Significance Test for the Difference of Two Proportions Section 8.4 Significance Test for the Difference of Two Proportions

Critical Values For a 2-sided test, what are the critical values for a significance level of α = 0.05?

Critical Values For a 2-sided test, what are the critical values for a significance level of α = 0.05?

Critical Values For a 2-sided test, what are the critical values for a significance level of α = 0.05?

Critical Values For a 2-sided test, what are the critical values for a significance level of α = 0.05? invNorm (.025) = -1.96 invNorm (.975) = 1.96 z* = ± 1.96

Critical Values For a one-sided test, what is the critical value for a significance level of α = 0.05?

Critical Values For a one-sided test, what is the critical value for a significance level of α = 0.05?

Critical Values For a one-sided test, what is the critical value for a significance level of α = 0.05? If Ha: p < po, invNorm (.05 ) = - 1.64

Critical Values For a one-sided test, what is the critical value for a significance level of α = 0.05?

Critical Values For a one-sided test, what is the critical value for a significance level of α = 0.05? If Ha: p > po, invNorm (.95 ) = 1.64

Components of a Significance Test for the Difference of Two Proportions for Surveys Four parts:

Components of a Significance Test for the Difference of Two Proportions for Surveys Four parts: 1. Name test and check conditions

Components of a Significance Test for the Difference of Two Proportions for Surveys Four parts: 1. Name test and check conditions 2. Write null and alternative hypotheses

Components of a Significance Test for the Difference of Two Proportions for Surveys Four parts: 1. Name test and check conditions 2. Write null and alternative hypotheses 3. Compute test statistic and P-value

Components of a Significance Test for the Difference of Two Proportions for Surveys Four parts: 1. Name test and check conditions 2. Write null and alternative hypotheses 3. Compute test statistic and P-value 4. Write a conclusion in context

Name Test & Check Conditions Name: Use the whole name! One-sided significance test for the difference of two proportions or Two-sided significance test for the difference of two proportions

Check Conditions First condition: Random samples selected independently from two different populations

Check Conditions Second condition: normal? All these quantities must be at least 5. Show all 4 calculations and state significance of results

Check Conditions Third condition: Each population is at least 10 times as large as its sample size. Remember to explain why this is reasonable.

Null Hypothesis Null hypothesis almost always one of “no difference” or “no effect”

Null Hypothesis Null hypothesis almost always one of “no difference” or “no effect” Several choices as long as define p1 and p2 Ho: The proportion of successes p1 in the first population is equal to the proportion of successes p2 in the second population. Write Ho in context of situation.

Null Hypothesis Null hypothesis almost always one of “no difference” or “no effect” Ho: p1 = p2, where p1 is the proportion of successes in the first population and p2 is the proportion of successes in the second population. Write Ho in context of situation.

Null Hypothesis Null hypothesis almost always one of “no difference” or “no effect” Ho: p1 - p2 = 0, where p1 is the proportion of successes in the first population and p2 is the proportion of successes in the second population. Write Ho in context of situation.

Alternative Hypothesis Form depends on whether you need a two-sided or one-sided test.

Alternative Hypothesis Two-sided test Ha: The proportion of successes p1 in the first population is not equal to the proportion of successes p2 in the second population. In symbols: Ha: p1 ≠ p2 or Ha: p1 – p2 ≠ 0

Alternative Hypothesis One-sided test Ha: The proportion of successes p1 in the first population is greater than the proportion of successes p2 in the second population. In symbols: Ha: p1 > p2 or Ha: p1 – p2 > 0

Alternative Hypothesis One-sided test Ha: The proportion of successes p1 in the first population is less than the proportion of successes p2 in the second population. In symbols: Ha: p1 < p2 or Ha: p1 – p2 < 0

Compute Test Statistic and P-value If we assume p1 = p2, then use

Compute Test Statistic and P-value If we assume p1 = p2, then use

Compute Test Statistic and P-value If we assume p1 = p2, then use

is called “pooled estimate” of the common proportion of successes

Compute Test Statistic and P-value or use 2-PropZTest Only valid if we assume p1 = p2

Compute Test Statistic and P-value If do not assume p1 = p2, then use:

Compute Test Statistic and P-value The P-value is the probability of getting a value of z as extreme or more extreme than that from your samples if Ho is true. Include a sketch.

Write a Conclusion in Context State whether you reject or do not reject the null hypothesis.

