https://file.aiquickdraw.com/mathbot/file/1730001283703vwjxhjwc.jpg

To solve this hypothesis testing problem, let's go through the steps:

### Step 1: Define the Hypotheses
Based on the problem, the engineer wants to test if employees at start-up companies work more hours on average than the national average of 47 hours. This indicates a one-tailed test.

1. **Null Hypothesis (H₀):** The mean work week at start-ups is equal to or less than 47 hours.
   - \( H_0: \mu = 47 \)
2. **Alternative Hypothesis (H₁):** The mean work week at start-ups is more than 47 hours.
   - \( H_1: \mu > 47 \)

### Step 2: Calculate the Sample Mean and Standard Deviation
The data provided:
- **Hours:** 49, 48, 60, 50, 49, 68, 57, 60, 49, 46, 55, 50

Using this dataTo solve this hypothesis testing problem, let's go through the steps:

### Step 1: Define the Hypotheses
Based on the problem, the engineer wants to test if employees at start-up companies work more hours on average than the national average of 47 hours. This indicates a one-tailed test.

1. **Null Hypothesis (H₀):** The mean work week at start-ups is equal to or less than 47 hours.
   - \( H_0: \mu = 47 \)
2. **Alternative Hypothesis (H₁):** The mean work week at start-ups is more than 47 hours.
   - \( H_1: \mu > 47 \)

### Step 2: Calculate the Sample Mean and Standard Deviation
The data provided:
- **Hours:** 49, 48, 60, 50, 49, 68, 57, 60, 49, 46, 55, 50

Using this 
1. Calculate the **sample mean ( \( \bar{x} \) )**.
2. Calculate the **sample standard deviation ( \( s \) )**.

### Step 3: Compute the Test Statistic
Since the population standard deviation is not given, we'll use the **t-test** statistic for a one-sample test:
\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
\]
where:
- \( \bar{x} \) = sample mean,
- \( \mu = 47 \) (hypothesized population mean),
- \( s \) = sample standard deviation,
- \( n = 12 \) (sample size).

### Step 4: Find the p-value and Make a Decision
With a 5% significance level ( \( \alpha = 0.05 \) ), we’ll compare the computed t-statistic against the critical value from the t-distribution with \( n - 1 = 11 \) degrees of freedom.

If \( p \)-value \( < 0.05 \), we reject the null hypothesis.

I'll now perform these calculations.Here are the calculated values for the hypothesis test:

1. **Sample Mean ( \( \bar{x} \) )**: 53.4167 hours
2. **Sample Standard Deviation ( \( s \) )**: 6.6121 hours
3. **Test Statistic ( \( t \) )**: 3.3617
4. **p-value**: 0.0032

### Decision
Since the p-value (0.0032) is less than the significance level of 0.05, we reject the null hypothesis.

### Conclusion
The test supports the claim that the mean number of hours worked by employees at start-ups is greater than the US mean of 47 hours.

Would you like more details on any part of the process? 

### Related Questions:
1. What is the difference between a one-tailed and two-tailed test?
2. Why do we use the t-test instead of the z-test in this scenario?
3. How does the p-value help in decision-making?
4. What would be the implications of a Type I error in this context?
5. How would results change if we used a 1% significance level instead?

### Tip:
When performing hypothesis testing, always check if the test is one-tailed or two-tailed, as it affects the interpretation of the p-value.

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 5% level of significance.

Conduct a hypothesis test to determine if employees at start-up companies work more hours on average than the national mean of 47 hours, using a 5% level of significance. This problem includes sample mean calculation, t-test formula, and p-value interpretation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation