A simple random sample of size 15 has mean x=41.8 and the population standard deviation is o=3.72. The population is normally distributed. Can you conclude that the population mean is less than 40?

Question

The population standard deviation [is or isnt?] known.
The sample size n [is or isnt] greater than 30.
The population [is or is not?] approximately normal.
The correct decision is to [perform the z-test, perform the t-test or consult to a statistician?]

Answer 1

Given the information you provided, here are the answers to your questions:

1. The population standard deviation is: known (σ = 3.72).
2. The sample size n: is not greater than 30 (n = 15).
3. The population: is approximately normal (as stated).

Since the population standard deviation is known and the sample size is less than 30, we would generally use the z-test for hypothesis testing. This is because the population is normally distributed.

To conclude whether the population mean is less than 40, you would typically perform a one-sample z-test:

1. Null Hypothesis (H0): μ ≥ 40

2. Alternative Hypothesis (H1): μ < 40

3. Calculate the z-value using the formula:

\[ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \]

Where:

• \( \bar{x} = 41.8 \) (sample mean)
• \( \mu_0 = 40 \) (hypothesized mean)
• \( \sigma = 3.72 \) (population standard deviation)
• \( n = 15 \)

5. Calculate:

\[ z = \frac{41.8 - 40}{3.72 / \sqrt{15}} \approx \frac{1.8}{0.962} \approx 1.87 \]

Next, you would compare this z-value to the critical z-value for your chosen significance level (commonly α = 0.05) for a one-tailed test.

If the calculated z-value is greater than the critical z-value (which would normally be about -1.645 for α = 0.05), you would fail to reject the null hypothesis.

Thus, the correct decision is to perform the z-test.

In summary:

• The population standard deviation is known.
• The sample size n is not greater than 30.
• The population is approximately normal.
• The correct decision is to perform the z-test.