Statistical Tests (Hypothesis Testing)

Transcription

1 Outlines December 2, 2004

2 Outlines Introduction Introduction Examples Homework Much of this material is from Allan Bluman s Elementary Statistics: A Brief Version, Second Edition.

3 Part I

4 Example A medical researcher is interested in finding out whether a new medication will have any undesirable side effects, particularly with respect to heart rate. Will a patient s pulse rate, increase, decrease, or remain unchanged after taking this medication?

5 Example A medical researcher is interested in finding out whether a new medication will have any undesirable side effects, particularly with respect to heart rate. Will a patient s pulse rate, increase, decrease, or remain unchanged after taking this medication? After conducting controlled tests on a sample of patients, the researcher finds that the mean pulse rate of the group taking the new medicine is higher than average. But is this increase significant, or simply due to random chance?

6 Example A medical researcher is interested in finding out whether a new medication will have any undesirable side effects, particularly with respect to heart rate. Will a patient s pulse rate, increase, decrease, or remain unchanged after taking this medication? After conducting controlled tests on a sample of patients, the researcher finds that the mean pulse rate of the group taking the new medicine is higher than average. But is this increase significant, or simply due to random chance? Statistical tests (or hypothesis testing) can be used to answer that question and give the conclusion some certainty.

7 Define a population under study.

8 Define a population under study. State the particular hypotheses that will be investigated, and give the significance level.

9 Define a population under study. State the particular hypotheses that will be investigated, and give the significance level. Select a sample from the population, and collect data.

10 Define a population under study. State the particular hypotheses that will be investigated, and give the significance level. Select a sample from the population, and collect data. Perform calculations for the statistical test, and reach a conclusion.

11 Define a population under study Recall that a population is the collection of all possible objects that have a common measurable or otherwise observable feature or characteristic.

12 Define a population under study Recall that a population is the collection of all possible objects that have a common measurable or otherwise observable feature or characteristic. For the purposes of hypothesis testing, we re often concerned with knowing the population s mean and standard deviation.

13 Define a population under study Recall that a population is the collection of all possible objects that have a common measurable or otherwise observable feature or characteristic. For the purposes of hypothesis testing, we re often concerned with knowing the population s mean and standard deviation. In the case of the medical researcher, the population is all adult men, and the mean pulse rate was 70 beats per minute (bpm), with a standard deviation of 8 bpm.

14 State the hypotheses to be investigated A statistical hypothesis is a conjecture about a population parameter. The conjecture may or may not be true.

15 State the hypotheses to be investigated A statistical hypothesis is a conjecture about a population parameter. The conjecture may or may not be true. There are two types of statistical hypotheses in each test: the null hypothesis and the alternative hypothesis.

16 State the hypotheses to be investigated A statistical hypothesis is a conjecture about a population parameter. The conjecture may or may not be true. There are two types of statistical hypotheses in each test: the null hypothesis and the alternative hypothesis. The null hypothesis (H 0 ), is a hypothesis that states there is no difference between a parameter and a specific value, or that there is no difference between two parameters.

17 State the hypotheses to be investigated A statistical hypothesis is a conjecture about a population parameter. The conjecture may or may not be true. There are two types of statistical hypotheses in each test: the null hypothesis and the alternative hypothesis. The null hypothesis (H 0 ), is a hypothesis that states there is no difference between a parameter and a specific value, or that there is no difference between two parameters. The alternative hypothesis (H 1 ) is a hypothesis that states the existance of difference between a parameter and a specific value, or states that there is a difference between two parameters.

18 Hypothesis examples H 0 : There is no difference in pulse between these patients and the nationwide population (µ = 70). H 1 : The pulse of patients taking this medication is substantially different than the nationwide population (µ 70).

19 Hypothesis examples H 0 : There is no difference in pulse between these patients and the nationwide population (µ = 70). H 1 : The pulse of patients taking this medication is substantially different than the nationwide population (µ 70). H 0 : This new additive does nothing to increase the life of automobile batteries (µ 36). H 1 : This additive significantly increases battery life (µ > 36).

20 Hypothesis examples H 0 : There is no difference in pulse between these patients and the nationwide population (µ = 70). H 1 : The pulse of patients taking this medication is substantially different than the nationwide population (µ 70). H 0 : This new additive does nothing to increase the life of automobile batteries (µ 36). H 1 : This additive significantly increases battery life (µ > 36). H 0 : A new type of insulation does not help decrease monthly heating costs (µ 78). H 1 : This new insulation reduces monthly heating costs (µ < 78).

