Find the test statistic, P-value, and critical value. Round to 3 decimal places.
Claim: The mean IQ score of statistics professors is greater than 118.
Sample data: n = 50, x = 120. Assume that σ = 15 and the significance level is α = .05
test statistic =
P-value =
critical value =
Use a one-sample z-test. Calculate to find the test statistic. Find the critical value using a z-table for .05 significance level for a one-tailed test. The p-value is the actual level of the test statistic and is found using a z-table as well.
To find the test statistic, P-value, and critical value, we need to perform a hypothesis test using the given information.
Step 1: State the hypotheses.
- Null hypothesis (H0): The mean IQ score of statistics professors is 118 or less. (μ ≤ 118)
- Alternative hypothesis (Ha): The mean IQ score of statistics professors is greater than 118. (μ > 118)
Step 2: Set the significance level (α).
The significance level is given as α = .05. This represents a 5% level of significance.
Step 3: Calculate the test statistic.
Since we know the population standard deviation (σ) and the sample size (n), we can use the Z-test.
The formula for the Z-test statistic is:
Z = (x - μ) / (σ / sqrt(n))
Given data:
Sample mean (x) = 120
Population standard deviation (σ) = 15
Sample size (n) = 50
Z = (120 - 118) / (15 / sqrt(50))
Z = 2 / (15 / sqrt(50))
Z ≈ 1.344
So the test statistic is approximately 1.344.
Step 4: Find the P-value.
To find the P-value, we need to find the area to the right of the test statistic in the standard normal distribution.
At a significance level of α = .05, we want to find P(Z > 1.344).
Using a Z-table or a calculator, we find that P(Z > 1.344) ≈ 0.089.
So the P-value is approximately 0.089.
Step 5: Determine the critical value.
The critical value is the Z-score that corresponds to a cumulative probability of (1 - α) or 95% in this case.
Using a Z-table or a calculator, we find that the critical value is approximately 1.645.
So the critical value is approximately 1.645.
Summary:
- Test statistic ≈ 1.344
- P-value ≈ 0.089
- Critical value ≈ 1.645