1. Hypothesis Testing

2. Terminology Hypothesis Testing: A decision making process for evaluating claims about a population. Every situation begins with a statement of a hypothesis. Statistical Hypothesis: A conjecture about a population parameter. The conjecture may or may not be true. Two types of statistical hypotheses: Null hypothesis and Alternative hypothesis. Null: Symbolized by H ₀, is a statistical hypothesis that states that there is no difference between parameter and a specific value. Alternative: Symbolized by H ₁, is statistical hypothesis that states a specific difference between parameter and a specific value.

3. Stating Hypotheses

6. Your Turn State the null and alternative hypotheses for each conjecture: #1. The average income of a CEO is $85,500. #2. A SHS teacher hypothesizes that grades will increase on a student’s SAT score if they enroll in a prep class. The average SAT score of students enrolled in a SAT Prep class is less than 1200. #3. A homeowner hypothesizes that if his township hires a land surveyor he might be able to lower his yearly flood insurance premium. The average amount of flood insurance paid in his neighborhood exceeds $3,200 a year.

7. Designing the Study After the hypotheses are formulated the researcher now selects the correct statistical test, chooses an appropriate level of significance and formulates a plan for conducting the study. Using first example: A group of patients using ADHD drug are selected and after a few weeks their weights are measured again. If the mean weight turns out to be 178 pounds then researcher may conclude difference due to chance and not reject null hypothesis. If avg. is 185 pounds then researcher would conclude it does increase weight and reject the null hypothesis. Question is now “Where do you draw the line?”

8. Statistical Test Statistical test: Uses the data obtained from a sample to make a decision about whether or not to reject the null hypothesis. Test value: The numerical value obtained from the statistical test. In this type of test the mean is computed from the sample and compared with the population mean. A decision is then made to keep the null hypothesis or reject it.

9. Types of Errors The are four possible outcomes from the testing situation. You will notice that there are 2 correct possibilities and 2 incorrect possibilities.

10. Types of Errors Type 1 Error: Occurs if one rejects the null hypothesis and it is true. Type 2 Error: Occurs if one does not reject the null hypothesis and it is not true. In our example: The medication may not change the weight of all users in the population but it might, by chance, change the weight in the sample causing the researcher to reject the null when it’s true. This is type 1 error. On the other hand the meds might not change the weight much for the sample but when given to population it might cause significant increases or decreases in weight. The researcher therefore does not reject the null and this is considered a type 2 error. Decision to reject or not reject does not prove anything. The decision is made based on probability. So when there is a large difference between parameters then null hypothesis is probably not true. So now the question is how large a difference is necessary to reject it?

11. Level of Significance

12. Critical Value Critical Value: Separates the critical region from the noncritical region. The symbol for critical value is C.V. The critical or rejection region: 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 non rejection region: 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. The critical values can be on the right side, left side, or both sides of the mean depending on the inequality sign of the alternative hypothesis.

13. One and Two-Tailed Tests One tailed test: Indicates that the null hypothesis should be rejected when the test value is in the critical region on one side of the mean. Depending on direction of inequality symbol they are classified as right-tailed or left-tailed. Two-tailed test: The null hypothesis should be rejected when the test value is in either of the critical regions.

14. How to find C.V.

15. C.V. Examples

16. High Five

17. The Z-Test

18. Z-Test Example Going to follow the 5 steps that were mentioned in previous slide to solve problem. #1: A researcher reports that the average yearly income for a firefighter is more than $44,000. A sample of 40 firefighters has a mean of $45,650. At ∝ =.05 test the claim that the firefighters earn more than $44,000 a year. The standard deviation is $5,100. Step 1: State hypotheses and identify the claim. Step 2: Find critical value/s. Step 3: Compute test value by using Z-Test. Step 4: Make decision to reject or to not reject the null. Step 5: Summarize results.

19. Z-Test Example #2: A newspaper claims that the average college student watches less television than the general public. The national average is 25.7 hours per week with a standard deviation of 3 hours. A sample of 30 teenagers has a mean of 28 hours. Is there enough evidence to support the claim at ∝ =.01? Step 1: State hypotheses and identify the claim. Step 2: Find critical value/s. Step 3: Compute test value by using Z-Test. Step 4: Make decision to reject or to not reject the null. Step 5: Summarize results.

