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HYPOTHESIS TESTING

HYPOTHESIS TESTING. CHAPTER 4 BCT 2053 APPLIED STATISTICS. CONTENT. 4.1 Introduction to Hypothesis Testing 4.2 Hypothesis Testing for Mean with known and unknown Variance 4.3 Hypothesis Testing for Difference Means with known and unknown Population Variance

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HYPOTHESIS TESTING

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  1. HYPOTHESIS TESTING CHAPTER 4 BCT 2053 APPLIED STATISTICS

  2. CONTENT • 4.1 Introduction to Hypothesis Testing • 4.2 Hypothesis Testing for Mean with known and unknown Variance • 4.3 Hypothesis Testing for Difference Means with known and unknown Population Variance • 4.4 Hypothesis Testing for Proportion • 4.5 Hypothesis Testing for the Difference between Two Proportions • 4.6 Hypothesis Testing for Variances and Standard Deviations • 4.7 Hypothesis Testing for Two Variances and Standard Deviations

  3. 4.1 Introduction to Hypothesis Testing • OBJECTIVES : After completing this chapter, you should be able to • Describe the meaning of terms used in hypothesis testing. • State the Null and Alternative Hypothesis

  4. General Terms in Hypothesis Testing • Hypothesis • A statement that something TRUE • Statistical Hypothesis • A statement about the parameters of one or more populations. • Null Hypothesis (Ho) • A hypothesis to be tested • Alternative Hypothesis (H1) • A hypothesis to be considered as an alternative to the null hypothesis

  5. How to make Null hypothesis • Should have an ‘equal’ sign • Generally, parameter Avalue

  6. How to make Alternative hypothesis • Should reflect the purpose of the hypothesis test and different from the null hypothesis • Generally (3 types);

  7. Basic logic of Hypothesis Testing • Accept null hypothesis • if the sample data are consistent with the null hypothesis • Reject null hypothesis • if the sample data are inconsistent with the null hypothesis, so accept Alternative hypothesis • Test statistics • the statistics used as a basis for deciding whether the null hypothesis should be rejected (z, t, Chi, F)

  8. Basic logic of Hypothesis Testing • Rejection (Critical) Region, (α) • the set of values for the test statistics that leads to rejection of the null hypothesis • Nonrejection (NonCritical) Region,(1 – α) • the set of values for the test statistics that leads to nonrejection of the null hypothesis • Critical values • the values of the test statistics that separate the rejection and nonrejection regions. • A critical value is considered part of the rejection region • In general, Reject Ho if test statistics > critical value

  9. Rejection Region Reject Ho Reject Ho Accept Ho 1 – α Critical value Critical value Reject Ho Accept Ho 1 – α Critical value Reject Ho Accept Ho 1 – α Critical value

  10. Type I and II Error • Type I Error • - Rejecting the null hypothesis when it is in fact true • Type II Error • -Not rejecting the null hypothesis when it is in fact false

  11. Hypothesis Testing Common Phrase H1 H1 H0 H0 H1 H0

  12. EXERCISE 4.1 State the null and alternative hypotheses for each conjecture. • A researcher thinks that if expectant mothers use vitamin pills, the birth weight of the babies will increase. The average birth weight of the population is 8.6 pounds. • An engineer hypothesizes that the mean number of defects can be decreased in a manufacturing process of compact disks by using robots instead of humans for certain tasks. The mean number of defective disks per 1000 is 18. • A psychologist feels that playing soft music during a test will change the results of the test. The psychologist is not sure whether the grades will be higher or lower. In the past, the mean of the scores was 73.

  13. Steps In Hypothesis Testing • Define the parameter used • Define the null and alternative hypothesis • Define all the given information • Chose appropriate Test Statistics (z,t,chi,F) • Find Critical value • Test the hypothesis (rejection region) • Make conclusion – there is enough evidence to reject/accept the claim at α

  14. 4.2 Hypothesis Testing for Mean with known and unknown Variance • OBJECTIVES : After completing this chapter, you should be able to • Test mean when σ² is known. • Test mean when σ² is unknown.

