test the claim, identify null hypothesis, alternative hypothesis, test statistic, p-value and critical values, and conclusion
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 the sigma is known to be 121.8lb. use a 0.10 significance level to test the claim that the population mean of all such here weights is less than 200lbs.
To test the claim, we need to follow a series of steps in hypothesis testing. Let's go through each step:
Step 1: State the null hypothesis and the alternative hypothesis:
The null hypothesis (H0) in this case would be that there is no significant difference between the population mean bear weight (μ) and 200 lbs.
H0: μ = 200 lbs
The alternative hypothesis (Ha) would be that the population mean bear weight is less than 200 lbs.
Ha: μ < 200 lbs
Step 2: Select the appropriate test statistic:
Since the sample size is greater than 30 and we know the population standard deviation (σ), we can use the Z-test statistic. The formula for the test statistic is calculated as:
Z = (X̄ - μ) / (σ / √n)
Where:
X̄ = sample mean
μ = population mean
σ = population standard deviation
n = sample size
Step 3: Determine the significance level:
The significance level, denoted by α, is stated in the problem as 0.10 or 10%. This is the probability of rejecting the null hypothesis when it is actually true.
Step 4: Calculate the test statistic:
Given the information in the problem:
X̄ (sample mean) = 182.9 lbs
μ (population mean) = 200 lbs (from the null hypothesis)
σ (population standard deviation) = 121.8 lbs
n (sample size) = 54
Now we can calculate the test statistic:
Z = (182.9 - 200) / (121.8 / √54) ≈ -1.991
Step 5: Determine the critical value(s) and p-value:
Since the alternative hypothesis is "less than 200 lbs," we need to find the critical value for a one-tailed test with a significance level of 0.10 (10%).
Using a Z-table or statistical software, we can find the critical value for a left-tailed test to be approximately -1.282.
The p-value is the probability of observing a test statistic as extreme as -1.991 (or more extreme) assuming the null hypothesis is true. To find this value, we can use the Z-table or statistical software. In this case, the p-value is approximately 0.0261.
Step 6: Make a decision and conclude:
We compare the test statistic (-1.991) to the critical value (-1.282).
Since the test statistic is smaller than the critical value, we can reject the null hypothesis.
Furthermore, since the p-value (0.0261) is less than the significance level of 0.10, we also have evidence to reject the null hypothesis.
Therefore, based on the given data and with a 10% significance level, there is enough evidence to conclude that the population mean bear weight in Yellowstone National Park is less than 200 lbs.