20. Your Turn #3: A shoe-store manager claims that the average price for a pair of sneakers is $89.95. A sample of 68 sneakers has an average of $90.26. The standard deviation of the sample is $3.00. At ∝ =.05, is there enough evidence to reject the manager’s claim.

21. Outcomes of Hypothesis Testing If the claim is the null hypothesis there are 2 possibilities: #1: If rejected: There is enough evidence to reject the claim. #2 If not rejected: There is not enough evidence to reject the claim. If the claim is the alternative hypothesis there are 2 possibilities: #1 If null is rejected than there is enough evidence to support the claim. #2 If null is not rejected than there is not enough evidence to support the claim.

22. P-Values

23. Steps to finding P - Values Step 1: State the hypotheses and identify the claim. Step 2: Compute the test value for the problem. Step 3: Use Table to find corresponding area for the z-value. Step 4: Subtract the area from.5000 to find area in the correct tail. This step determines the P – Value. Step 5: Make a decision to reject the null hypothesis or not reject it. Step 6: Summarize the results.

24. P – Value Examples A researcher wishes to test the claim that the average age of doctors in a certain city is 37 years. She selects a sample of 40 doctors and finds the mean to be 37.8 years with a standard deviation of 2 years. Is there evidence to support the claim at ∝ = 0.05? Find the P-Value. Step 1: State the hypotheses and identify the claim. Step 2: Compute the test value for the problem. Step 3: Use Table to find corresponding area for the z- value. Step 4: Subtract the area from.5000 to find area in the correct tail. This step determines the P – Value. Step 5: Make a decision to reject the null hypothesis or not reject it. Step 6: Summarize the results.

25. Your Turn A college professor claims that the average cost of a book is greater than $27.50. A sample of 50 books has an average of $29.30. Standard deviation of the sample is $5.00. Find the P – Value for the test. On the basis of the P – Value should the null be rejected at ∝ = 0.05?

26. ∝ Value vs. P-Value There is a clear distinction between ∝ value and P – Value. ∝ value: chosen before the statistical test is conducted. P – Value: Is computed after the sample mean has been found. 2 schools of thought on P – Values. – 1. Some researchers do not choose an ∝ value but report the P – Value and allow the reader to decide whether to reject the null or not. – 2. Others decide on the ∝ value in advance and use the P – Value to make the decision.

27. The T- test

28. Finding critical t value Step 1: Find the correct ∝ using the top row and determine if the test is a one-tailed test or a two-tailed test. Step 2: Determine the d.f and where they intersect each other that become the critical value for the problem. If test is right-tailed answer is positive, if left- tailed answer is negative, if two-tailed answer is both positive and negative.

29. T – test examples Going to use same procedures as z – test. Example 1: A machine is supposed to fill jars with 16 ounces of coffee. A consumer suspects that the machine is not filling the jars completely. A sample of 8 jars has mean of 15.6 ounces and a standard deviation of 0.3 ounces. Is there enough evidence to support the claim at ∝ = 0.10?

30. T – test example Example 2: A physician claims that a joggers’ maximal volume oxygen uptake is greater than the average of all adults. A sample of 15 joggers has a mean of 43.6 milliliters per kilogram and a standard deviation of 6 ml/kg. If the average of all adults is 36.7 ml/kg, is there enough evidence to support the physician’s claim at ∝ = 0.01?

31. Your turn The average amount of rainfall for the Northeast, during the summer, is 11.52 inches. A researcher selects a random sample of 10 cities and finds that the average amount of rainfall was 7.42 inches. The standard deviation for the sample is 1.3 inches. At ∝ = 0.05, can it be concluded that for the summer the mean rainfall was below 11.52 inches?

