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."
