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Chapter. 10. Hypothesis Tests Regarding a Parameter. Section. 10. 2. Hypothesis Tests for a Population Proportion. Objectives. Explain the logic of hypothesis testing Test the hypotheses about a population proportion

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  1. Chapter 10 Hypothesis Tests Regarding a Parameter

  2. Section 10.2 Hypothesis Tests for a Population Proportion

  3. Objectives Explain the logic of hypothesis testing Test the hypotheses about a population proportion Test hypotheses about a population proportion using the binomial probability distribution.

  4. Objective 1 Explain the Logic of Hypothesis Testing

  5. A researcher obtains a random sample of 1000 people and finds that 534 are in favor of the banning cell phone use while driving, so = 534/1000. Does this suggest that more than 50% of people favor the policy? Or is it possible that the true proportion of registered voters who favor the policy is some proportion less than 0.5 and we just happened to survey a majority in favor of the policy? In other words, would it be unusual to obtain a sample proportion of 0.534 or higher from a population whose proportion is 0.5? What is convincing, or statistically significant, evidence?

  6. When observed results are unlikely under the assumption that the null hypothesis is true, we say the result is statistically significant. When results are found to be statistically significant, we reject the null hypothesis.

  7. To determine if a sample proportion of 0.534 is statistically significant, we build a probability model. Since np(1 – p) = 100(0.5)(1 – 0.5) = 250 ≥ 10 and the sample size (n = 1000) is sufficiently smaller than the population size, we can use the normal model to describe the variability in . The mean of the distribution of is and the standard deviation is

  8. Sampling distribution of the sample proportion

  9. We may consider the sample evidence to be statistically significant (or sufficient) if the sample proportion is too many standard deviations, say 2, above the assumed population proportion of 0.5. The Logic of the Classical Approach

  10. Recall that our simple random sample yielded a sample proportion of 0.534, so standard deviations above the hypothesized proportion of 0.5. Therefore, using our criterion, we would reject the null hypothesis.

  11. Why does it make sense to reject the null hypothesis if the sample proportion is more than 2 standard deviations away from the hypothesized proportion? The area under the standard normal curve to the right ofz= 2 is 0.0228.

  12. If the null hypothesis were true (population proportion is 0.5), then 1 – 0.0228 = 0.9772 = 97.72% of all sample proportions will be less than .5 + 2(0.016) = 0.532 and only 2.28% of the sample proportions will be more than 0.532.

  13. If the sample proportion is too many standard deviations from the proportion stated in the null hypothesis, we reject the null hypothesis. Hypothesis Testing Usingthe Classical Approach

  14. If the sample proportion of getting a sample proportion as extreme or more extreme than the one obtained is small under the assumption the statement in the null hypothesis is true, reject the null hypothesis. Hypothesis Testing Usingthe P-Value Approach

  15. A second criterion we may use for testing hypotheses is to determine how likely it is to obtain a sample proportion of 0.534 or higher from a population whose proportion is 0.5. If a sample proportion of 0.534 or higher is unlikely (or unusual), we have evidence against the statement in the null hypothesis. Otherwise, we do not have sufficient evidence against the statement in the null hypothesis. The Logic of the P-Value Approach

  16. We can compute the probability of obtaining a sample proportion of 0.534 or higher from a population whose proportion is 0.5 using the normal model.

  17. Recall So, we compute The value 0.0158 is called the P-value, which means about 2 samples in 100 will give a sample proportion as high or higher than the one we obtained if the population proportion really is 0.5. Because these results are unusual, we take this as evidence against the statement in the null hypothesis.

  18. If the probability of getting a sample proportion as extreme or more extreme than the one obtained is small under the assumption the statement in the null hypothesis is true, reject the null hypothesis. Hypothesis Testing Using the P-value Approach

  19. Objective 2 Test Hypotheses about a Population Proportion.

  20. The best point estimate of p, the proportion of the population with a certain characteristic, is given by where x is the number of individuals in the sample with the specified characteristic and n is the sample size. Recall:

  21. The sampling distribution of is approximately normal, with mean and standard deviation provided that the following requirements are satisfied: The sample is a simple random sample. np(1-p) ≥ 10. The sampled values are independent of each other. Recall:

  22. To test hypotheses regarding the population proportion, we can use the steps that follow, provided that: The sample is obtained by simple random sampling. np0(1 – p0) ≥ 10. The sampled values are independent of each other. Testing Hypotheses Regarding a Population Proportion, p

  23. Step 1:Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways:

  24. Step 2:Select a level of significance, α, based on the seriousness of making a Type I error.

  25. Classical Approach Step 3:Compute the test statistic Note: We use p0 in computing the standard error rather than . This is because, when we test a hypothesis, the null hypothesis is always assumed true.

