Final answer:
The t-statistic for the student's data is approximately 4.798, indicating a likely rejection of the null hypothesis. The critical t-value for a 95% confidence interval is approximately 2.021, and the calculated margin of error is approximately 0.660.
Step-by-step explanation:
A student's question regards calculating a t-statistic, determining a critical t-value, stating a conclusion based on the p-value, and calculating the margin of error for a two-tailed, single-sample t-test.
To calculate the t-statistic, use the formula:
t = (m - μ) / (s / sqrt(n))
Where m = sample mean, μ = population mean, s = sample standard deviation, and n = sample size.
In your case:
- μ = 7
- m = 8.5
- s = 2.1
- n = 41
So, the t-statistic is:
t = (8.5 - 7) / (2.1 / sqrt(41)) ≈ 4.798
For part b, the critical t-value for a 95% confidence interval with degrees of freedom (df = n - 1 = 40) can be found using a t-distribution table or statistical software. Let's assume it is approximately 2.021 for 40 df.
For part c, without an exact p-value provided, a t-statistic of approximately 4.798 likely indicates a p-value < 0.05, suggesting that the null hypothesis should be rejected.
Lastly, the margin of error (ME) can be calculated with the formula:
ME = tcritical × (s / sqrt(n))
Using the assumed critical t-value:
ME = 2.021 × (2.1 / sqrt(41)) ≈ 0.660