Testing Hypotheses about a Population Proportion Lecture 31 Sections 9.1 – 9.3 Mon, Oct 31, 2005

Discovering Characteristics of a Population Any question about a population must first be described in terms of a population parameter. Then the question about that parameter generally falls into one of two categories. What is the value of the parameter? That is, estimate its value. Does the evidence support or refute a claim about the value of 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 to be a girl. However, some people believe that boys are more likely. Suppose a random sample of 1000 live births shows that 520 are boys and 480 are girls. Use the data to estimate the proportion of male births. Does this evidence support the claim that a greater proportion of births are boys?

Two Approaches for Hypothesis Testing Classical approach. Determine the critical value and the rejection region. See whether the statistic falls in the rejection region. Report the decision. p-Value approach. Compute the p-value of the statistic. See whether the p-value is less than the significance level. Report the p-value.

Classical Approach H0 

Classical Approach H0  

Classical Approach H0   z c

Classical Approach H0   z c Acceptance Region Rejection Region

Classical Approach H0   z c Acceptance Region Rejection Region

Classical Approach H0   Reject z c z Acceptance Region c z Acceptance Region Rejection Region

Classical Approach H0   z c Acceptance Region Rejection Region

Classical Approach H0  Accept  z z c Acceptance Region z c Acceptance Region Rejection Region

p-Value Approach H0 

p-Value Approach H0  

p-Value Approach H0   z

p-Value Approach H0   z Acceptance Region Rejection Region

p-Value Approach H0   z Acceptance Region Rejection Region

p-Value Approach H0   z z Acceptance Region Rejection Region

p-Value Approach H0  p-value <   Reject z z Acceptance Region z Acceptance Region Rejection Region

p-Value Approach H0   z Acceptance Region Rejection Region

p-Value Approach H0   z z Acceptance Region Rejection Region

p-Value Approach H0 p-value >   Accept  z z Acceptance Region z Acceptance Region Rejection Region

The Steps of Testing a Hypothesis (p-Value Approach) The basic steps are 1. State the null and alternative hypotheses. 2. State the significance level. 3. Compute the value of the test statistic. 4. Compute the p-value. 5. Make a decision. 6. State the conclusion. See page 566. (The above steps are modified from what is in the book.)

Step 1: State the Null and Alternative Hypotheses Let p = proportion of live births that are boys. The null and alternative hypotheses are H0: p = 0.50. H1: p > 0.50.

State the Null and Alternative Hypotheses The null hypothesis should state a hypothetical value p0 for the population proportion. H0: p = p0. The alternative hypothesis must contradict the null hypothesis in one of three ways: H1: p < p0. (Direction of extreme is left.) H1: p  p0. (Direction of extreme is left and right.) H1: p > p0. (Direction of extreme is right.)

Explaining the Data The observation is 520 males out of 1000 births, or 52%. That is, p^ = 0.52. Since we did not observe 50%, how do we explain the discrepancy? Chance, or 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. If it isn’t, then use  = 0.05. 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 . If we were using the classical approach, we would use  to find the critical value(s).  is the probability that the value of the test statistic is 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 H0 is true, i.e., that p = p0.

The Sampling Distribution of p^ We know that the sampling distribution of p^ is normal with mean p and standard deviation Thus, we assume that p^ has mean p0 and standard deviation:

Step 3: The Test Statistic Test statistic – The z-score of p^, under the assumption that H0 is true. Thus, Write this

The Test Statistic In our example, we compute 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^.

Computing the Test Statistic In the sample, p^ = 0.52. Thus, z = (0.52 – 0.50)/0.0158 = 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. 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.

Compute the p-value In this example, the test is one-tailed, with the direction of extreme to the right. So we compute p-value = P(Z > 1.26) = 0.1038.

Compute the p-value An alternative is to evaluate normalcdf(0.52, E99, 0.50, 0.0158) on the TI-83. It should give the same answer (except for round-off).

Step 5: State the Conclusion Since the p-value is greater than , our decision is: Do not reject the null hypothesis. 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 conclusion must relate the decision to the context of the problem.

Summary 1. H0: p = 0.50 H1: p > 0.50 2.  = 0.05. 2.  = 0.05. 3. Test statistic: 4. z = (0.52 – 0.50)/0.0158 = 1.26. 5. p-value = P(Z > 1.26) = 0.1038. 6. Do not reject H0. 7. The data do not indicate that the proportion of boys is greater than the proportion of girls among newborns.

Summary 1. H0: p = 0.50 H1: p > 0.50 2.  = 0.05. 2.  = 0.05. 3. Test statistic: 4. z = (0.52 – 0.50)/0.0158 = 1.26. 5. p-value = P(Z > 1.26) = 0.1038. 6. Do not reject H0. 7. The data do not indicate that the proportion of boys is greater than the proportion of girls among newborns. Before collecting data

Summary 1. H0: p = 0.50 H1: p > 0.50 2.  = 0.05. 2.  = 0.05. 3. Test statistic: 4. z = (0.52 – 0.50)/0.0158 = 1.26. 5. p-value = P(Z > 1.26) = 0.1038. 6. Do not reject H0. 7. The data do not indicate that the proportion of boys is greater than the proportion of girls among newborns. Before collecting data After collecting data

Let’s Do It! Let’s Do It! 9.2, p. 575 – Improved Process? Let’s Do It! 9.3, p. 576 – ESP. Let’s Do It! 9.4, p. 577 – Working Part Time.

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

Testing Hypotheses on the TI-83 Enter the value of p0. 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. Select the type of alternative hypothesis. Press the down arrow. Select Calculate. Press ENTER. (You may select Draw to see a picture.)

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

The Classical Approach The seven steps 1. State the null and alternative hypotheses. 2. State the significance level. 3. Define the test statistic to be used. 4. State the decision rule. 5. Compute the value of the test statistic. 6. State the decision. 7. State the conclusion. (Do not compute the p-value.)

The Classical Approach The seven steps 1. State the null and alternative hypotheses. 2. State the significance level. 3. Define the test statistic to be used. 4. State the decision rule. 5. Compute the value of the test statistic. 6. State the decision. 7. State the conclusion. (Do not compute the p-value.) Before collecting data

The Classical Approach The seven steps 1. State the null and alternative hypotheses. 2. State the significance level. 3. Define the test statistic to be used. 4. State the decision rule. 5. Compute the value of the test statistic. 6. State the decision. 7. State the conclusion. (Do not compute the p-value.) Before collecting data After collecting data

Example of the Classical Approach Test the hypothesis that there are more male births than female births. Let p = the proportion of live births that are male. Step 1: State the hypotheses. H0: p = 0.50 H1: p > 0.50

Example of the Classical Approach Step 2: State the significance level. Let  = 0.05. Step 3: Define the test statistic.

Example of the Classical Approach Step 4: State the decision rule. Find the critical value. On the standard scale, the value z0 = 1.645 cuts off an upper tail of area 0.05. This is a normal percentile problem. Use invNorm(0.95) on the TI-83 or use the table. Therefore, we will reject H0 if z > 1.645. The decision rule

Example of the Classical Approach Step 5: Compute the value of the test statistic.

Example of the Classical Approach Step 6: State the decision. Since z < 1.645, our decision is to accept H0. Step 7: State the conclusion. Our conclusion is that the proportion of male births is the same as the proportion of female births.