Write a Conclusion in Context State whether you reject or do not reject the null hypothesis. Link this conclusion to your computations by (1) comparing z to critical value z* reject null hypothesis if z is more extreme than z* or

Write a Conclusion in Context State whether you reject or do not reject the null hypothesis. Link this conclusion to your computations by (1) comparing z to critical value z* reject null hypothesis if z is more extreme than z* or (2) comparing P-value to level of significance α reject null hypothesis if P-value is smaller than α

Write a Conclusion in Context State whether you reject or do not reject the null hypothesis. Link this conclusion to your computations by (1) comparing z to critical value z*, reject null hypothesis if z is more extreme than z* or (2) comparing P-value to level of significance α, reject null hypothesis if P-value is smaller than α Write sentence giving conclusion in context related to alternative hypothesis.

Page 536, P50

Page 536, P50 Give appropriate statistical evidence to support your answer means complete all 4 steps for a significance test.

Page 536, P50 Name: one-sided significance test for the difference of two proportions because you are asked whether the data support the conclusion that there was a decrease in voter support for the candidate.

Page 536, P50 We are told that we have two random samples from a population of probable voters in some city. It’s reasonable to assume that the samples are independent because the random samples were taken about 2 weeks apart.

Page 536, P50 Let n1 be 1st sample and n2 the 2nd sample. Each of the following is at least 5:

Page 536, P50 The number of probable voters at both times should be larger than 10 times the sample size for both samples, and it is unknown whether this is the case.

Page 536, P50 Using symbols, what is the null hypothesis?

Page 536, P50 Ho: p1 = p2

Page 536, P50 Ho: p1 = p2, where p1 is the proportion of all probable voters who favored the candidate at the time of the first survey and p2 is the proportion of all probable voters who favored the candidate one week before the election.

Page 536, P50 Ha: p1 > p2 We want to see if there was a decrease in voter support from 1st survey to 2nd survey.

Test Statistic and P-value 2-PropZTest x1: 321 n1: 600 x2: 382 n2: 750 p1 > p2 Calculate Survey 3 weeks before start of campaign 2nd survey

Test Statistic and P-value 2-PropZTest x1: 321 n1: 600 x2: 382 n2: 750 p1 > p2 Calculate z = 0.938 and P-value = 0.1741 Survey 3 weeks before start of campaign 2nd survey

Page 536, P50

Write Conclusion in Context I do not reject the null hypothesis because the P-value of 0.1741 is greater than the significance level of α = 0.05.

Write Conclusion in Context I do not reject the null hypothesis because the P-value of 0.1741 is greater than the significance level of α = 0.05. There is not sufficient evidence to conclude that there was a decrease in voter support for the new candidate after the parking tickets were revealed.

For this one-sided test: What critical value(s) of z would you use to compare against the test statistic of z = 0.938 for a 95% confidence level?

Z* What critical value(s) of z would you use to compare against the test statistic of z = 0.938 for a 95% confidence level? One-sided test at 95% confidence level: z* = invNorm(0.95) = 1.64

Write Conclusion in Context I do not reject the null hypothesis because the test statistic of z = 0.938 is less extreme than the critical value of z* = 1.64. There is not sufficient evidence to conclude that there was a decrease in voter support for the new candidate after the parking tickets were revealed.

Problem # 1 If the P-value of a test is less than the level of significance, then which of these conclusions is correct?

C. The null hypothesis is true. If the P-value of a test is less than the level of significance, then which of these conclusions is correct? A. The value of the test statistic is in the rejection region for this test. B. The sample size should be increased to decrease the margin of error. C. The null hypothesis is true. D. The corresponding confidence interval will contain the hypothesized value of the parameter in the null hypothesis. E. None of these is a valid conclusion.

C. The null hypothesis is true. If the P-value of a test is less than the level of significance, then which of these conclusions is correct? A. The value of the test statistic is in the rejection region for this test. B. The sample size should be increased to decrease the margin of error. C. The null hypothesis is true. D. The corresponding confidence interval will contain the hypothesized value of the parameter in the null hypothesis. E. None of these is a valid conclusion.

Problem # 1 A. If the P-value is less than α, then the result is “statistically significant.” Reject H0. This corresponds to the test statistic falling in the rejection region.

If all else remains the same, which of these will make a confidence interval for the difference of two proportions wider? I. Increase the confidence level II. Increase the sample size III. Increase the margin of error IV. Increase the probability of a Type I error

If all else remains the same, which of these will make a confidence interval for the difference of two proportions wider? I. Increase the confidence level II. Increase the sample size III. Increase the margin of error IV. Increase the probability of a Type I error

One-sided or Two-sided? Page 537 E64 E66 E67 E68

One-sided or Two-sided? E64: Two-sided E66: E67: E68:

One-sided or Two-sided? E64: Two-sided E66: Two-sided E67: E68:

One-sided or Two-sided? E64: Two-sided E66: Two-sided E67: One-sided E68:

One-sided or Two-sided? E64: Two-sided E66: Two-sided E67: One-sided E68: One-sided

Questions?