  15. Hypothesis Testing for Mean μ Where: NOTE: Ztestand ttest are test statistics

  16. Example 1: Hypothesis testing for mean μ with known σ² • A lecturer state that the IQ score for IPT students must be more higher than other people IQ’s which is known to be normally distributed with mean 110 and standard deviation 10. To prove his hypothesis, 25 IPT students were chosen and were given an IQ test. The result shows that the mean IQ score for 25 IPT students is 114. Can we accept his hypothesis at significance level, α = 0.05?

  17. Example 2: Hypothesis testing for mean μ with unknown σ²and n ≥ 30 • UMP students said that they have no enough time to sleep. A sample of 36 students give that, the mean of sleep time is 6 hours and the standard deviation is 0.9 hours. It is known that the mean sleep time for adult is 6.5 hours. Can we accept their hypothesis at significance level, α = 0.01?

  18. Example 3: Hypothesis testing for mean μ with unknown σ²and n < 30 • In a wood cutting process to produce rulers, the mean of rulers height is set to be equal 100 cm at all times. If the mean height of rulers is not equal to 100 cm, the process will stop immediately. The height for a sample of 10 rulers produces by the process shows below: 100.13 100.11 100.02 99.99 99.98 100.14 100.03 100.10 99.97 100.21 Can we stop the process at significance level, α = 0.05?

  19. 4.3 Hypothesis Testing for Difference Means with known and unknown Population Variance • OBJECTIVES : After completing this chapter, you should be able to • Test the difference between two means when σ’s are known. • Test the difference between two means when σ’s are unknown and equal. • Test the difference between two means when σ’s are unknown and not equal.

  20. Hypothesis Testing for Different Mean with known and unknown Variance

  21. Example 4: Hypothesis testing for μ1 – μ2 with known σ1² andσ2² • The mean lifetime for 30 battery type A is 5.3 hours while the mean lifetime for 35 battery type B is 4.8 hours. If the lifetime standard deviation for the battery type A is 1 and the lifetime standard deviation for the battery type B is 0.7 hours, can we conclude that the lifetime for both batteries type A and type B are same at significance level, α = 0.05?

  22. Example 5: Hypothesis testing forμ1 – μ2withunknownσ1²&σ2², σ1² ≠ σ2², n1 ≥ 30&n2 ≥ 30 • The mean price of 30 acre of land in Cerok before a highway is build is RM20,000 per acre with standard deviation RM 3000 per acre. The mean price of 36 acre of land in Cerok after the highway is build is RM50,000 per acre with standard deviation RM 4000 per acre. Test a hypothesis that the new highway will increases the land price in Cerok among RM35,000 at significance level, α = 0.05. Assume that the variances of land price in Cerok are not same before and after the highway is build.

  23. Example 6: Hypothesis testing forμ1 – μ2withunknownσ1²&σ2², σ1² ≠ σ2², n1 <30 & n2 < 30 • The average size of a farm in Indiana is 191 acres. The average size of a farm in Greene is 199 acres. Assume the data were obtained from two samples with standard deviations of 38 and 12 acres, respectively, and sample sizes of 8 and 10, respectively. Can we conclude that the atα= 0.05, the average size of the farms in Indiana is less than Greene? Assume that the variance for both countries are different.

  24. Example 7: Hypothesis testing forμ1 – μ2withunknownσ1²&σ2², σ1² = σ2² , n1 ≥ 30 & n2 ≥ 30 • Many studies have been conducted to test the effects of marijuana use on mental abilities. In a study, groups of light and heavy users of marijuana were tested for memory recall, with the result below. • Item sort correctly by light marijuana users: • Item sort correctly by heavy marijuana users: Use 0.01 significance level to test the claim that the population of heavy marijuana users has a lowermean than the light users if the variance population for both users are same.

  25. Example 8: Hypothesis testing forμ1 – μ2withunknownσ1²&σ2², σ1² = σ2² , n1 <30 & n2 <30 • Two catalyst are being analyzed to determine the mean yield of a chemical process. A test is run in the pilot plant and results are shown below. catalyst 1: 91.50 94.18 92.18 95.39 91.79 89.07 94.72 89.21 catalyst 2: 89.19 90.95 90.46 93.21 97.19 97.04 91.07 92.75 Is there any different between the mean yield? Use α = 0.05 and assume the variances population are equal.

