MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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![### Hypothesis Testing: Determining Test Statistics and Drawing Conclusions
---
#### Task 1: Determine the Test Statistic
Determine the test statistic for the hypothesis test. Round the solution to two decimal places.
*Input Field for Test Statistic:*
\[ \_\_\_\_ \]
---
#### Task 2: Determine the p-value
Determine the \( p \)-value for the hypothesis test. Round the solution to four decimal places.
*Input Field for \( p \)-value:*
\[ \_\_\_\_ \]
---
#### Task 3: Conclude the Hypothesis Test
Determine the appropriate conclusion for this hypothesis test from the options provided:
- **Option 1:**
- The sample data **do not** provide sufficient evidence to reject the null hypothesis that the mean lifespan of the 60-watt LED light bulbs produced by this company is 27,000 hours, and thus we conclude that the company's claim that the average lifespan of the 60-watt LED light bulbs is 27,000 hours is likely true.
- **Option 2:**
- The sample data **do not** provide sufficient evidence to reject the alternative hypothesis that the mean lifespan of the 60-watt LED light bulbs produced by this company is less than 27,000 hours, and thus we conclude that the company's claim that the light bulbs last an average of 27,000 hours is likely false.
- **Option 3:**
- The sample data **provide** sufficient evidence to reject the null hypothesis that the mean lifespan of the 60-watt LED light bulbs produced by this company is 27,000 hours, and thus we conclude that the company's claim that the average lifespan of the 60-watt LED light bulbs is 27,000 hours is likely false.
- **Option 4:**
- The sample data **provide** sufficient evidence to reject the alternative hypothesis that the mean lifespan of the 60-watt LED light bulbs produced by this company is less than 27,000 hours, and thus we conclude that the company's claim that the light bulbs last an average of 27,000 hours is likely true.](https://content.bartleby.com/qna-images/question/af14cf2c-4c6b-44b0-8c19-b82da3615fd0/cecfbcf8-15d7-4b50-8ac4-0e8f648d14c3/8vs1d7t_processed.jpeg)
Transcribed Image Text:### Hypothesis Testing: Determining Test Statistics and Drawing Conclusions
---
#### Task 1: Determine the Test Statistic
Determine the test statistic for the hypothesis test. Round the solution to two decimal places.
*Input Field for Test Statistic:*
\[ \_\_\_\_ \]
---
#### Task 2: Determine the p-value
Determine the \( p \)-value for the hypothesis test. Round the solution to four decimal places.
*Input Field for \( p \)-value:*
\[ \_\_\_\_ \]
---
#### Task 3: Conclude the Hypothesis Test
Determine the appropriate conclusion for this hypothesis test from the options provided:
- **Option 1:**
- The sample data **do not** provide sufficient evidence to reject the null hypothesis that the mean lifespan of the 60-watt LED light bulbs produced by this company is 27,000 hours, and thus we conclude that the company's claim that the average lifespan of the 60-watt LED light bulbs is 27,000 hours is likely true.
- **Option 2:**
- The sample data **do not** provide sufficient evidence to reject the alternative hypothesis that the mean lifespan of the 60-watt LED light bulbs produced by this company is less than 27,000 hours, and thus we conclude that the company's claim that the light bulbs last an average of 27,000 hours is likely false.
- **Option 3:**
- The sample data **provide** sufficient evidence to reject the null hypothesis that the mean lifespan of the 60-watt LED light bulbs produced by this company is 27,000 hours, and thus we conclude that the company's claim that the average lifespan of the 60-watt LED light bulbs is 27,000 hours is likely false.
- **Option 4:**
- The sample data **provide** sufficient evidence to reject the alternative hypothesis that the mean lifespan of the 60-watt LED light bulbs produced by this company is less than 27,000 hours, and thus we conclude that the company's claim that the light bulbs last an average of 27,000 hours is likely true.
