Question

What is meant by the term significance? How is significance level related to what you learned in previous chapters about probability? Please provide an example using significance in your work or life.

What is the main use of a one sample Z-Test? Please provide an example.

Answer 1

The level of significance is defined as the probability of rejecting a null hypothesis by the test when it is really true, which is denoted as α. That is, P (Type I error) = α.

For any hypothesis testing,the rejection rule is as follows:

The rejection region for two-tailed test is shown below:

The rejection region for one-tailed test is :

• In the left-tailed test, the rejection region is shaded in left side.

• In the right-tailed test, the rejection region is shaded in right side.

The one-sample z-test is used to test whether the mean of a population is greater than, less than, or not equal to a specific value. Because the standard normal distribution is used to calculate critical values for the test, this test is often called the one-sample z-test.

Step 1: State the Null Hypothesis.

Step 2: State the alternate hypothesis

Step 3: State your alpha level

Step 4: Find the z-score associated with your alpha level called as z_tab

Step 5: Use the z-formula to find a z-score i.e z_tab

Step 6: If at Step 5 z_cal > z_tab, reject the null hypothesis.

Example:

1,500 women followed the Atkin’s diet for a month. A random sample of 29 women gained an average of 6.7 pounds. Test the hypothesis that the average weight gain per woman for the month was over 5 pounds. The standard deviation for all women in the group was 7.1.

Null hypothesis: average weight gain per woman for the month was 5 pounds

Alternative hypothesis: average weight gain per woman for the month was over 5 pounds

choose alpha = 0.05 so, z_tab = 1.96

the value of z_cal is
Z = 6.7 – 5 / (7.1/√29) = 1.289.

since z_cal < z_tab, so we do not reject the null hypothesis.