Testing Hypotheses about a Population Proportion Lecture 29 Sections 9.1 – 9.3 Fri, Nov 12, 2004

Discovering Characteristics of a Population Our question about a population must first be described in terms of a population parameter. Our question about a population must first be described in terms of a population parameter. Then our question about that parameter generally falls into one of two categories. Then our question about that parameter generally falls into one of two categories. What is the value of the parameter? What is the value of the parameter? That is, estimate its value. That is, estimate its value. Does the evidence tend to support or refute a claim about the value of the parameter? Does the evidence tend to support or refute a claim about the value of the parameter? That is, test a hypothesis concerning the parameter. That is, test a hypothesis concerning the parameter.

Example A standard assumption is that a newborn baby is as likely to be a boy as it is to be a girl. A standard assumption is that a newborn baby is as likely to be a boy as it is to be a girl. However, some believe that boys are more likely. However, some believe that boys are more likely. A random sample of 1000 live births shows that 520 are boys and 480 are girls. A random sample of 1000 live births shows that 520 are boys and 480 are girls. Does this evidence support or refute the standard assumption? Does this evidence support or refute the standard assumption?

The Steps of Testing a Hypothesis The basic steps are The basic steps are 1. State the null and alternative hypotheses. 1. State the null and alternative hypotheses. 2. State the significance level. 2. State the significance level. 3. Compute the value of the test statistic. 3. Compute the value of the test statistic. 4. Compute the p-value. 4. Compute the p-value. 5. State the conclusion. 5. State the conclusion. See page 519. See page 519.

Step 1: State the Null and Alternative Hypotheses Let p = proportion of live births that are boys. Let p = proportion of live births that are boys. The null and alternative hypotheses are The null and alternative hypotheses are H 0 : p = H 0 : p = H 1 : p > H 1 : p > 0.50.

State the Null and Alternative Hypotheses The null hypothesis should state a hypothetical value p 0 for the population proportion. The null hypothesis should state a hypothetical value p 0 for the population proportion. H 0 : p = p 0. H 0 : p = p 0. The alternative hypothesis must contradict the null hypothesis in one of three ways: The alternative hypothesis must contradict the null hypothesis in one of three ways: H 1 : p < p 0. (Direction of extreme is left.) H 1 : p < p 0. (Direction of extreme is left.) H 1 : p  p 0. (Direction of extreme is left and right.) H 1 : p  p 0. (Direction of extreme is left and right.) H 1 : p > p 0. (Direction of extreme is right.) H 1 : p > p 0. (Direction of extreme is right.)

Explaining the Data The observation is 520 males out of 1000 births, or 52%. The observation is 520 males out of 1000 births, or 52%. Since we did not observe 50%, how do we explain the discrepancy? Since we did not observe 50%, how do we explain the discrepancy? Chance, or Chance, or The true proportion is not 50%, but something larger, maybe 52%. The true proportion is not 50%, but something larger, maybe 52%.

Step 2: State the Significance Level The significance level  should be given in the problem. The significance level  should be given in the problem. If it isn’t, then use  = If it isn’t, then use  = In this example, we will use  = In this example, we will use  = 0.05.

The Sampling Distribution of p^ To decide whether the sample evidence is significant, we will compare the p-value to . To decide whether the sample evidence is significant, we will compare the p-value to . From the value of , we may find the critical value(s). From the value of , we may find the critical value(s).  is the probability that the sample data are at least as extreme as the critical value(s), if the null hypothesis is true.  is the probability that the sample data are at least as extreme as the critical value(s), if the null hypothesis is true.

The Sampling Distribution of p^ Therefore, when we compute the p-value, we do it under the assumption that H 0 is true, i.e., that p = p 0. Therefore, when we compute the p-value, we do it under the assumption that H 0 is true, i.e., that p = p 0.

The Sampling Distribution of p^ We know that the sampling distribution of p^ is normal with mean p and standard deviation We know that the sampling distribution of p^ is normal with mean p and standard deviation Thus, we assume that p^ has mean p 0 and standard deviation: Thus, we assume that p^ has mean p 0 and standard deviation:

Step 3: The Test Statistic Test statistic – The z-score of p^, under the assumption that H 0 is true. Test statistic – The z-score of p^, under the assumption that H 0 is true. Thus, Thus,

The Test Statistic In our example, we compute In our example, we compute Therefore, the test statistic is Therefore, the test statistic is Now, to find the value of the test statistic, all we need to do is to collect the sample data and substitute the value of p^. Now, to find the value of the test statistic, all we need to do is to collect the sample data and substitute the value of p^.

Computing the Test Statistic In the sample, p^ = In the sample, p^ = Thus, Thus, z = (0.52 – 0.50)/ = z = (0.52 – 0.50)/ = 1.26.

Step 4: Compute the p-value To compute the p-value, we must first check whether it is a one-tailed or a two-tailed test. To compute the p-value, we must first check whether it is a one-tailed or a two-tailed test. We will compute the probability that Z would be at least as extreme as the value of our test statistic. We will compute the probability that Z would be at least as extreme as the value of our test statistic. If the test is two-tailed, then we must take into account both tails of the distribution to get the p- value. If the test is two-tailed, then we must take into account both tails of the distribution to get the p- value.

Compute the p-value In this example, the test is one-tailed, with the direction of extreme to the right. In this example, the test is one-tailed, with the direction of extreme to the right. So we compute So we compute P(Z > 1.26) =

Compute the p-value An alternative is to evaluate An alternative is to evaluate normalcdf(0.52, E99, 0.50, ) on the TI-83. It should give the same answer (except for round-off). It should give the same answer (except for round-off).

Step 5: State the Conclusion Since the p-value is greater than , we should not reject the null hypothesis. Since the p-value is greater than , we should not reject the null hypothesis. State the conclusion in a sentence. State the conclusion in a sentence. “The data do not support the claim, at the 5% level of significance, that more than 50% of live births are male.” “The data do not support the claim, at the 5% level of significance, that more than 50% of live births are male.”

Testing Hypotheses on the TI-83 The TI-83 has special functions designed for hypothesis testing. The TI-83 has special functions designed for hypothesis testing. Press STAT. Press STAT. Select the TESTS menu. Select the TESTS menu. Select 1-PropZTest… Select 1-PropZTest… Press ENTER. Press ENTER. A window with several items appears. A window with several items appears.

Testing Hypotheses on the TI-83 Enter the value of p 0. Press ENTER and the down arrow. Enter the value of p 0. Press ENTER and the down arrow. Enter the numerator x of p^. Press ENTER and the down arrow. Enter the numerator x of p^. Press ENTER and the down arrow. Enter the sample size n. Press ENTER and the down arrow. Enter the sample size n. Press ENTER and the down arrow. Select the type of alternative hypothesis. Press the down arrow. Select the type of alternative hypothesis. Press the down arrow. Select Calculate. Press ENTER. Select Calculate. Press ENTER. (You may select Draw to see a picture.) (You may select Draw to see a picture.)

Testing Hypotheses on the TI-83 The display shows The display shows The title “1-PropZTest” The title “1-PropZTest” The alternative hypothesis. The alternative hypothesis. The value of the test statistic Z. The value of the test statistic Z. The p-value. The p-value. The value of p^. The value of p^. The sample size. The sample size. We are interested in the p-value. We are interested in the p-value.