A textbook company states that the average time a student needs to take a quiz from its book is 30 minutes with a standard deviation of 3 minutes. A teacher using the book is not sure that this is correct for her classes and wants to check. She collects data on 10 random students and finds that the average time to take the quiz was only 25 minutes. As a result, the teacher performs a two-tailed hypothesis test with a significance level of 5%. Which conclusion is valid based on the results of the test?
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What conclusions are listed?
Her students, on average, do not take 30 minutes on the quiz, contrary to what the textbook company stated.
To determine which conclusion is valid based on the results of the hypothesis test, we need to follow a series of steps.
Step 1: State the null hypothesis (H0) and alternative hypothesis (Ha)
The null hypothesis (H0) states that there is no significant difference between the average time a student needs to take a quiz (μ) and the claimed average time (μ0) of 30 minutes.
H0: μ = μ0
The alternative hypothesis (Ha) states that there is a significant difference between the average time a student needs to take a quiz (μ) and the claimed average time (μ0) of 30 minutes.
Ha: μ ≠ μ0
Step 2: Determine the significance level (α)
In this case, the significance level is given as 5%, which corresponds to α = 0.05.
Step 3: Set up the test statistic and its distribution
Since the population standard deviation (σ) is known and the sample size (n) is small (10 students), we can use the t-distribution.
Step 4: Calculate the test statistic
The test statistic, t, is calculated using the formula:
t = (x̄ - μ0) / (σ / sqrt(n))
In this case, x̄ is the sample mean (25 minutes), μ0 is the claimed average time (30 minutes), σ is the population standard deviation (3 minutes), and n is the sample size (10 students).
t = (25 - 30) / (3 / sqrt(10))
Step 5: Determine the critical region
Since this is a two-tailed test, we need to split the significance level (α) between the two tails. For a 5% significance level, each tail would have an equal 2.5% (0.025) area under the curve.
Using a t-table or statistical software, we can find the critical t-values for a two-tailed test with 0.025 area in each tail and degrees of freedom (df) equal to n-1 (10-1 = 9).
In this case, the critical t-values are approximately -2.262 and 2.262.
Step 6: Make a decision
If the calculated test statistic falls within the critical region (outside the range between -2.262 and 2.262), we reject the null hypothesis (H0). If the calculated test statistic falls outside the critical region, we fail to reject the null hypothesis (H0).
Step 7: Draw a conclusion
Based on the results of the test, if the calculated test statistic falls within the critical region, we reject the null hypothesis (H0) and conclude that there is a significant difference between the average time a student needs to take a quiz and the claimed average time. If the calculated test statistic falls outside the critical region, we fail to reject the null hypothesis (H0) and conclude that there is not enough evidence to suggest a significant difference.
In this case, we calculate the test statistic as t ≈ -3.162, which falls outside the critical region (-2.262 to 2.262) for a 5% significance level. Therefore, we reject the null hypothesis (H0) and conclude that there is a significant difference between the average time a student needs to take a quiz and the claimed average time.