32. P – Values for t - tests Are computed the same way as they are in the z-test but with a twist. Table F gives t – values for only selected value of ∝. To compute the exact P – Values for a t- test one would need a table similar to Table E for each d.f Suppose test value is 2.056 for a sample size of 11. To get the P – Value you need to look in Table F across the row with d.f. 10 and find 2.056 falls between 1.812 and 2.228. Since it is a one-tailed test look at row labeled “One tail, ∝ ” and 2.056 falls between 0.05 and 0.25. Therefore the P – Value would be contained in the interval 0.025 < P – Value <0.05.

33. The Proportion Test

34. Steps for Proportion Problems

35. Proportion Test Examples

36. Your turn A statistician read that at least 77% of the population oppose replacing the $1 bills with $1 coins. To see if the claim is valid, he selected a sample of 80 people and found that 55 were opposed to replacing the $1 bills. At ∝ = 0.01, can it be concluded that at least 77% of the population are opposed to the change?

37. Variance or Standard Deviation Test In Chapter 8 we used the Chi-Square distribution to construct confidence intervals for a single variance or standard deviation. It can also be used to test a claim about them as well. 3 cases to consider for these types of problems. Finding the chi-square critical value for a specific ∝ when they hypothesis test is right-tailed. Finding the chi-square critical value for a specific ∝ when the hypothesis test is left-tailed. Finding the chi-square critical value for a specific ∝ when the hypothesis test is two-tailed.

38. Chi-Square C.V. for Right-Tailed Test Example: Find the critical chi-square value for 16 d.f. when ∝ = 0.05. Find the ∝ value at the top of the Table and find the d.f. along the left column. Where they intersect becomes our C.V. What is it for our problem?

39. Chi-Square C.V. for Left-Tailed Test When we have a left tailed test we are going to have to subtract the ∝ from 1. The reason for this is because the Chi-Square Table gives the area to the right of the C.V. and the Chi-Square statistic can’t be negative. Example: Find the critical chi-square value for 11 d.f. when ∝ = 0.01. Subtract 1 – 0.01 = 0.99, which is the area to the right of the C.V. Now we look for.99 on top of table and the d.f. = 11 on the side. Intersection becomes our C.V. What is our C.V.?

40. Chi-Square C.V. for Two-Tailed Test When a two-tailed test is conducted the area must be split. Example: Find the critical chi-square value for 18 d.f. when ∝ = 0.10. Therefore the area to the right of the larger value is 0.05 and the area to the right of the smaller value is 0.95 (1 – 0.05). Now look at intersections of d.f. = 18 and 0.05 and 0.95. What are our C.V.?

41. Larger than 30? After d.f. reach 30, the table only gives values at multiples of 10 (40,50,60, etc.) When d.f. are not specifically given we will use the closest SMALLER value. Example: d.f. = 47. We will then use the d.f. = 40.

42. Chi-Square Test for a Single Variance

43. Why test Variances or Standard Deviations? A couple of reasons: 1.In any situation, where consistency is required, one would like to have the smallest variation possible in the products. Example: If we manufacture car parts we need to make sure that the holes vary in diameter minimally or other pieces may not come together properly. 2.In education if a test is given to a group of students, to check for understanding of the subject, the overall deviation should be large so that we can determine which students have learned the subject and which students have not.

44. Variance or Standard Deviation Examples Going to use the same 5 steps we have been using for hypothesis testing. Example 1: An instructor wishes to see if the variation in scores of the 23 students in her class is less than the variance of the population. The variance of the class is 198. Is there enough evidence to support the claim that the variation of the students is less than the population variance of 225 at ∝ = 0.05? Assume scores are normally distributed.

45. Variance or Standard Deviation Examples Example 2: A medical researcher believes that the standard deviation of the temperatures of newborn infants is greater than 0.6 ℉. A sample of 15 infants was found to have a standard deviation of 0.8 ℉. At ∝ = 0.10, does the evidence support the researcher’s belief? Assume that the variable is normally distributed.

46. Your turn A manufacturer claims that the standard deviation of the strength of wrapping cord is 9 pounds. A sample of 10 wrapping cords produced a standard deviation of 11 pounds. Can it be concluded that at ∝ = 0.05, the claim is correct?