Hypothesis Testing Introduction Bluman 5th Ed. Slides © McGraw Hill With enhancements by the Darton State U. / Cordele staff Bluman, Chapter 8

Hypothesis Testing Researchers are interested in answering many types of questions. For example, Is the earth warming up? Does a new medication lower blood pressure? Does the public prefer a certain color in a new fashion line? Is a new teaching technique better than a traditional one? Do seat belts reduce the severity of injuries? These types of questions can be addressed through statistical hypothesis testing, which is a decision-making process for evaluating claims about a population. Bluman, Chapter 8

Hypothesis Testing Three methods used to test hypotheses: 1. The traditional method - TODAY 2. The P-value method - SOMETIME 3. The confidence interval method – WE WON’T Bluman, Chapter 8

8.1 Steps in Hypothesis Testing-Traditional Method A statistical hypothesis is a conjecture about a population parameter. This conjecture may or may not be true. Our problems involve two hypotheses The null hypothesis, symbolized by H0, The alternative hypothesis is symbolized by H1. Bluman, Chapter 8

8.1 Steps in Hypothesis Testing-Traditional Method The null hypothesis, symbolized by H0, is a statistical hypothesis that states that there is no difference between a parameter and a specific value, or that there is no difference between two parameters. You can think of it as the status quo, what’s currently believed, the conventional wisdom, etc….. Bluman, Chapter 8

Steps in Hypothesis Testing-Traditional Method The alternative hypothesis, symbolized by H1, is a statistical hypothesis that states the existence of a difference between a parameter and a specific value, or states that there is a difference between two parameters. The Alternative Hypothesis is the new kid, challenging the Null Hypothesis. Bluman, Chapter 8

Situation A A medical researcher is interested in finding out whether a new medication will have any undesirable side effects. The researcher is particularly concerned with the pulse rate of the patients who take the medication. Will the pulse rate increase, decrease, or remain unchanged after a patient takes the medication? The researcher knows that the mean pulse rate for the population under study is 82 beats per minute. The hypotheses for this situation are This is called a two-tailed hypothesis test. Bluman, Chapter 8

Situation A The researcher is particularly concerned with the pulse rate of the patients who take the medication… The hypotheses for this situation are This is called a two-tailed hypothesis test because there’s TWO ways to reject the null hypothesis: 1) Left tail, it’s significantly lower than 82 2) Right tail, it’s significantly higher than 82 Some other tests are only one-tailed. Bluman, Chapter 8

Situation B A chemist invents an additive to increase the life of an automobile battery. The mean lifetime of the automobile battery without the additive is 36 months. In this book, the null hypothesis is always stated using the equals sign. The hypotheses for this situation are This is called a right-tailed hypothesis test. We “reject the null hypothesis” only if the sample mean is significantly HIGHER than 36. Lower than 36 doesn’t affect us. We’re wondering about HIGHER. Bluman, Chapter 8

Situation C A contractor wishes to lower heating bills by using a special type of insulation in houses. If the average of the monthly heating bills is $78, her hypotheses about heating costs with the use of insulation are The hypotheses for this situation are This is called a left-tailed hypothesis test. Only significantly LOWER sample mean causes us to reject the null hypothesis. Bluman, Chapter 8

Claim When a researcher conducts a study, he or she is generally looking for evidence to support a claim. Therefore, the researcher’s claim should be stated as the alternative hypothesis, or research hypothesis. Bluman, Chapter 8

Hypothesis Testing After stating the hypotheses, the researcher’s next step is to design the study. The researcher selects the correct statistical test, chooses an appropriate level of significance, and formulates a plan for conducting the study. Bluman, Chapter 8

Kinds of Statistical Tests Look at the lesson titles in Chapters 8, 9, 10. Early on, it’s easy – because we’ve only studied one or two. The longer you study statistics, the more tests you learn about And the more there is to choose from. Bluman, Chapter 8

