Testing Hypotheses about a Population Proportion Lecture 31 Sections 9 1 9 3 Wed Mar 21 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 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 Specify Two Approaches for Hypothesis Testing p Value approach Compute the p value of the statistic Report the p value If is specified then report the decision Classical Approach H0 Classical Approach H0 Classical Approach H0 z 0 c Critical value Classical Approach H0 z 0 Acceptance Region c Rejection Region Classical Approach H0 z 0 Acceptance Region c Rejection Region Classical Approach H0 0 Acceptance Region Reject c z Rejection Region z Classical Approach H0 z 0 Acceptance Region c Rejection Region Classical Approach H0 Accept 0 z z c Acceptance Region Rejection Region p Value Approach H0 p Value Approach H0 p Value Approach H0 p Value Approach H0 0 z z p Value Approach H0 p value 0 Reject z z p Value Approach H0 z 0 p Value Approach H0 0 z z p Value Approach H0 p value Accept 0 z 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 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 State the Null and Alternative Hypotheses 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 right p p0 Direction of extreme is left and right H1 Explaining the Data The observation is 520 males out of 1000 births or 52 That is p 0 52 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 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 is the probability that the value that we observe is at least as extreme as the critical value s if the null hypothesis is true 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 p 1 p p n Thus we assume that p has mean p0 and standard deviation p p0 1 p0 n Step 3 The Test Statistic Test statistic The z score of p under the assumption that H0 is true Thus Z p p p p p0 p0 1 p0 n The Test Statistic In our example we compute 50 1 50 p 0 01581 1000 Therefore the test statistic is p 0 50 Z 0 01581 The Test Statistic 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 Step 4 Compute the Test Statistic In the sample p 0 52 Thus 0 52 0 50 Z 1 265 0 01581 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 pvalue 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 0 1029 Compute the p value An alternative method is to evaluate normalcdf 0 52 E99 0 50 0 01581 on the TI 83 It should give the same answer except for round off 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 H0 p 0 50 H1 p 0 50 2 0 05 3 Test statistic Z p p0 p0 1 p0 n 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 It is not true that more than 50 of live births are male Summary 1 H0 p 0 50 H1 p 0 50 2 0 05 3 Test statistic Z p p0 p0 1 p0 n Before collecting data 4 z 0 52 0 50 0 0158 1 26 5 …
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