Hypothesis Test _One-sample t-test, Z-test, Proportion Z-test
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This document discusses hypothesis testing concepts including the null and alternative hypotheses, type I and II errors, and the hypothesis testing process. It provides examples of hypothesis testing for a mean where the population standard deviation is known (z-test) and unknown (t-test). The document outlines the 6 steps in hypothesis testing and provides examples using both the critical value approach and p-value approach. It discusses the relationship between hypothesis testing and confidence intervals.
Hypothesis Test _One-sample t-test, Z-test, Proportion Z-test
- 1. MBMG-7104/ ITHS-2202/ IMAS-3101/ IMHS-3101 @Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
- 2. Learning Objectives In this chapter, you learn: The basic principles of hypothesis testing How to use hypothesis testing to test a mean or proportion The assumptions of each hypothesis-testing procedure, how to evaluate them, and the consequences if they are seriously violated How to avoid the pitfalls involved in hypothesis testing The ethical issues involved in hypothesis testing Chap 9-2
- 3. What is a Hypothesis? Hypothesis is a claim made by a person/organization. A hypothesis is a claim (assertion) about a population parameter such as mean or proportion and we seek evidence from a sample for the support of the claim. for example, claim could be that the average salary of analytics experts is at least USD 1,00,000. Hypothesis testing is a process used for either rejecting or retaining a null hypothesis Chap 9-3
- 4. Chap 9-4 What is a Hypothesis? Suppose : population mean population proportion Chap 9-4 Example: The mean monthly cell phone bill in this city is μ = $42 Example: The proportion of adults in this city with cell phones is π = 0.68
- 5. The Null Hypothesis, H0 States the claim or assertion to be tested Example: The average diameter of a manufactured bolt is 30mm ( ) Is always about a population parameter, not about a sample statistic Chap 9-5 30 μ : H0 30 μ : H0 30 X : H0
- 6. The Null Hypothesis, H0 Begin with the assumption that the null hypothesis is true Similar to the notion of innocent until proven guilty Refers to the status quo or historical value Null always contains “=“sign May or may not be rejected Chap 9-6 (continued)
- 7. The Alternative Hypothesis, H1 Is the opposite of the null hypothesis e.g., The average diameter of a manufactured bolt is not equal to 30mm ( H1: μ ≠ 30 ) Challenges the status quo Alternative never contains the “=”sign May or may not be proven Is generally the hypothesis that the researcher is trying to prove Chap 9-7
- 8. Chap 9-8 The Hypothesis Testing Process Claim: The population mean age is 50. H0: μ = 50, H1: μ ≠ 50 Sample the population and find sample mean. Chap 9-8 Population Sample
- 9. The Hypothesis Testing Process Suppose the sample mean age was X = 20. This is significantly lower than the claimed mean population age of 50. If the null hypothesis were true, the probability of getting such a different sample mean would be very small, so you reject the null hypothesis. In other words, getting a sample mean of 20 is so unlikely if the population mean was 50, you conclude that the population mean must not be 50. Chap 9-9 (continued)
- 10. The Hypothesis Testing Process Chap 9-10 Sampling Distribution of X μ = 50 If H0 is true If it is unlikely that you would get a sample mean of this value ... ... then you reject the null hypothesis that μ = 50. 20 ... When in fact this were the population mean… X (continued)
- 11. Chap 9-11 The Test Statistic and Critical Values If the sample mean is close to the stated population mean, the null hypothesis is not rejected. If the sample mean is far from the stated population mean, the null hypothesis is rejected. How far is “far enough” to reject H0? The critical value of a test statistic creates a “line in the sand” for decision making -- it answers the question of how far is far enough. Chap 9-11
- 12. Chap 9-12 The Test Statistic and Critical Values Chap 9-12 Critical Values “Too Far Away” From Mean of Sampling Distribution Sampling Distribution of the test statistic Region of Rejection Region of Rejection Region of Non-Rejection
