Question

A member of the Islip Urban Renewal Taskforce claimed, after the installation of internal heating facilities in homes, the proportion of homes with fireplaces fell below 50%. 1. State the direction of the alternative hypothesis used to test the taskforce claim. Type gt (greater than), ge (greater than or equal to), lt (less than), le (less than or equal to) or ne (not equal to) as appropriate in the box. Blank 1 Another committee member, who did not originally believe the claim, took a random sample of 84 houses in the area to test the validity of the claim. He found that 33 of them had fireplaces.

2. Use the tables in the textbook to determine the critical value of the test statistic using a 2% level of significance. If there are two critical values, state only the positive value.

3. Calculate the test statistic, correct to two decimal places.

4. Is the null hypothesis rejected for this test? Type yes or no.

5. Regardless of your answer for 4, if the null hypothesis was rejected, can we conclude after the installation of internal heating facilities in homes, the proportion of homes with fireplaces fell below 50% at the 2% level of significance? Type yes or no.

Answer 1

Given Information:

A member of the Islip Urban Renewal Taskforce claimed, after the installation of internal heating facilities in homes, the proportion of homes with fireplaces fell below 50%.

The null hypothesis and alternate hypothesis for this test is denoted by respectively and is given by:

The test is a left tailed test.

Let x be number of houses with fireplaces and n be the total number of houses sampled.

For this problem, we have:

1) The direction of the alternative hypothesis used to test the taskforce claim is lt (less than) that is a lower tailed z - test for a population proportion.

2) The left tailed critical value of the test statistic using a 2% level of significance is the tabulated value of Z at 2% significance level and is calculated using standard normal table and is given by:

3) The best point estimate of population proportion p is sample proportion and is given by:

Under Null Hypothesis:

The test statistic can be calculated as:

Hence, the test statistic is -1.96.

4) The calculated value of test statistic is less than the tabulated value of test statistic. (-1.96>--2.05) that is (1.96<2.05), the decision is to accept the null hypothesis.

No, the null hypothesis is not rejected for this test.

5) If the null hypothesis was rejected the decision made is that the alternative is true that is and Yes the conclusion is that after the installation of internal heating facilities in homes, the proportion of homes with fireplaces fell below 50% at the 2% level of significance.