Solve your doubts and expand your knowledge with IDNLearn.com's extensive Q&A database. Our experts provide timely, comprehensive responses to ensure you have the information you need.
Sagot :
Let's go through the solution step-by-step to test the claim using a [tex]\(1\% \)[/tex] level of significance.
1. Formulating the Hypotheses:
The null hypothesis ([tex]\(H_0\)[/tex]) and the alternative hypothesis ([tex]\(H_a\)[/tex]) are:
[tex]\[ H_0: \mu \leq 47 \text{ hours} \][/tex]
[tex]\[ H_a: \mu > 47 \text{ hours} \][/tex]
Here, [tex]\( \mu \)[/tex] represents the mean number of hours employees at start-up companies work per week.
2. Given Data:
[tex]\[ \text{Sample Data (in hours): } 50, 40, 50, 53, 47, 68, 45, 59, 45, 48, 52, 55 \][/tex]
[tex]\[ \text{Sample Size (n): } 12 \][/tex]
[tex]\[ \text{Population Mean (} \mu_0 \text{): } 47 \text{ hours} \][/tex]
[tex]\[ \text{Level of Significance (} \alpha \text{): } 0.01 \][/tex]
3. Calculating Sample Mean:
[tex]\[ \text{Sample Mean (} \bar{x} \text{): } 51.0 \text{ hours} \][/tex]
4. Calculating Sample Standard Deviation:
[tex]\[ \text{Sample Standard Deviation (s): } 7.3485 \][/tex]
5. Calculating the Test Statistic:
The test statistic for a one-sample t-test is calculated by:
[tex]\[ t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} \][/tex]
Plugging in the values:
[tex]\[ t = \frac{51.0 - 47}{\frac{7.3485}{\sqrt{12}}} = 1.8856 \][/tex]
6. Degrees of Freedom:
[tex]\[ \text{Degrees of Freedom (df): } n - 1 = 12 - 1 = 11 \][/tex]
7. Finding the p-value:
Using the test statistic and the degrees of freedom, we can find the p-value associated with a one-tailed test.
[tex]\[ p\text{-value} = 0.0430 \][/tex]
8. Decision Making:
Since the p-value [tex]\( (0.0430) \)[/tex] is greater than the level of significance [tex]\( (0.01) \)[/tex]:
- We fail to reject the null hypothesis [tex]\( H_0 \)[/tex].
The correct decision is to "Fail to reject the null hypothesis."
9. Summary:
Based on the results, "There is insufficient evidence to conclude that the mean number of hours of all employees at start-up companies work more than the US mean of 47 hours."
### Summary of the Solution:
- Hypotheses:
[tex]\( H_0: \mu = 47 \text{ hours} \)[/tex]
[tex]\( H_a: \mu > 47 \text{ hours} \)[/tex]
- Test Statistic: [tex]\( t = 1.8856 \)[/tex]
- p-value: [tex]\( 0.0430 \)[/tex]
- Decision: "Fail to reject the null hypothesis."
- Correct Summary: "There is insufficient evidence to conclude that the mean number of hours of all employees at start-up companies work more than the US mean of 47 hours."
1. Formulating the Hypotheses:
The null hypothesis ([tex]\(H_0\)[/tex]) and the alternative hypothesis ([tex]\(H_a\)[/tex]) are:
[tex]\[ H_0: \mu \leq 47 \text{ hours} \][/tex]
[tex]\[ H_a: \mu > 47 \text{ hours} \][/tex]
Here, [tex]\( \mu \)[/tex] represents the mean number of hours employees at start-up companies work per week.
2. Given Data:
[tex]\[ \text{Sample Data (in hours): } 50, 40, 50, 53, 47, 68, 45, 59, 45, 48, 52, 55 \][/tex]
[tex]\[ \text{Sample Size (n): } 12 \][/tex]
[tex]\[ \text{Population Mean (} \mu_0 \text{): } 47 \text{ hours} \][/tex]
[tex]\[ \text{Level of Significance (} \alpha \text{): } 0.01 \][/tex]
3. Calculating Sample Mean:
[tex]\[ \text{Sample Mean (} \bar{x} \text{): } 51.0 \text{ hours} \][/tex]
4. Calculating Sample Standard Deviation:
[tex]\[ \text{Sample Standard Deviation (s): } 7.3485 \][/tex]
5. Calculating the Test Statistic:
The test statistic for a one-sample t-test is calculated by:
[tex]\[ t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} \][/tex]
Plugging in the values:
[tex]\[ t = \frac{51.0 - 47}{\frac{7.3485}{\sqrt{12}}} = 1.8856 \][/tex]
6. Degrees of Freedom:
[tex]\[ \text{Degrees of Freedom (df): } n - 1 = 12 - 1 = 11 \][/tex]
7. Finding the p-value:
Using the test statistic and the degrees of freedom, we can find the p-value associated with a one-tailed test.
[tex]\[ p\text{-value} = 0.0430 \][/tex]
8. Decision Making:
Since the p-value [tex]\( (0.0430) \)[/tex] is greater than the level of significance [tex]\( (0.01) \)[/tex]:
- We fail to reject the null hypothesis [tex]\( H_0 \)[/tex].
The correct decision is to "Fail to reject the null hypothesis."
9. Summary:
Based on the results, "There is insufficient evidence to conclude that the mean number of hours of all employees at start-up companies work more than the US mean of 47 hours."
### Summary of the Solution:
- Hypotheses:
[tex]\( H_0: \mu = 47 \text{ hours} \)[/tex]
[tex]\( H_a: \mu > 47 \text{ hours} \)[/tex]
- Test Statistic: [tex]\( t = 1.8856 \)[/tex]
- p-value: [tex]\( 0.0430 \)[/tex]
- Decision: "Fail to reject the null hypothesis."
- Correct Summary: "There is insufficient evidence to conclude that the mean number of hours of all employees at start-up companies work more than the US mean of 47 hours."
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.
