2. At a certain college, it is estimated that at most 25% of the students ride bicycles to class. Does this seem to be a valid estimate at a = 0.05 if, in a random sample of 90 students, 28 are found to ride bicycles to class? Please show all steps of the classical approach clearly and interpret your conclusion.
2. At a certain college, it is estimated that at most 25% of the students ride bicycles to class. Does this seem to be a valid estimate at a = 0.05 if, in a random sample of 90 students, 28 are found to ride bicycles to class? Please show all steps of the classical approach clearly and interpret your conclusion.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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![### Hypothesis Testing Problem
**Question 2:**
At a certain college, it is estimated that at most 25% of the students ride bicycles to class. Does this seem to be a valid estimate at \( \alpha = 0.05 \) if, in a random sample of 90 students, 28 are found to ride bicycles to class? Please show all steps of the classical approach clearly and interpret your conclusion.
---
#### Step-by-Step Solution Using the Classical Approach
**Step 1: State the hypotheses.**
- Null Hypothesis (\(H_0\)): \( p \leq 0.25 \) (The proportion of students who ride bicycles to class is at most 25%)
- Alternative Hypothesis (\(H_1\)): \( p > 0.25 \) (The proportion of students who ride bicycles to class is greater than 25%)
**Step 2: Determine the significance level.**
- Significance level (\( \alpha \)): 0.05
**Step 3: Identify the appropriate test statistic.**
For proportions, the test statistic is \( z \) and is given by:
\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \]
where:
- \( \hat{p} \) is the sample proportion
- \( p_0 \) is the hypothesized population proportion
- \( n \) is the sample size
**Step 4: Calculate the test statistic.**
From the given data:
- Sample size (\( n \)): 90
- Number of students who ride bicycles (\( x \)): 28
- Sample proportion (\( \hat{p} \)): \( \frac{28}{90} = 0.3111 \)
- Hypothesized proportion (\( p_0 \)): 0.25
Substituting these into the formula:
\[ z = \frac{0.3111 - 0.25}{\sqrt{\frac{0.25 \cdot (1 - 0.25)}{90}}} \]
\[ z = \frac{0.0611}{\sqrt{\frac{0.25 \cdot 0.75}{90}}} \]
\[ z = \frac{0.0611}{\sqrt{0.1875 /](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F322ab9b5-8b73-42ca-a79a-5c7ee3611144%2F6d5a2ed4-76c0-4b70-964d-2aedaff1a447%2Fwz5sh1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Hypothesis Testing Problem
**Question 2:**
At a certain college, it is estimated that at most 25% of the students ride bicycles to class. Does this seem to be a valid estimate at \( \alpha = 0.05 \) if, in a random sample of 90 students, 28 are found to ride bicycles to class? Please show all steps of the classical approach clearly and interpret your conclusion.
---
#### Step-by-Step Solution Using the Classical Approach
**Step 1: State the hypotheses.**
- Null Hypothesis (\(H_0\)): \( p \leq 0.25 \) (The proportion of students who ride bicycles to class is at most 25%)
- Alternative Hypothesis (\(H_1\)): \( p > 0.25 \) (The proportion of students who ride bicycles to class is greater than 25%)
**Step 2: Determine the significance level.**
- Significance level (\( \alpha \)): 0.05
**Step 3: Identify the appropriate test statistic.**
For proportions, the test statistic is \( z \) and is given by:
\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \]
where:
- \( \hat{p} \) is the sample proportion
- \( p_0 \) is the hypothesized population proportion
- \( n \) is the sample size
**Step 4: Calculate the test statistic.**
From the given data:
- Sample size (\( n \)): 90
- Number of students who ride bicycles (\( x \)): 28
- Sample proportion (\( \hat{p} \)): \( \frac{28}{90} = 0.3111 \)
- Hypothesized proportion (\( p_0 \)): 0.25
Substituting these into the formula:
\[ z = \frac{0.3111 - 0.25}{\sqrt{\frac{0.25 \cdot (1 - 0.25)}{90}}} \]
\[ z = \frac{0.0611}{\sqrt{\frac{0.25 \cdot 0.75}{90}}} \]
\[ z = \frac{0.0611}{\sqrt{0.1875 /
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