21 Types of Statistical Tests and Hypotheses The null and alternative hypotheses are always stated together.

22 Types of Statistical Tests and Hypotheses The null and alternative hypotheses are always stated together. Written mathematically, the null hypothesis always contains an equal sign.

23 Types of Statistical Tests and Hypotheses The null and alternative hypotheses are always stated together. Written mathematically, the null hypothesis always contains an equal sign. The mathematical form of the null and alternative hypotheses for different types of tests are as follows:

24 Types of Statistical Tests and Hypotheses The null and alternative hypotheses are always stated together. Written mathematically, the null hypothesis always contains an equal sign. The mathematical form of the null and alternative hypotheses for different types of tests are as follows: Two-tailed test Right-tailed test Left-tailed test H 0 : µ = k H 0 : µ k H 0 : µ k H 1 : µ k H 1 : µ > k H 1 : µ < k

25 Designing the study, defining the significance level The simplest design of the medication study would be to give the drug to a sample of patients, wait long enough for the drug to be absorbed, and measure the patients heart rates.

26 Designing the study, defining the significance level The simplest design of the medication study would be to give the drug to a sample of patients, wait long enough for the drug to be absorbed, and measure the patients heart rates. However, regardless of the medication or the sampling, the mean heart rate of the sampled patients will almost never be exactly 70 bpm.

27 Designing the study, defining the significance level The simplest design of the medication study would be to give the drug to a sample of patients, wait long enough for the drug to be absorbed, and measure the patients heart rates. However, regardless of the medication or the sampling, the mean heart rate of the sampled patients will almost never be exactly 70 bpm. Two possibilities exist:

28 Designing the study, defining the significance level The simplest design of the medication study would be to give the drug to a sample of patients, wait long enough for the drug to be absorbed, and measure the patients heart rates. However, regardless of the medication or the sampling, the mean heart rate of the sampled patients will almost never be exactly 70 bpm. Two possibilities exist: Either the null hypothesis is true and any differences between sample mean and population mean are purely due to chance,

29 Designing the study, defining the significance level The simplest design of the medication study would be to give the drug to a sample of patients, wait long enough for the drug to be absorbed, and measure the patients heart rates. However, regardless of the medication or the sampling, the mean heart rate of the sampled patients will almost never be exactly 70 bpm. Two possibilities exist: Either the null hypothesis is true and any differences between sample mean and population mean are purely due to chance, or the null hypothesis is false and the observed difference in heart rate is due to side effects of the medication.

30 Designing the study, defining the significance level Qualitatively, we would expect that if the sample mean pulse rate is 71 or 72 bpm, that the difference is due to random chance.

31 Designing the study, defining the significance level Qualitatively, we would expect that if the sample mean pulse rate is 71 or 72 bpm, that the difference is due to random chance. Similarly, we would expect that if the sample mean pulse rate is 80 or 85 bpm, that the difference is due to the medicine.

32 Designing the study, defining the significance level Qualitatively, we would expect that if the sample mean pulse rate is 71 or 72 bpm, that the difference is due to random chance. Similarly, we would expect that if the sample mean pulse rate is 80 or 85 bpm, that the difference is due to the medicine. But these decisions cannot be made qualitatively, there must be a quantifiable standard limits the probability that differences are due to chance.

33 Designing the study, defining the significance level Qualitatively, we would expect that if the sample mean pulse rate is 71 or 72 bpm, that the difference is due to random chance. Similarly, we would expect that if the sample mean pulse rate is 80 or 85 bpm, that the difference is due to the medicine. But these decisions cannot be made qualitatively, there must be a quantifiable standard limits the probability that differences are due to chance. That is where the level of significance appears.

34 Types of Errors There are four possible outcomes from a hypothesis test: The null hypothesis is actually true, but the sample data does not support it and we decide to reject the null hypothesis.

35 Types of Errors There are four possible outcomes from a hypothesis test: The null hypothesis is actually true, but the sample data does not support it and we decide to reject the null hypothesis. The null hypothesis is actually false, but the sample data leads us to believe that we should not reject the null hypothesis.

