Question

A. A statistics practitioner randomly sampled 10 observations and found x = 103 and s = 17. Is there sufficient evidence at the 10% significance level to conclude that the population mean is less than 110?

b. Repeat part (a) assuming that you know that the population standard deviation is 17.

c. Explain why the conclusions produced in parts (a) and (b) differ.

Answer 1

Yes, there is sufficient evidence to conclude that the population mean is less than 110 at the 10% significance level.

Step-by-step explanation:

To find out if there is sufficient evidence to conclude that the population mean is less than 110, we need to perform a hypothesis test.

Step 1: State the null and alternate hypotheses:

Null hypothesis (H0): The population mean is equal to 110.

Alternate hypothesis (Ha): The population mean is less than 110.

Step 2: Choose the significance level:

Given that the significance level is 10% (or 0.10), the critical value for a one-tailed test is -1.282 (using a z-table).

Step 3: Calculate the test statistic:

We can calculate the test statistic using the formula:

test statistic (z) = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))

Using the given information, we have:

sample mean (x) = 103

sample standard deviation (s) = 17

sample size (n) = 10

population mean (µ) = 110

Plugging in these values, we get:

test statistic (z) = (103 - 110) / (17 / sqrt(10))

Simplifying, we get:

test statistic (z) = -2.051

Step 4: Make a decision:

Comparing the test statistic (-2.051) with the critical value (-1.282), we can see that the test statistic falls in the critical region. Therefore, we reject the null hypothesis.

Step 5: Make a conclusion:

Since we reject the null hypothesis, there is sufficient evidence to conclude that the population mean is less than 110 at the 10% significance level.