Hypothesis Testing A statistical test uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected. The numerical value obtained from a statistical test is called the test value. The Test Value is compared to a Critical Value. We make a Decision. Bluman, Chapter 8

Test Value, Critical Value, and The Decision We’ll get a TEST VALUE from our sample. We’ll get a CRITICAL VALUE from the way we frame the experiment. Usually these are both z values or t values Compare those z values (or t values) If the Test Value is extreme, beyond the critical value(s), “Reject the Null Hypothesis” Bluman, Chapter 8

Test Value, Critical Value, and The Decision (p-value method) We’ll get a TEST VALUE from our sample. We’ll use the TEST VALUE to come up with a p VALUE, which is short for Probability Value. If the p Value is smaller than the experiment’s “Level of Significance”, we “Reject the Null Hypothesis” Bluman, Chapter 8

Hypothesis Testing In reality, the null hypothesis may or may not be true, and a decision is made to reject or not to reject it on the basis of the data obtained from a sample. A type I error occurs if one rejects the null hypothesis when it is true. A type II error occurs if one does not reject the null hypothesis when it is false. Bluman, Chapter 8

You get one of these 4 outcomes Bluman, Chapter 8

Four outcomes, illustrated. Bluman, Chapter 8

Four outcomes, illustrated. Bluman, Chapter 8

Example: You wonder if the cheaper store brand of detergent performs just as well as the more costly name brand detergent. Formulate hypotheses. Analyze the four possible outcomes * Whether the null hypothesis was really true * Whether it was rejected or not rejected Bluman, Chapter 8

Four outcomes, illustrated. Bluman, Chapter 8

Hypothesis Testing Likewise, The level of significance is the maximum probability of committing a type I error. This probability is symbolized by a (alpha). That is, P(type I error) = a. Likewise, P(type II error) = b (beta). Bluman, Chapter 8

Level of significance, 𝛼 Our problems will almost all have a Level Of Significance, designated by the Greek letter 𝛼 means strong evidence is required to reject a Null Hypothesis, it’s a higher standard of proof. In Intro Stats, 𝛽 gets only a brief mention. Bluman, Chapter 8

Level of significance, 𝛼 Larger 𝛼 means weaker evidence is required to reject a Null Hypothesis. Larger 𝛼 means a larger critical area means more likelihood that we’ll reject an H0. Bluman, Chapter 8

Hypothesis Testing Typical significance levels are: 0.10, 0.05, and 0.01 For example, when a = 0.10, there is a 10% chance of rejecting a true null hypothesis. Bluman, Chapter 8

Hypothesis Testing The critical value, C.V., separates the critical region from the noncritical region. The critical or rejection region is the range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected. The noncritical or nonrejection region is the range of values of the test value that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected. Bluman, Chapter 8

Hypothesis Testing Finding the Critical Value for α = 0.01 (Right-Tailed Test) z = 2.33 for α = 0.01 (Right-Tailed Test) Bluman, Chapter 8

TI-84 to find the Critical Value Finding the Critical Value for α = 0.01 (Right-Tailed Test) z = 2.33 for α = 0.01 (Right-Tailed Test) Bluman, Chapter 8

Hypothesis Testing Finding the Critical Value for α = 0.01 (Left-Tailed Test) z Because of symmetry, z = -2.33 for α = 0.01 (Left-Tailed Test) Bluman, Chapter 8

Hypothesis Testing Finding the Critical Value for α = 0.01 (Two-Tailed Test) z = ±2.58 Bluman, Chapter 8

TI-84 Critical Value for 𝛼=0.01, two-tailed test. Total area 1.0000 Area in two tails 0.0100 total Area in one tail is 0.0100 ÷ 2 = 0.0050 Three regions .0050 + .9900 + .0050 Bluman, Chapter 8

Finding the Critical Values for Specific α Values, Using Table E Procedure Table Finding the Critical Values for Specific α Values, Using Table E Step 1 Draw the figure and indicate the appropriate area. a. If the test is left-tailed, the critical region, with an area equal to α, will be on the left side of the mean. b. If the test is right-tailed, the critical region, with an area equal to α, will be on the right side of the mean. c. If the test is two-tailed, α must be divided by 2; one- half of the area will be to the right of the mean, and one-half will be to the left of the mean. Bluman, Chapter 8

