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. Should you accept the manufacturer's claim? 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. Should you accept the manufacturer's claim? Yes. No.
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 in Quality Control: Should the Manufacturer's Claim Be Accepted?**
**Scenario:**
A manufacturer claims that the average lifetime of its lightbulbs is equal to 36 months. To verify this claim, a random sample of 64 bulbs is tested. The sample data reveals a mean lifetime of 32 months with a standard deviation of 11 months.
**Objective:**
We'll utilize a z-test for the population mean to check the manufacturer's claim at a significance level of α = 0.05.
**Steps for Hypothesis Testing:**
1. **Formulate the Hypotheses:**
- Null Hypothesis (H0): The average lifetime of the lightbulbs is 36 months (H0: μ = 36).
- Alternative Hypothesis (H1): The average lifetime of the lightbulbs is not 36 months (H1: μ ≠ 36).
2. **Calculate the Test Statistic:**
The test statistic for a z-test is calculated using the formula:
\[ Z = \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}} \]
where:
- \(\overline{X}\) = sample mean = 32
- \(\mu\) = population mean (under H0) = 36
- \(\sigma\) = sample standard deviation = 11
- \(n\) = sample size = 64
3. **Determine the Rejection Region:**
For a two-tailed test at α = 0.05, the critical values are ±1.96.
4. **Decision Rule:**
If the calculated z-value falls outside the range of -1.96 to 1.96, reject the null hypothesis.
**Question:**
Should you accept the manufacturer's claim?
- [ ] Yes.
- [ ] No.
**Consider:**
Performing the above calculations and testing steps will lead to a conclusion regarding whether the manufacturer's claim about the average lifetime of its lightbulbs should be accepted or rejected.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8aa4fe98-75dd-42fc-98d3-625019e98cb1%2F2784fd42-d9a9-4dd7-8599-ca46c423f46b%2Fwxvorm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Hypothesis Testing in Quality Control: Should the Manufacturer's Claim Be Accepted?**
**Scenario:**
A manufacturer claims that the average lifetime of its lightbulbs is equal to 36 months. To verify this claim, a random sample of 64 bulbs is tested. The sample data reveals a mean lifetime of 32 months with a standard deviation of 11 months.
**Objective:**
We'll utilize a z-test for the population mean to check the manufacturer's claim at a significance level of α = 0.05.
**Steps for Hypothesis Testing:**
1. **Formulate the Hypotheses:**
- Null Hypothesis (H0): The average lifetime of the lightbulbs is 36 months (H0: μ = 36).
- Alternative Hypothesis (H1): The average lifetime of the lightbulbs is not 36 months (H1: μ ≠ 36).
2. **Calculate the Test Statistic:**
The test statistic for a z-test is calculated using the formula:
\[ Z = \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}} \]
where:
- \(\overline{X}\) = sample mean = 32
- \(\mu\) = population mean (under H0) = 36
- \(\sigma\) = sample standard deviation = 11
- \(n\) = sample size = 64
3. **Determine the Rejection Region:**
For a two-tailed test at α = 0.05, the critical values are ±1.96.
4. **Decision Rule:**
If the calculated z-value falls outside the range of -1.96 to 1.96, reject the null hypothesis.
**Question:**
Should you accept the manufacturer's claim?
- [ ] Yes.
- [ ] No.
**Consider:**
Performing the above calculations and testing steps will lead to a conclusion regarding whether the manufacturer's claim about the average lifetime of its lightbulbs should be accepted or rejected.
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