Testing Hypotheses about a Population Proportion Lecture 29 Sections 9.1 – 9.3 Tue, Oct 23, 2007

Discovering Characteristics of a Population Any question about a population must first be described in terms of a population parameter. We will work with the population mean  and the population proportion p.

Discovering Characteristics of a Population Then the question about that parameter generally falls into one of two categories.  Estimation What is the value of the parameter?  Hypothesis testing Does the evidence support or refute a claim about the value of the parameter?

Examples If we want to learn about voters’ preferences, how do we phrase the question?  What parameter do we use?  Do we estimate a parameter or test a hypothesis?

Example If we want to learn about the effectiveness of a new drug, how do we phrase the question?  What parameter do we use?  Do we estimate a parameter or test a hypothesis?

Example If we want to find out whether a newborn child is more likely to be male than female, how do we phrase the question?  What parameter do we use?  Do we estimate a parameter or test a hypothesis?

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. We will test the hypothesis that male births are as likely as female births, using these data.

p-Value Approach  H0H0

  H0H0

  H0H0 Observed value

p-Value Approach   0 z H0H0 z Observed value

p-Value Approach   0 z Reject p-value <  H0H0 z

p-Value Approach   0 z H0H0 Observed value

p-Value Approach   0 z H0H0 z Observed value

p-Value Approach   0 z p-value >  H0H0 Accept z

The Steps of Testing a Hypothesis (p-Value Approach) The seven steps:  1. State the null and alternative hypotheses.  2. State the significance level.  3. State the formula for the test statistic.  4. Compute the value of the test statistic.  5. Compute the p-value.  6. Make a decision.  7. State the conclusion.

The Steps of Testing a Hypothesis (p-Value Approach) See page 566. (Our seven 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  H 0 : 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.  H 0 : p = p 0.

State the Null and Alternative Hypotheses 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 right.)  H 1 : p  p 0. (Direction of extreme is left and right.)

Explaining the Data The observation is 520 males out of 1000 births, or 52%. That is, p ^ = Since we observed 52%, not 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  = 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 p-value < , then we reject H 0. If p-value > , then we reject H 0.

The Sampling Distribution of p ^ We know that the sampling distribution of p ^ is normal with mean p and standard deviation Thus, under H 0 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. Thus,

The Test Statistic In our example, we compute Therefore, the test statistic is

The Test Statistic Now, to find the value of the test statistic, all we need to do is to collect the sample data, find p ^, and substitute it into the formula for z.

Step 4: Compute the Test Statistic In the sample, p ^ = Thus,

Step 5: 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. (Double the value in one tail.)

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 > 1.265) =

Compute the p-value To find this value, we evaluate normalcdf(0.52, E99, 0.50, ) on the TI-83.

Step 6: Make a Decision Since the p-value is greater than , our decision is: Do not reject the null hypothesis. The decision is stated in statistical jargon.

Step 7: State the Conclusion State the conclusion in a sentence:  It is not true that more than 50% of live births are male. The conclusion must state the decision in the language of the original problem. It should not use statistical jargon.

Summary 1.H 0 : p = 0.50 H 1 : p >  = Test statistic: 4.z = (0.52 – 0.50)/ = p-value = P(Z > 1.26) = Do not reject H 0. 7.It is not true that more than 50% of live births are male.

Before collecting data Summary 1.H 0 : p = 0.50 H 1 : p >  = Test statistic: 4.z = (0.52 – 0.50)/ = p-value = P(Z > 1.26) = Do not reject H 0. 7.It is not true that more than 50% of live births are male.

After collecting data Summary 1.H 0 : p = 0.50 H 1 : p >  = Test statistic: 4.z = (0.52 – 0.50)/ = p-value = P(Z > 1.26) = Do not reject H 0. 7.It is not true that more than 50% of live births are male.