Transcript Lecture20
Hypothesis Tests In statistics a hypothesis is a statement that something is true. • Selecting the population parameter being tested (mean, proportion, variance, ect.) • Using p-values for hypothesis tests • Using Critical Regions for hypothesis tests • One tailed vs. two tailed tests Hypothesis Tests In a hypothesis test: 1. Identify H0 and HA 2. Select a level of significance ( ) 3. Assume the null hypothesis is true 4. Take a sample and determine the probability of that occurring. This is called the p-value. 5. Reject or Fail to reject H0 Error in Hypothesis Tests Type I vs Type II Error Conclusions and Consequences for a Test of Hypothesis True State of Nature Conclusion H0 True Ha True Accept H0 (Assume H0 True) Correct decision Type II error (probability ) Reject H0 (Assume Ha True) Type I error (probability ) Correct decision Population Proportion We have seen how to conduct hypothesis tests for a mean and we now give some attention to proportions. The process is completely analogous. We use the z-score (for large samples) and we will need to use the standard deviation formula for a proportion. E.g. pq n Example The CEO of a large electric utility claims that at least 80 percent of his 1,000,000 customers are very satisfied with the service they receive. To test this claim, the local newspaper surveyed 100 customers, using simple random sampling. Among the sampled customers, 73 percent say they are very satisfied. The lawyers for the newspaper says to avoid a law suit they can accuse the CEO of misrepresentation if they are 97% certain she is wrong. Should they print the story? Example - Two tailed The CEO of a large electric utility claims that exactly 80 percent of his 1,000,000 customers are very satisfied with the service they receive, but has a reputation for just making up data. To test this claim, the local newspaper surveyed 100 customers, using simple random sampling. Among the sampled customers, 73 percent say they are very satisfied. The lawyers for the newspaper says to avoid a law suit they can once again accuse the CEO of misrepresentation if they are 97% certain she is wrong. Should they print the story? What if the lawyer had said 95%? Tests of Hypothesis about a Population Variance • Hypotheses about the variance use the Chi-Square distribution and statistic n 1s • The quantity has a sampling distribution that follows the chi-square distribution assuming the population the sample is drawn from is normally distributed. 2 2 Properties of c2 • Continuous • Right of vertical axis • Shape varies with = n-1 = degrees of freedom • Skewed (n < 30) • Nearly normal for n>30 • See page 897/898 in text or formula sheet on line. Sketch of Chi-squared distribution c 2 25, 0.01 44.3141 9 Example Franklin’s Aviation Parts Unlimited manufactures aircraft altimeters with errors normally distributed with mean of 0 ft and a standard deviation of 43.7 ft. After installation of a new production line, 30 altimeters were randomly selected for a quality control test. This sample group had errors with standard deviation of 57.4 ft. Use a 0.05 significance level to test the claim the new altimeters have a different standard deviation from the old ones. Example Franklin’s Aviation Parts Unlimited manufactures aircraft altimeters with errors normally distributed with mean of 0 ft and a standard deviation of 43.7 ft. After installation of a new production line, 30 altimeters were randomly selected for a quality control test. This sample group had errors with standard deviation of 57.4 ft. Use a 0.05 significance level to test the claim the new altimeters have a higher standard deviation from the old ones. Exercises • #8.74, 8.78 on page 403 • #8.110, 8.112 on page 416 Problems from supplementary exercises: • # 8.125, 8.127 on page 419 • # 8.138, 8.144 on page 421 Homework • Review Chapters 8.1-8.5, 8.7 • Read Chapter 9.1-9.5 13