A researcher believes that the percentage of people who exercise in California is greater than the national exercise rate. The national rate is 20%. The researcher gathers a random sample of 120 individuals who live in California and finds that the number who exercise regularly is 31 out of 120.

· a. What is ÷2obt?
· b. What is df for this test?
· c. What is ÷2cv?
· d. What conclusion should be drawn from these results?

To answer these questions, we need to perform a hypothesis test using the information provided.

a. To calculate the ÷2obt (observed ÷2 value), we first need to calculate the expected values. The expected value is the proportion of people who exercise regularly in California assuming that the national exercise rate (p) is 20%. The expected value, E, can be calculated as E = n * p, where n is the sample size (120) and p is the national exercise rate (20%). Therefore, E = 120 * 0.20 = 24.

Now, we can calculate the ÷2obt using the formula ÷2obt = (O - E)^2 / E, where O is the observed value (31) and E is the expected value (24). Plugging in the values, we get ÷2obt = (31 - 24)^2 / 24 = 7^2 / 24 = 49 / 24 = 2.04.

b. To calculate the df (degrees of freedom) for this test, we use the formula df = (number of categories - 1) = (2 - 1) = 1. Since we are comparing the California exercise rate to the national exercise rate, there are two categories: exercise and no exercise.

c. To find the ÷2cv (critical ÷2 value), we need to specify the significance level (usually denoted as α). Let's say the significance level is 0.05 (or 5%). We can look up the critical ÷2 value from the ÷2 distribution table using the given df (1) and α (0.05). Based on the table, the ÷2cv is approximately 3.841.

d. To draw a conclusion from these results, we compare the ÷2obt value (2.04) with the ÷2cv value (3.841). If ÷2obt < ÷2cv, we fail to reject the null hypothesis (which states that the California exercise rate is not significantly different from the national exercise rate). If ÷2obt > ÷2cv, we reject the null hypothesis and conclude that the California exercise rate is significantly different from the national exercise rate.

In this case, since ÷2obt (2.04) is less than ÷2cv (3.841), we fail to reject the null hypothesis. Therefore, we do not have enough evidence to conclude that the percentage of people who exercise in California is greater than the national exercise rate.