![### Hypothesis Testing for the Mean Lifespan of 60-Watt LED Light Bulbs
An LED light bulb manufacturer claims that the average lifespan of its 60-watt LED light bulbs is 27,000 hours.
A corporate watchdog group is suspicious of the company's claim and thinks that the true average lifespan of the 60-watt LED light bulbs produced by the company may be less than the advertised 27,000 hours. The group collected and tested a random sample of 211 light bulbs and found the average lifespan of the sample was 26,941 hours.
Using the p-value method, we will test the hypothesis that the mean lifespan of this brand of 60-watt LED light bulb is less than 27,000 hours, with a significance level (α) of 0.1. Assume the standard deviation of the lifespan of all such light bulbs is known to be 370 hours.
#### Step-by-Step Process:
1. **State the null and alternative hypothesis for this test.**
- Null Hypothesis \(( H_0 )\): \( \mu = 27,000 \)
- Alternative Hypothesis \(( H_1 )\): \( \mu < 27,000 \)
2. **Determine if this test is left-tailed, right-tailed, or two-tailed.**
- Since the watchdog group suspects the mean lifespan is less than the advertised 27,000 hours, this is a left-tailed test.
\[
\begin{align*}
&\quad \quad \circ\text{ right-tailed} \\
&\quad \quad \circ\text{ two-tailed} \\
&\quad \quad \bullet\text{ left-tailed}
\end{align*}
\]
3. **Should the standard normal \(( z )\) distribution or Student's \(( t )\) distribution be used for this test?**
- Given that the standard deviation of the population is known, the standard normal \(( z )\) distribution should be used.
\[
\begin{align*}
&\quad \quad \circ\text{ The Student’s } t\text{ distribution should be used} \\
&\quad \quad \bullet\text{ The standard normal } ( z )\text{ distribution should be used}
\end{align*}](https://content.bartleby.com/qna-images/question/af14cf2c-4c6b-44b0-8c19-b82da3615fd0/cecfbcf8-15d7-4b50-8ac4-0e8f648d14c3/ki32fek_processed.jpeg)
Transcribed Image Text:### Hypothesis Testing for the Mean Lifespan of 60-Watt LED Light Bulbs
An LED light bulb manufacturer claims that the average lifespan of its 60-watt LED light bulbs is 27,000 hours.
A corporate watchdog group is suspicious of the company's claim and thinks that the true average lifespan of the 60-watt LED light bulbs produced by the company may be less than the advertised 27,000 hours. The group collected and tested a random sample of 211 light bulbs and found the average lifespan of the sample was 26,941 hours.
Using the p-value method, we will test the hypothesis that the mean lifespan of this brand of 60-watt LED light bulb is less than 27,000 hours, with a significance level (α) of 0.1. Assume the standard deviation of the lifespan of all such light bulbs is known to be 370 hours.
#### Step-by-Step Process:
1. **State the null and alternative hypothesis for this test.**
- Null Hypothesis \(( H_0 )\): \( \mu = 27,000 \)
- Alternative Hypothesis \(( H_1 )\): \( \mu < 27,000 \)
2. **Determine if this test is left-tailed, right-tailed, or two-tailed.**
- Since the watchdog group suspects the mean lifespan is less than the advertised 27,000 hours, this is a left-tailed test.
\[
\begin{align*}
&\quad \quad \circ\text{ right-tailed} \\
&\quad \quad \circ\text{ two-tailed} \\
&\quad \quad \bullet\text{ left-tailed}
\end{align*}
\]
3. **Should the standard normal \(( z )\) distribution or Student's \(( t )\) distribution be used for this test?**
- Given that the standard deviation of the population is known, the standard normal \(( z )\) distribution should be used.
\[
\begin{align*}
&\quad \quad \circ\text{ The Student’s } t\text{ distribution should be used} \\
&\quad \quad \bullet\text{ The standard normal } ( z )\text{ distribution should be used}
\end{align*}
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