The mean of 15 sample IQ scores of statistic instructors is 118. The sample standard deviation is 11. Given the sample data appear to come from a normally distributed population, find the test statistic for a claim that the population mean IQ of statistic instructors is greater than 110.

Question

A) t = 9.342
B) t = 2.817
C) t = 3.767
D) t = 10.909

Answer 1

To determine the test statistic for the claim that the population mean IQ of statistic instructors is greater than 110, we can use the t-test formula for the mean. The formula for the t-statistic is as follows:

$$t=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}$$

Where:

• $\overline{x}$ = Sample mean
• ${\mu }_0$ = Population mean under the null hypothesis
• $s$ = Sample standard deviation
• $n$ = Sample size

Given:

• Sample mean ($\overline{x}$) = 118
• Hypothesized population mean (${\mu }_0$) = 110
• Sample standard deviation ($s$) = 11
• Sample size ($n$) = 15

Now we can plug in the values into the formula:

1. Calculate $s/\sqrt{n}$:

$$s/\sqrt{n}=11/\sqrt{15}$$

First, calculate $\sqrt{15}$:

$$\sqrt{15}\approx 3.872$$

Now, calculate $s/\sqrt{n}$:

$$s/\sqrt{n}\approx 11/3.872\approx 2.840$$

2. Plug this value into the t-statistic formula:

$$t=\frac{118-110}{2.840}\approx \frac{8}{2.840}\approx 2.816$$

Thus, the calculated t-statistic is approximately:

$$t\approx 2.816$$

Comparing this result with the provided options:

A) t = 9.342
B) t = 2.817
C) t = 3.767
D) t = 10.909

The closest answer, rounding to three decimal places, suggests that the correct answer is:

B) t = 2.817.