- 13. Chap 9-13 Possible Errors in Hypothesis Test Decision Making Type I Error Reject a true null hypothesis Considered a serious type of error The probability of a Type I Error is Called level of significance of the test Set by researcher in advance Type II Error Failure to reject a false null hypothesis The probability of a Type II Error is β Chap 9-13
- 14. Chap 9-14 Possible Errors in Hypothesis Test Decision Making Possible Hypothesis Test Outcomes Actual Situation Decision H0 True H0 False Do Not Reject H0 No Error Probability 1 - α Type II Error Probability β Reject H0 Type I Error Probability α No Error Probability 1 - β Chap 9-14 (continued)
- 15. Chap 9-15 Possible Errors in Hypothesis Test Decision Making The confidence coefficient (1-α) is the probability of not rejecting H0 when it is true. The confidence level of a hypothesis test is (1-α)*100%. The power of a statistical test (1-β) is the probability of rejecting H0 when it is false. Chap 9-15 (continued)
- 16. Chap 9-16 Type I & II Error Relationship Chap 9-16 Type I and Type II errors cannot happen at the same time A Type I error can only occur if H0 is true A Type II error can only occur if H0 is false If Type I error probability () , then Type II error probability (β)
- 17. Chap 9-17 Factors Affecting Type II Error All else equal, β when the difference between hypothesized parameter and its true value β when β when σ β when n Chap 9-17
- 18. Chap 9-18 Level of Significance and the Rejection Region Chap 9-18 Level of significance = This is a two-tail test because there is a rejection region in both tails H0: μ = 30 H1: μ ≠ 30 Critical values Rejection Region /2 30 /2
- 19. Chap 9-19 Hypothesis Tests for the Mean Chap 9-19 Known Unknown Hypothesis Tests for (Z test) (t test)
- 20. Chap 9-20 Z Test of Hypothesis for the Mean (σ Known) Convert sample statistic ( ) to a ZSTAT test statistic Chap 9-20 X The test statistic is: n σ μ X ZSTAT σ Known σ Unknown Hypothesis Tests for Known Unknown (Z test) (t test)
- 21. Chap 9-21 Critical Value Approach to Testing For a two-tail test for the mean, σ known: Convert sample statistic ( ) to test statistic (ZSTAT) Determine the critical Z values for a specified level of significance from a table or computer Decision Rule: If the test statistic falls in the rejection region, reject H0 ; otherwise do not reject H0 Chap 9-21 X
- 22. Chap 9-22 Two-Tail Tests Chap 9-22 Do not reject H0 Reject H0 Reject H0 There are two cutoff values (critical values), defining the regions of rejection /2 -Zα/2 0 H0: μ = 30 H1: μ 30 +Zα/2 /2 Lower critical value Upper critical value 30 Z X
- 23. 6 Steps in Hypothesis Testing 1. State the null hypothesis, H0 and the alternative hypothesis, H1 2. Choose the level of significance, , and the sample size, n 3. Determine the appropriate test statistic and sampling distribution 4. Determine the critical values that divide the rejection and non-rejection regions Chap 9-23
- 24. 6 Steps in Hypothesis Testing 5. Collect data and compute the value of the test statistic 6. Make the statistical decision and state the managerial conclusion. If the test statistic falls into the nonrejection region, do not reject the null hypothesis H0. If the test statistic falls into the rejection region, reject the null hypothesis. Express the managerial conclusion in the context of the problem Chap 9-24 (continued)
- 25. Chap 9-25 Hypothesis Testing Example Chap 9-25 Test the claim that the true mean diameter of a manufactured bolt is 30mm. (Assume σ = 0.8) 1. State the appropriate null and alternative hypotheses H0: μ = 30 H1: μ ≠ 30 (This is a two-tail test) 2. Specify the desired level of significance and the sample size Suppose that = 0.05 and n = 100 are chosen for this test
- 26. Chap 9-26 Hypothesis Testing Example Chap 9-26 2.0 0.08 .16 100 0.8 30 29.84 n σ μ X ZSTAT 3. Determine the appropriate technique σ is assumed known so this is a Z test. 4. Determine the critical values For = 0.05 the critical Z values are ±1.96 5. Collect the data and compute the test statistic Suppose the sample results are n = 100, X = 29.84 (σ = 0.8 is assumed known) So the test statistic is: (continued)
- 27. Chap 9-27 6. Is the test statistic in the rejection region? Hypothesis Testing Example Chap 9-27 Reject H0 Do not reject H0 /2 = 0.025 -Zα/2 = -1.96 0 Reject H0 if ZSTAT < -1.96 or ZSTAT > 1.96; otherwise do not reject H0 (continued) /2 = 0.025 Reject H0 +Zα/2 = +1.96 Here, ZSTAT = -2.0 < -1.96, so the test statistic is in the rejection region
