To test the claim that the mean weight loss is greater than 0 lb, we can perform a one-sample z-test. The following information is provided:
- Sample mean (\( \bar{x} \)) = 3.0 lb
- Sample standard deviation (\( s \)) = 4.9 lb
- Sample size (\( n \)) = 40
- Hypothesized mean (\( \mu_0 \)) = 0 lb
- Significance level = 0.01
Step 1: Calculate the z-statistic
The formula for the z-statistic is:
\[ z = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]
Substituting the values:
\[ z = \frac{3.0 - 0}{4.9 / \sqrt{40}} = \frac{3.0}{4.9 / 6.3246} \approx \frac{3.0}{0.7746} \approx 3.87 \]
Step 2: Find the P-value
Now we need to find the P-value associated with the z-statistic of 3.87. Since this is a one-tailed test (we are testing if the mean is greater than 0), we will look at the area to the right of our z-value.
Referring to z-tables or using normal distribution calculators, a z-value of 3.87 corresponds to a very small P-value, generally less than 0.001.
Step 3: Choose the correct range for the P-value
Given that our calculated z-value leads to a P-value less than 0.005, we can define the following ranges:
- A) P-value < 0.01
- B) 0.01 < P-value < 0.02
- C) P-value < 0.005
- D) 0.005 < P-value < 0.01
Based on our calculations:
The correct answer is C) P-value < 0.005.