A random sample of 150 pumpkins is obtained and the mean circumference is found to be
45 cm. Assuming that the population standard deviation is known to be 1.8 cm, use a 0.03
significance level to test the claim that the mean circumference of all pumpkins is greater
than 38 cm.
Do you have typos?
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Relate to .03.
To test the claim that the mean circumference of all pumpkins is greater than 38 cm, we can use a one-sample z-test.
Here are the steps to perform the test:
Step 1: State the null and alternative hypotheses.
Null hypothesis (H0): The mean circumference of all pumpkins is not greater than 38 cm.
Alternative hypothesis (Ha): The mean circumference of all pumpkins is greater than 38 cm.
Step 2: Determine the significance level (α).
The significance level given in the question is 0.03. This represents the maximum probability of making a Type I error (rejecting the null hypothesis when it is actually true).
Step 3: Compute the test statistic.
We can calculate the test statistic using the formula:
z = (x̄ - μ) / (σ / √n)
where:
x̄ = sample mean
μ = population mean (hypothesized value)
σ = population standard deviation
n = sample size
Given:
x̄ = 45 cm (sample mean)
μ = 38 cm (hypothesized population mean)
σ = 1.8 cm (population standard deviation)
n = 150 (sample size)
Substituting the values, the test statistic becomes:
z = (45 - 38) / (1.8 / √150)
Step 4: Determine the critical value.
Since we are conducting a one-tailed test (looking for a greater than condition), we need to find the critical value from the z-table at the given significance level α = 0.03.
The critical value is the z-value that corresponds to the cumulative probability of 1 - α or 0.97. By looking up the z-table, we find that the critical value is approximately 1.88.
Step 5: Make a decision.
If the test statistic is greater than the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
Step 6: Calculate the p-value.
The p-value corresponds to the probability of obtaining a test statistic as extreme as the observed data, assuming the null hypothesis is true. Since we are conducting a one-tailed test and looking for a greater than condition, the p-value is the area under the normal distribution curve to the right of the test statistic.
We can calculate the p-value using the z-table or a statistical calculator. For the given problem, the p-value is the probability P(Z > z), where Z is the standard normal random variable.
Step 7: Compare the p-value with the significance level.
If the p-value is less than the significance level (α), we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
Note: In this case, the p-value should be less than 0.03 to reject the null hypothesis.
By following these steps, you can test the claim that the mean circumference of all pumpkins is greater than 38 cm.