Finding the Critical Values for Specific α Values, Using Table E Procedure Table Finding the Critical Values for Specific α Values, Using Table E Step 2 Find the z value in Table E. a. For a left-tailed test, use the z value that corresponds to the area equivalent to α in Table E. b. For a right-tailed test, use the z value that corresponds to the area equivalent to 1 – α. c. For a two-tailed test, use the z value that corresponds to α / 2 for the left value. It will be negative. For the right value, use the z value that corresponds to the area equivalent to 1 – α / 2. It will be positive. Bluman, Chapter 8

Procedure for Critical Value, TI-84 You still need to draw a normal curve You still need to label your diagram with the areas of the two or three regions. Use TI-84 invNorm(area to left) = the critical value. Two-tailed tests have two critical values that are opposites of each other. Bluman, Chapter 8

Example 8-2: Using Table E Using Table E in Appendix C, find the critical value(s) for each situation and draw the appropriate figure, showing the critical region. a. A left-tailed test with α = 0.10. Step 1 Draw the figure and indicate the appropriate area. Step 2 Find the area closest to 0.1000 in Table E. In this case, it is 0.1003. The z value is 1.28. Bluman, Chapter 8

Example 8-2: TI-84 method Using Table E in Appendix C, find the critical value(s) for each situation and draw the appropriate figure, showing the critical region. a. A left-tailed test with α = 0.10. Step 1 Draw the figure and indicate the appropriate area. Bluman, Chapter 8

Example 8-2: Using Table E Using Table E in Appendix C, find the critical value(s) for each situation and draw the appropriate figure, showing the critical region. b. A two-tailed test with α = 0.02. Step 1 Draw the figure with areas α /2 = 0.02/2 = 0.01. Step 2 Find the areas closest to 0.01 and 0.99. The areas are 0.0099 and 0.9901. The z values are -2.33 and 2.33. Bluman, Chapter 8

Example 8-2: TI-84 Method Using Table E in Appendix C, find the critical value(s) for each situation and draw the appropriate figure, showing the critical region. b. A two-tailed test with α = 0.02. Step 1 Draw the figure with areas α /2 = 0.02/2 = 0.01. Bluman, Chapter 8

Example 8-2: Using Table E Using Table E in Appendix C, find the critical value(s) for each situation and draw the appropriate figure, showing the critical region. c. A right-tailed test with α = 0.005. Step 1 Draw the figure and indicate the appropriate area. Step 2 Find the area closest to 1 – α = 0.995. There is a tie: 0.9949 and 0.9951. Average the z values of 2.57 and 2.58 to get 2.575 or 2.58. Bluman, Chapter 8

Example 8-2: TI-84 Method Using Table E in Appendix C, find the critical value(s) for each situation and draw the appropriate figure, showing the critical region. c. A right-tailed test with α = 0.005. Step 1 Draw the figure and indicate the appropriate area. Bluman, Chapter 8

Solving Hypothesis-Testing Problems Procedure Table Solving Hypothesis-Testing Problems (Traditional Method) Step 1 State the hypotheses and identify the claim. Step 2 Find the critical value(s) from the appropriate table in Appendix C. Step 3 Compute the test value. Step 4 Make the decision to reject or not reject the null hypothesis. Step 5 Summarize the results. Bluman, Chapter 8

Traditional and p-Value Two methods compared side by side State hypotheses, the null, H0, and the alt, H1. Find the critical value(s) (based on the 𝛼). Compute the test value. Make decision: compare test value to critical value(s). Results in plain English. State hypotheses, the null, H0, and the alt, H1. Compute the test value. Find p value for that test value. Make decision: compare p value to 𝛼 (or 𝛼/2). Results in plain English. Bluman, Chapter 8