  26. 4.4 Hypothesis Testing for Proportion • OBJECTIVES : After completing this chapter, you should be able to • Test proportions using z-test.

  27. Hypothesis testing for proportion p Where:

  28. Example 9: Hypothesis testing for proportion p • An attorney claims that more than 25% of all lawyers advertise. A sample of 200 lawyers in a certain city showed that 63 had used some form of advertising. At α = 0.01, is there enough evidence to support the attorney’s claim?

  29. Example 10: Hypothesis testing for proportion p • A group of scientist believes that their new medicine can heal 40% of patients. The current medicine in market can only heal 30% of patients. A research is done to test the hypothesis made by the scientists. The new medicine is given to the 100 patients and it shows that only 26 patients are recovered. Can we accept their hypothesis at significance level α = 0.05?

  30. 4.5 Hypothesis Testing for the Difference between two Proportions • OBJECTIVES : After completing this chapter, you should be able to • Test the difference between two proportions.

  31. Hypothesis testing for the difference between two proportions p1 – p2 Where:

  32. Example 11: Hypothesis testing for difference proportion p1 – p2 • In a sample of 200 surgeons, 15% thought the government should control health care. In a sample of 200 practitioners, 21% felt the same way. At α = 0.01, is there a difference in the proportions between surgeons and practitioners?

  33. Example 12: Hypothesis testing for difference proportion p1 – p2 • Random samples of 747 Malaysian men and 434 Malaysian women were taken. Of those sampled, 276 men and 195 women said that they sometimes ordered dish without meat or fish when they eat out. Do the data provide sufficient evidence to conclude that, in Malaysia, the percentage of men who sometimes order a dish without meat or fish is smaller than the percentage of women who sometimes order a dish without meat or fish at significance level α= 0.05?

  34. 4.6 Hypothesis Testing for Variances and Standard Deviations • OBJECTIVES : After completing this chapter, you should be able to • Test single variance and standard deviation

  35. Hypothesis testing for variance σ² Where:

  36. Example 13: Hypothesis Testing for σ² • In a wood cutting process to produce rulers, the variance of ruler’s height is set to be equal 2 cm² at all times. If the variance of ruler’s height is not equal to 2 cm², the process will stop immediately. The height for a sample of 10 rulers produces by the process shows below: 100.23 100.11 100.42 99.66 99.68 100.14 100.33 100.10 99.50 100.21 Can we stop the process at significance level α = 0.05?

  37. Example 14: Hypothesis Testing for σ² • A hospital administrator believes that the standard deviation of the number of people using outpatient surgery per day is greater than 8. A random sample of 15 days is selected and the standard deviation is 11.2. At α = 0.05, is there enough evidence to support the administrator’s claim?

  38. 4.7 Hypothesis Testing for Two Variances and Standard Deviations • OBJECTIVES : After completing this chapter, you should be able to • Test the difference between two variances.

  39. Hypothesis testing for variance ratio σ1²/ σ2²

  40. Example 15: Hypothesis Testing for difference proportions σ1²/ σ2² • Before service, a machine can packed 10 packets of sugar with variance weight 64 g² while after service the variance weight for 5 packets of sugar are 25 g². Do the services improve the packaging process at significance level, α = 0.05?

  41. Example 16: Hypothesis Testing for difference proportions σ1²/ σ2² • A medical researcher whishes to see whether the variance of the heart rates (in beats per minute) of smokers is different from the variance of heart rates of people do not smoke. Two samples are selected, and the data are as shown below. Using α = 0.1, is there enough evidence to support the claim? • Smokers: Nonsmokers:

  42. 0.01, 0.05 and 0.1 significance levels are usually used in testing a hypothesis. Hypothesis test are closely related to confidence interval. Whenever a confidence interval can be computed, a hypothesis test can also be performed, and vice versa. The End Summary

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