A manufacturer claims that the average lifetime of its lightbulbs is equal to 36 months. A random sample of 64 bulbs has a mean lifetime of 32 months, and the sample standard deviation is 11 months. We will be using a z-test for the population mean at x = 0.05 to check the manufacturer's claim. Do we reject the null hypothesis? Yes. No.
A manufacturer claims that the average lifetime of its lightbulbs is equal to 36 months. A random sample of 64 bulbs has a mean lifetime of 32 months, and the sample standard deviation is 11 months. We will be using a z-test for the population mean at x = 0.05 to check the manufacturer's claim. Do we reject the null hypothesis? Yes. No.
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.4: Distributions Of Data
Problem 7PPS
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![### Statistical Hypothesis Testing - Example Problem
A manufacturer claims that the average lifetime of its lightbulbs is equal to 36 months. To test this claim, we will conduct a hypothesis test using the following sample data:
- A random sample of 64 bulbs has a mean lifetime of 32 months.
- The sample standard deviation is 11 months.
- We will use a z-test for the population mean with a significance level of \(\alpha = 0.05\).
**Hypothesis Testing Questions:**
1. Do we reject the null hypothesis?
**Answer Choices:**
- Yes.
- No.
### Detailed Explanation
**Step-by-Step Solution:**
1. **Specify the null and alternative hypotheses:**
- Null hypothesis (\(H_0\)): \(\mu = 36\) months
- Alternative hypothesis (\(H_1\)): \(\mu \neq 36\) months
2. **Calculate the standard error of the mean (SEM):**
\[
SEM = \frac{\sigma}{\sqrt{n}} = \frac{11}{\sqrt{64}} = \frac{11}{8} = 1.375
\]
3. **Compute the z-score:**
\[
z = \frac{\bar{x} - \mu}{SEM} = \frac{32 - 36}{1.375} = \frac{-4}{1.375} \approx -2.91
\]
4. **Determine the critical z-value for \(\alpha = 0.05\):**
For a two-tailed test at \(\alpha = 0.05\),
\[
\text{Critical z-value} = \pm 1.96
\]
5. **Make a decision:**
Compare the computed z-score to the critical z-values.
- If the computed z-score is less than -1.96 or greater than 1.96, we reject the null hypothesis.
- In this case, \( z \approx -2.91 \), which is less than -1.96.
**Conclusion:**
Given that the computed z-score is outside the range of \(-1.96\) to \(1.96\), we **reject the null hypothesis**.
**Correct Answer:**
- Yes.
This example shows how hypothesis testing can be used to verify claims about population](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8aa4fe98-75dd-42fc-98d3-625019e98cb1%2F50428609-9762-40e8-8a19-5d627a6f9c5e%2Fo4s1a9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Statistical Hypothesis Testing - Example Problem
A manufacturer claims that the average lifetime of its lightbulbs is equal to 36 months. To test this claim, we will conduct a hypothesis test using the following sample data:
- A random sample of 64 bulbs has a mean lifetime of 32 months.
- The sample standard deviation is 11 months.
- We will use a z-test for the population mean with a significance level of \(\alpha = 0.05\).
**Hypothesis Testing Questions:**
1. Do we reject the null hypothesis?
**Answer Choices:**
- Yes.
- No.
### Detailed Explanation
**Step-by-Step Solution:**
1. **Specify the null and alternative hypotheses:**
- Null hypothesis (\(H_0\)): \(\mu = 36\) months
- Alternative hypothesis (\(H_1\)): \(\mu \neq 36\) months
2. **Calculate the standard error of the mean (SEM):**
\[
SEM = \frac{\sigma}{\sqrt{n}} = \frac{11}{\sqrt{64}} = \frac{11}{8} = 1.375
\]
3. **Compute the z-score:**
\[
z = \frac{\bar{x} - \mu}{SEM} = \frac{32 - 36}{1.375} = \frac{-4}{1.375} \approx -2.91
\]
4. **Determine the critical z-value for \(\alpha = 0.05\):**
For a two-tailed test at \(\alpha = 0.05\),
\[
\text{Critical z-value} = \pm 1.96
\]
5. **Make a decision:**
Compare the computed z-score to the critical z-values.
- If the computed z-score is less than -1.96 or greater than 1.96, we reject the null hypothesis.
- In this case, \( z \approx -2.91 \), which is less than -1.96.
**Conclusion:**
Given that the computed z-score is outside the range of \(-1.96\) to \(1.96\), we **reject the null hypothesis**.
**Correct Answer:**
- Yes.
This example shows how hypothesis testing can be used to verify claims about population
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