  26. (critical value) Classical Approach Use Table V to determine the critical value. Two-Tailed

  27. (critical value) Classical Approach Left-Tailed

  28. (critical value) Classical Approach Right-Tailed

  29. Classical Approach Step 4:Compare the critical value with the test statistic:

  30. P-Value Approach By Hand Step 3: Compute the test statistic.

  31. P-Value Approach Use Table V to estimate the P-value. Two-Tailed

  32. P-Value Approach Left-Tailed

  33. P-Value Approach Right-Tailed

  34. P-Value Approach Technology Step 3:Use a statistical spreadsheet or calculator with statistical capabilities to obtain the P-value. The directions for obtaining the P-value using the TI-83/84 Plus graphing calculator, MINITAB, Excel, and StatCrunch are in the Technology Step-by-Step in the text.

  35. P-Value Approach Step 4:If the P-value < α, reject the null hypothesis.

  36. Step 5:State the conclusion.

  37. Parallel Example 1: Testing a Hypothesis about a Population Proportion: Large Sample Size In 1997, 46% of Americans said they did not trust the media “when it comes to reporting the news fully, accurately and fairly”. In a 2007 poll of 1010 adults nationwide, 525 stated they did not trust the media. At the α =0.05 level of significance, is there evidence to support the claim that the percentage of Americans that do not trust the media to report fully and accurately has increased since 1997? Source: Gallup Poll

  38. Solution We want to know if p>0.46. First, we must verify the requirements to perform the hypothesis test: • This is a simple random sample. • np0(1– p0)=1010(0.46)(1 – 0.46)=250.9>10 • Since the sample size is less than 5% of the population size, the assumption of independence is met.

  39. Solution Step 1:H0: p = 0.46 versus H1: p > 0.46 Step 2: The level of significance is α= 0.05. Step 3: The sample proportion is . The test statistic is then

  40. Solution: Classical Approach Step 4: Since this is a right-tailed test, we determine the critical value at the α= 0.05 level of significance to be z0.05 = 1.645. Step 5: Since the test statistic, z0 = 3.83, is greater than the critical value 1.645, we reject the null hypothesis.

  41. Solution: P-Value Approach Step 4: Since this is a right-tailed test, the P- value is the area under the standard normal distribution to the right of the test statistic z0=3.83. That is,P-value = P(Z > 3.83)≈0. Step 5: Since the P-value is less than the level of significance, we reject the null hypothesis.

  42. Solution Step 6: There is sufficient evidence at the α =0.05 level of significance to conclude that the percentage of Americans that do not trust the media to report fully and accurately has increased since 1997.

  43. Objective 3 Test hypotheses about a population proportion using the binomial probability distribution.

  44. For the sampling distribution of to be approximately normal, we require np(1– p) be at least 10. What if this requirement is not met? We stated that an event was unusual if the probability of observing the event was less than 0.05. This criterion is based on the P-value approach to testing hypotheses; the probability that we computed was the P-value. We use this same approach to test hypotheses regarding a population proportion for small samples.

  45. Parallel Example 4: Hypothesis Test for a Population Proportion: Small Sample Size In 2006, 10.5% of all live births in the United States were to mothers under 20 years of age. A sociologist claims that births to mothers under 20 years of age is decreasing. She conducts a simple random sample of 34 births and finds that 3 of them were to mothers under 20 years of age. Test the sociologist’s claim at the α = 0.01 level of significance.

  46. Parallel Example 4: Hypothesis Test for a Population Proportion: Small Sample Size Approach: Step 1: Determine the null and alternative hypotheses Step 2: Check whether np0(1–p0) is greater than or equal to 10, where p0 is the proportion stated in the null hypothesis. If it is, then the sampling distribution of is approximately normal and we can use the steps for a large sample size. Otherwise we use the following Steps 3 and 4.

  47. Parallel Example 4: Hypothesis Test for a Population Proportion: Small Sample Size Approach: Step 3: Compute the P-value. For right-tailed tests, the P-value is the probability of obtaining x or more successes. For left-tailed tests, the P- value is the probability of obtaining x or fewer successes. The P-value is always computed with the proportion given in the null hypothesis. Step 4: If the P-value is less than the level of significance, α, we reject the null hypothesis.

  48. Solution Step 1: H0: p=0.105 versus H1: p<0.105 Step 2: From the null hypothesis, we have p0=0.105. There were 34 mothers sampled, so np0(1– p0)=3.20 < 10. Thus, the sampling distribution of is not approximately normal.

  49. Solution Step 3: Let X represent the number of live births in the United States to mothers under 20 years of age. We have x = 3 successes in n = 34 trials so = 3/34= 0.088. We want to determine whether this result is unusual if the population mean is truly 0.105. Thus, P-value = P(X ≤ 3 assuming p=0.105 ) = P(X = 0) + P(X =1) + P(X =2) + P(X = 3) = 0.51

  50. Solution Step 4: The P-value = 0.51 is greater than the level of significance so we do not reject H0. There is insufficient evidence to conclude that the percentage of live births in the United States to mothers under the age of 20 has decreased below the 2006 level of 10.5%.

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