1,500 women followed the Atkin’s diet for a month. A random sample of 29 women gained
an average of 6.7 pounds. Test the hypothesis that the average weight gain per woman for
the month was over 5 pounds. The standard deviation for all women in the group was 7.1.

To test the hypothesis that the average weight gain per woman for the month was over 5 pounds, we can perform a one-sample \( t \)-test. Here's the step-by-step process:

### Step 1: State the Hypotheses
- **Null Hypothesis (\( H_0 \))**: The average weight gain per woman is 5 pounds or less.
  \[
  H_0: \mu \leq 5
  \]
- **Alternative Hypothesis (\( H_1 \))**: The average weight gain per woman is more than 5 pounds.
  \[
  H_1: \mu > 5
  \]

### Step 2: Identify the Test Statistic
Since the population standard deviation is known (or the sample size is large enough to approximate the population standard deviation), we can use the z-test instead of the t-test. The test statistic for a one-sample z-test is calculated as:
\[
z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
\]
Where:
- \( \bar{x} \) is the sample mean (6.7 pounds)
- \( \mu_0 \) is the hypothesized population mean (5 pounds)
- \( \sigma \) is the population standard deviation (7.1 pounds)
- \( n \) is the sample size (29)

### Step 3: Perform the Calculation
Substitute the given values into the formula:
\[
z = \frac{6.7 - 5}{\frac{7.1}{\sqrt{29}}}
\]
First, calculate the denominator:
\[
\frac{7.1}{\sqrt{29}} \approx \frac{7.1}{5.385} \approx 1.318
\]
Then calculate the z-value:
\[
z = \frac{1.7}{1.318} \approx 1.29
\]

### Step 4: Determine the Critical Value and Make a Decision
- **Significance Level (\( \alpha \))**: Typically, we use \( \alpha = 0.05 \) for hypothesis testing.
- **Critical Value**: For a one-tailed test at \( \alpha = 0.05 \), the critical z-value is approximately 1.645.

### Step 5: Conclusion
- Since the calculated z-value \( 1.29 \) is less than the critical value \( 1.645 \), we **fail to reject the null hypothesis**.
- This means that there is not enough evidence to support the claim that the average weight gain per woman was more than 5 pounds.

### Summary
Based on the analysis, we do not have sufficient evidence to conclude that the average weight gain per woman for the month was over 5 pounds.

---

Would you like more details on any part of this explanation? Here are some related questions you might find interesting:

1. How would the conclusion change if the sample size was larger?
2. What is the impact of increasing the significance level to 0.10?
3. How would the results differ if the sample mean was 7 pounds?
4. What is the difference between a z-test and a t-test?
5. How can we calculate the p-value for this test?

**Tip:** When performing hypothesis testing, always ensure the assumptions for the test you're using are met, such as normality or sample size considerations.

Explore hypothesis testing with a case study on the average weight gain of women following the Atkins diet. Analyze a sample of 1,500 women where a random sample of 29 gained an average of 6.7 pounds. Determine if there is sufficient evidence to support weight gain over 5 pounds using a one-sample z-test.

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