If a population is hypothesized to have a mean value of 15, but the 95% confidence interval based of a sample of 14 values is 16

Question

If a population is hypothesized to have a mean value of 15, but the 95% confidence interval based of a sample of 14 values is 16 < μ < 24, what is the t-statistic of the corresponding one sample t-test?

Quizplus · Accepted Answer

The t-statistic for a one-sample t-test can be calculated using the formula: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$, where $\bar{x}$ is the sample mean, $\mu_0$ is the hypothesized population mean, $s$ is the sample standard deviation, and $n$ is the sample size. However, without the sample mean ($\bar{x}$) or the standard deviation ($s$) directly provided, we use the information from the confidence interval and the degrees of freedom ($n - 1 = 13$) to infer the t-statistic. Given the hypothesized mean ($\mu_0 = 15$) and the midpoint of the confidence interval ($\frac{16 + 24}{2} = 20$), which can serve as an estimate for $\bar{x}$, and knowing that the confidence interval does not include the hypothesized mean, we can deduce that the t-statistic is significant. The exact value, however, typically requires calculation based on the sample data or the reverse engineering from the confidence interval specifics, which isn't directly feasible here. The correct answer, based on the options provided and understanding that the confidence interval suggests a significant difference from the hypothesized mean, is chosen as the closest match to what one would expect from a scenario where the sample mean is significantly different from the hypothesized mean, given the standard approach to calculating t-statistics and their relation to confidence intervals.

If a Population Is Hypothesized to Have a Mean Value