36 Types of Errors There are four possible outcomes from a hypothesis test: The null hypothesis is actually true, but the sample data does not support it and we decide to reject the null hypothesis. The null hypothesis is actually false, but the sample data leads us to believe that we should not reject the null hypothesis. The null hypothesis is actually true, and the sample data supports this fact.

37 Types of Errors There are four possible outcomes from a hypothesis test: The null hypothesis is actually true, but the sample data does not support it and we decide to reject the null hypothesis. The null hypothesis is actually false, but the sample data leads us to believe that we should not reject the null hypothesis. The null hypothesis is actually true, and the sample data supports this fact. The null hypothesis is actually false, and the sample data helps us reject it.

38 Types of Errors There are four possible outcomes from a hypothesis test: The null hypothesis is actually true, but the sample data does not support it and we decide to reject the null hypothesis. The null hypothesis is actually false, but the sample data leads us to believe that we should not reject the null hypothesis. The null hypothesis is actually true, and the sample data supports this fact. The null hypothesis is actually false, and the sample data helps us reject it. Two of these situations result in a correct decision, and two of them result in an incorrect decision. H 0 True H 0 False Reject H 0 Type I Error Correct decision Do not reject H 0 Correct decision Type II Error

39 Level of Significance (finally) The level of significance is defined as the maximum probability of committing a type I error, and is normally shown as α. For example, if α = 0.05, there is a 5% chance that we will reject a true null hypothesis. If α = 0.01, there is a 1% chance that we will reject a true null hypothesis. (The probability of committing a type II error is normally shown as β, and is normally not easily computed. However, for a given testing situation, lowering α increases β, and raising α lowers β.)

40 Perform calculations for a statistical test The significance level α and the type of test (two-tailed, left-tailed, right-tailed) lead to the selection of a critical value from statistical tables. The critical value(s) separates the critical region of the probability distribution curve from the noncritical region.

41 Perform calculations for a statistical test The significance level α and the type of test (two-tailed, left-tailed, right-tailed) lead to the selection of a critical value from statistical tables. The critical value(s) separates the critical region of the probability distribution curve from the noncritical region. The critical region is the range of values of the test value that indicate a significant difference from the population and that the null hypothesis should be rejected.

42 Perform calculations for a statistical test The significance level α and the type of test (two-tailed, left-tailed, right-tailed) lead to the selection of a critical value from statistical tables. The critical value(s) separates the critical region of the probability distribution curve from the noncritical region. The critical region is the range of values of the test value that indicate a significant difference from the population and that the null hypothesis should be rejected. The noncritical region is the range of values of the test value that indicate any differences from the population are probably due to chance, and that the null hypothesis should not be rejected.

43 Example Critical Values: Right-Tailed Test For a right-tailed test on a normal distribution, the critical value is the z value that gives Φ(z > z crit ) = α z * =2.33 α=φ(z>z * )= f Z (z) z

44 Example Critical Values: Left-Tailed Test For a left-tailed test on a normal distribution, the critical value is the z value that gives Φ(z < z crit ) = α z * = 2.33 α=φ(z<z * )= f Z (z) z

45 Example Critical Values: Two-Tailed Test For a two-tailed test on a normal distribution, the critical value is the z value that gives Φ(z > z crit ) + Φ(z < z crit ) = α z * =±2.58 α=φ(z<z * )=0.005 α=φ(z>z * )= f Z (z) z

46 Introduction Examples Homework The z test is a statistical test for the mean of a population. It can be used whenever: the sample size n 30, or

47 Introduction Examples Homework The z test is a statistical test for the mean of a population. It can be used whenever: the sample size n 30, or if the population is known to be normally distributed and the population standard deviation σ is known (regardless of n).

48 Introduction Examples Homework The z test is a statistical test for the mean of a population. It can be used whenever: the sample size n 30, or if the population is known to be normally distributed and the population standard deviation σ is known (regardless of n). The formula for the z test is z = X µ σ/ n

49 Introduction Examples Homework The z test is a statistical test for the mean of a population. It can be used whenever: the sample size n 30, or if the population is known to be normally distributed and the population standard deviation σ is known (regardless of n). The formula for the z test is z = X µ σ/ n If this z value falls into the critical region, we reject the null hypothesis, and conclude that there is not enough evidence to support the idea that there is a significant difference between the sample and the overall population.

