Testing Hypotheses about a Population Proportion Lecture 42 Sections 9.3 Wed, Apr 7, 2004

Example Example 9.3, p. 525. State the hypotheses. H0: p = 0.255. H1: p < 0.255. State the level of significance.  = 0.05.

Example Compute the value of the test statistic. Compute p^. p^ = (p0(1 – p0)/n) = ((0.255)(0.745)/2400) = 0.008897. The value of the test statistic is z = (p^ – p0)/p^ = (0.25 – 0.255)/0.008897 = –0.5620.

Example Compute the p-value. State the conclusion. p-value = P(Z < –0.5620) = 0.2871. State the conclusion. This is larger than , so we should not reject H0.

Example Our conclusion is, “The data do not support the claim, at the 5% level of significance, that the smoking rate has decreased.”

Let’s Do It! Let’s do it! 9.2, p. 529 – Improved Process? Let’s do it! 9.3, p. 530 – ESP. Let’s do it! 9.4, p. 531 – Working Part Time.

Variations on the Method We will consider two variations on the method previously described. Classical Approach State the hypotheses and . Compute the critical value and compare the test statistic to it. p-value approach Do not state . Report the p-value and let the reader decide.

The Classical Approach 1. State the hypotheses. 2. State the level of significance. 3. Find the critical value of the test statistic and the rejection region. 4. State the decision rule. 5. Compute the value of the test statistic. 6. State the conclusion.

The Critical Value and the Rejection Region In a one-tailed test, the critical value c is the value of the test statistic that cuts off a tail of area  in the direction of extreme. The rejection region is the interval that starts at c and goes in the direction of extreme. p0 c

The Critical Value and the Rejection Region In a one-tailed test, the critical value c is the value of the test statistic that cuts off a tail of area  in the direction of extreme. The rejection region is the interval that starts at c and goes in the direction of extreme. Rejection region p0 c

The Critical Value and the Rejection Region In a two-tailed test, the critical value c is the value that cuts off an upper tail of area /2. The rejection region is the two intervals that start at c and go in the directions of extreme. –c p0 c

The Critical Value and the Rejection Region In a two-tailed test, the critical value c is the value that cuts off an upper tail of area /2. The rejection region is the two intervals that start at c and go in the directions of extreme. Rejection region Rejection region –c p0 c

The Decision Rule The decision rule says to reject H0 if the test statistic is in the rejection region. One-tailed test, extreme to the right. Reject H0 if z > c. One-tailed test, extreme to the left. Reject H0 if z < c. Two-tailed test. Reject H0 if either z > c or z < –c.

Example Rework Example 9.3, p. 525, by this method.

The Modified p-value Approach This is like the p-value approach, except there is no level of significance. At the end of the test, we state the p-value. It is left to the reader to decide for himself. This requires that the reader be familiar with statistics and p-values.

Assignment Page 534: Exercises 17 – 18. * Show all steps in the hypothesis test.