Question

The health of the bear population in Yellowstone National Park is monitored by periodic measurements taken from anesthetized bears. A sample of 54 bears has a mean weight of 182.9 lb. Assuming that σ is known to be 121.8 lb, use a 0.05 significance level to test the claim that the population mean of all such bears is greater than 150 lb. Find the P-value and state your conclusion about the null hypothesis.

A) 0.0239; fail to reject the null hypothesis
B) 0.9761; reject the null hypothesis
C) 0.9761; fail to reject the null hypothesis
D) 0.0239; reject the null hypothesis

Answer 1

To test the hypothesis that the population mean weight of bears in Yellowstone National Park is greater than 150 lb, we can set up the following hypotheses:

- Null hypothesis (\(H_0\)): \(\mu \leq 150\) lb
- Alternative hypothesis (\(H_a\): \(\mu > 150\) lb

We are given the following data:
- Sample size (\(n\)) = 54
- Sample mean (\(\bar{x}\)) = 182.9 lb
- Population standard deviation (\(\sigma\)) = 121.8 lb
- Significance level (\(\alpha\)) = 0.05

### Step 1: Calculate the test statistic
We use the Z-test for this scenario, given that we know the population standard deviation. The test statistic \(Z\) can be calculated using the formula:

\[
Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}
\]

Where:
- \(\bar{x} = 182.9\)
- \(\mu_0 = 150\)
- \(\sigma = 121.8\)
- \(n = 54\)

Substituting the values into the formula:

\[
Z = \frac{182.9 - 150}{121.8 / \sqrt{54}}
\]

Calculating the denominator:

\[
\sigma / \sqrt{n} = \frac{121.8}{\sqrt{54}} \approx \frac{121.8}{7.348} \approx 16.56
\]

Now calculate \(Z\):

\[
Z = \frac{182.9 - 150}{16.56} \approx \frac{32.9}{16.56} \approx 1.988
\]

### Step 2: Find the P-value
To find the P-value for our test statistic, we look up the Z-score of 1.988 in the standard normal distribution table, or use a calculator.

For \(Z = 1.988\):

The P-value associated with \(Z = 1.988\) is approximately 0.0239.

### Step 3: Conclusion
Now we compare the P-value to the significance level (\(α = 0.05\)):

- If P-value < \(α\), we reject the null hypothesis.
- If P-value ≥ \(α\), we fail to reject the null hypothesis.

Since \(0.0239 < 0.05\), we reject the null hypothesis.

### Answer
The P-value is 0.0239 and we reject the null hypothesis. Therefore, the correct answer is:

D) 0.0239; reject the null hypothesis.