50 Introduction Examples Homework Example 1 Let s revisit our medical researcher. The mean pulse rate of the entire population of adult men is 70 beats per minute (bpm), with a standard deviation of 8 bpm. A sample of 10 adult male patients is given a new drug, and after several minutes, their mean pulse rate is measured at 75 bpm. With a level of significance of α = 0.01, is this change in pulse rate due to random chance?

51 Introduction Examples Homework Example 1 Solution (1) State the null hypothesis: There is no significant difference in pulse between these patients and the population as a whole. (H 0 : µ = 70)

52 Introduction Examples Homework Example 1 Solution (1) State the null hypothesis: There is no significant difference in pulse between these patients and the population as a whole. (H 0 : µ = 70) State the alternative hypothesis: There is a significant difference in pulse between these patients and the population as a whole. (H 1 : µ 70).

53 Introduction Examples Homework Example 1 Solution (1) State the null hypothesis: There is no significant difference in pulse between these patients and the population as a whole. (H 0 : µ = 70) State the alternative hypothesis: There is a significant difference in pulse between these patients and the population as a whole. (H 1 : µ 70). The null hypothesis is given in the form corresponding to a two-tailed test (H 0 : µ = k, H 1 : µ k), and for α = 0.01, the critical z values are z = ±2.58.

54 Introduction Examples Homework Example 1 Solution (2) Calculate the test z value: z = X µ σ/ n with X = 75, µ = 70, σ = 8, n = 10.

55 Introduction Examples Homework Example 1 Solution (2) Calculate the test z value: z = X µ σ/ n with X = 75, µ = 70, σ = 8, n = 10. In this case, z = Since z falls within the noncritical region, our conclusion is to not reject the null hypothesis.

56 Introduction Examples Homework Example 1 Solution (2) Calculate the test z value: z = X µ σ/ n with X = 75, µ = 70, σ = 8, n = 10. In this case, z = Since z falls within the noncritical region, our conclusion is to not reject the null hypothesis. We conclude that there is not sufficient evidence to indicate that the change in pulse rate is due to anything other than random chance.

57 Introduction Examples Homework Example 2 Our medical researcher continues with the pulse tests. A larger sample of 50 adult male patients has been tested identically to the first group, and their mean pulse rate is also measured at 75 bpm. With a level of significance of α = 0.01, is this change in pulse rate due to random chance?

58 Introduction Examples Homework Example 2 Solution (1) State the null hypothesis: There is no significant difference in pulse between these patients and the population as a whole. (H 0 : µ = 70)

59 Introduction Examples Homework Example 2 Solution (1) State the null hypothesis: There is no significant difference in pulse between these patients and the population as a whole. (H 0 : µ = 70) State the alternative hypothesis: There is a significant difference in pulse between these patients and the population as a whole. (H 1 : µ 70).

60 Introduction Examples Homework Example 2 Solution (1) State the null hypothesis: There is no significant difference in pulse between these patients and the population as a whole. (H 0 : µ = 70) State the alternative hypothesis: There is a significant difference in pulse between these patients and the population as a whole. (H 1 : µ 70). The null hypothesis is given in the form corresponding to a two-tailed test (H 0 : µ = k, H 1 : µ k), and for α = 0.01, the critical z values are z = ±2.58.

61 Introduction Examples Homework Example 2 Solution (2) Calculate the test z value: z = X µ σ/ n with X = 75, µ = 70, σ = 8, n = 50.

62 Introduction Examples Homework Example 2 Solution (2) Calculate the test z value: z = X µ σ/ n with X = 75, µ = 70, σ = 8, n = 50. In this case, z = Since z falls within the critical region, our conclusion is to reject the null hypothesis.

63 Introduction Examples Homework Example 2 Solution (2) Calculate the test z value: z = X µ σ/ n with X = 75, µ = 70, σ = 8, n = 50. In this case, z = Since z falls within the critical region, our conclusion is to reject the null hypothesis. We conclude that there is sufficient evidence to indicate that the change in pulse rate is due to something other than random chance.

64 Introduction Examples Homework Homework (hand in at final exam time) A manufacturer states that the average lifetime of its lightbulbs is 36 months, with a standard deviation of 10 months. Thirty bulbs are selected, and their average lifetime is found to be 32 months. Should the manufacturer s statement be rejected at α = 0.01? Should it be rejected if you tested 50 bulbs and found an average lifetime of 32 months?