \( n(1 - p) \) must be greater than or equal to 10 for the Z-test to be valid.
3. **Performing the Z-test:**
- Calculate the test statistic which is given by:
\[
Z = \frac{\hat{p} - p}{\sqrt{\frac{p(1 - p)}{n}}}
\]
Where \(\hat{p}\) is the sample proportion.
4. **Finding the p-value:**
- The p-value represents the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct.
- Typically, the p-value is derived from a standard normal distribution graph and is two times the area under the curve to the left of the Z value.
**Graphs and Diagrams Explanation:**
- There is a visual representation to help select one-tailed or two-tailed tests, and decide whether the critical area lies in one or two tails of the normal distribution.
- The bell-shaped curve shown represents a standard normal distribution S
Transcribed Image Text:### Understanding Standard Normal Distribution for Hypothesis Testing
#### Step-by-Step Guide:
**Step 1: Select one-tailed or two-tailed.**
- Options:
- One-tailed
- Two-tailed
**Step 2: Enter the test statistic. (Round to 3 decimal places.)**
**Step 3: Shade the area represented by the *p*-value.**
- This involves indicating the region under the normal curve that corresponds to the *p*-value.
**Step 4: Enter the *p*-value. (Round to 3 decimal places.)**
#### Example
A chart illustrating a Standard Normal Distribution is provided. The curve peaks at the mean (0) and has symmetrical tails extending to -3 and 3. The height of the curve represents the probability density.
#### Interpretation Based on the *p*-value:
**(d) Based on your answer to part (c), choose what can be concluded, at the 0.10 level of significance, about the claim made in the school’s reports:**
1. **Option A:**
- 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.
2. **Option B:**
- 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.
3. **Option C:**
- 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.
4. **Option D:**
- 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.
**Remember:**
- If the *p*-value ≤ significance level (0.10 in this case), reject the null hypothesis.
- If the *p*-value > significance level (0.10 in this case), do
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