- 28. Chap 9-28 6 (continued). Reach a decision and interpret the result Hypothesis Testing Example Chap 9-28 -2.0 Since ZSTAT = -2.0 < -1.96, reject the null hypothesis and conclude there is sufficient evidence that the mean diameter of a manufactured bolt is not equal to 30 (continued) Reject H0 Do not reject H0 = 0.05/2 -Zα/2 = -1.96 0 = 0.05/2 Reject H0 +Zα/2= +1.96
- 29. Chap 9-29 p-Value Approach to Testing p-value: Probability of obtaining a test statistic equal to or more extreme than the observed sample value given H0 is true The p-value is also called the observed level of significance H0 can be rejected if the p-value is less than α Chap 9-29
- 30. Chap 9-30 p-Value Approach to Testing: Interpreting the p-value Compare the p-value with If p-value < , reject H0 If p-value , do not reject H0 Remember If the p-value is low then H0 must go Chap 9-30
- 31. The 5 Step p-value approach to Hypothesis Testing 1. State the null hypothesis, H0 and the alternative hypothesis, H1 2. Choose the level of significance, , and the sample size, n 3. Determine the appropriate test statistic and sampling distribution 4. Collect data and compute the value of the test statistic and the p-value 5. Make the statistical decision and state the managerial conclusion. If the p-value is < α then reject H0, otherwise do not reject H0. State the managerial conclusion in the context of the problem Chap 9-31
- 32. Chap 9-32 p-value Hypothesis Testing Example Chap 9-32 Test the claim that the true mean diameter of a manufactured bolt is 30mm. (Assume σ = 0.8) 1. State the appropriate null and alternative hypotheses H0: μ = 30 H1: μ ≠ 30 (This is a two-tail test) 2. Specify the desired level of significance and the sample size Suppose that = 0.05 and n = 100 are chosen for this test
- 33. Chap 9-33 p-value Hypothesis Testing Example Chap 9-33 2.0 0.08 .16 100 0.8 30 29.84 n σ μ X ZSTAT 3. Determine the appropriate technique σ is assumed known so this is a Z test. 4. Collect the data, compute the test statistic and the p-value Suppose the sample results are n = 100, X = 29.84 (σ = 0.8 is assumed known) So the test statistic is: (continued)
- 34. Chap 9-34 p-Value Hypothesis Testing Example: Calculating the p-value 4. (continued) Calculate the p-value. How likely is it to get a ZSTAT of -2 (or something farther from the mean (0), in either direction) if H0 is true? Chap 9-34 p-value = 0.0228 + 0.0228 = 0.0456 P(Z < -2.0) = 0.0228 0 -2.0 Z 2.0 P(Z > 2.0) = 0.0228
- 35. Chap 9-35 5. Is the p-value < α? Since p-value = 0.0456 < α = 0.05 Reject H0 5. (continued) State the managerial conclusion in the context of the situation. There is sufficient evidence to conclude the average diameter of a manufactured bolt is not equal to 30mm. p-value Hypothesis Testing Example Chap 9-35 (continued)
- 36. Chap 9-36 Connection Between Two-Tail Tests and Confidence Intervals For X = 29.84, σ = 0.8 and n = 100, the 95% confidence interval is: 29.6832 ≤ μ ≤ 29.9968 Since this interval does not contain the hypothesized mean (30), we reject the null hypothesis at = 0.05 100 0.8 (1.96) 29.84 to 100 0.8 (1.96) - 29.84
- 37. Chap 9-37 Do You Ever Truly Know σ? Probably not! In virtually all real world business situations, σ is not known. If there is a situation where σ is known then µ is also known (since to calculate σ you need to know µ.) If you truly know µ there would be no need to gather a sample to estimate it. Chap 9-37
- 38. Chap 9-38 Hypothesis Testing: σ Unknown If the population standard deviation is unknown, you instead use the sample standard deviation S. Because of this change, you use the t distribution instead of the Z distribution to test the null hypothesis about the mean. When using the t distribution you must assume the population you are sampling from follows a normal distribution. All other steps, concepts, and conclusions are the same. Chap 9-38
- 39. Chap 9-39 t Test of Hypothesis for the Mean (σ Unknown) Chap 9-39 The test statistic is: Hypothesis Tests for σ Known σ Unknown Known Unknown (Z test) (t test) Convert sample statistic ( ) to a tSTAT test statistic The test statistic is: Hypothesis Tests for σ Known σ Unknown Known Unknown (Z test) (t test) X The test statistic is: n S μ X tSTAT Hypothesis Tests for σ Known σ Unknown Known Unknown (Z test) (t test)
- 40. Chap 9-40 Example: Two-Tail Test ( