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To solve this hypothesis testing problem, let’s go through each part step-by-step.

### 1. Setting up the Hypotheses
We have the following information:
- The population mean (\(\mu\)) of work hours for the U.S. is 47 hours.
- The sample data consists of hours worked by employees at start-up companies.

We want to test if employees at start-ups work more hours than the U.S. average, so our hypotheses are:

- **Null Hypothesis (H₀):** \(\mu = 47\)
- **Alternative Hypothesis (H₁):** \(\mu > 47\)

This will be a **one-tailed test** since we are only interested in whether the mean is greater than 47 hours.

### 2. Calculating the Sample Mean and Standard Deviation
Let’s calculate the mean and standard deviation of the sample. The data provided is:

\[
45, 48, 56, 51, 45, 54, 55, 58, 49, 49, 49, 54
\]

**Sample Mean (x̄)** and **Sample Standard Deviation (s)** can be calculated using these values.

- **Mean (x̄):**
  \[
  x̄ = \frac{\sum X}{n} = \frac{45 + 48 + 56 + 51 + 45 + 54 + 55 + 58 + 49 + 49 + 49 + 54}{12}
  \]

- **Sample Standard Deviation (s):**
  \[
  s = \sqrt{\frac{\sum (X - x̄)^2}{n - 1}}
  \]

I will calculate these values next.

### 3. Test Statistic
We use the **t-test** because the population standard deviation is unknown and the sample size is relatively small (\(n = 12\)).

The t-statistic formula is:
\[
t = \frac{x̄ - \mu}{\frac{s}{\sqrt{n}}}
\]
where:
- \(x̄\) = sample mean
- \(\mu\) = population mean (47 hours)
- \(s\) = sample standard deviation
- \(n\) = sample size (12)

### 4. Finding the p-value
With a 1% significance level (\(\alpha = 0.01\)), we will compare the calculated t-value to the critical value from the t-distribution table with \(n - 1 = 11\) degrees of freedom.

If the p-value is less than 0.01, we reject the null hypothesis. Otherwise, we fail to reject it.

Let’s calculate each of these values.

The work week for adults in the US that work full time is normally distributed with a mean of 47 hours. A newly hired engineer at a start-up company believes that employees at start-up companies work more on average than most working adults in the US. She asks 12 engineering friends at start-ups for the lengths in hours of their work week. Their responses are shown in the table below. Test the claim using a 1% level of significance. Give answer to at least 4 decimal places.

Perform hypothesis testing on work hours to determine if start-up employees work more than the average 47 hours per week for US workers. Using a one-sample t-test with a 1% significance level, this problem involves calculating sample mean, standard deviation, and t-statistic.

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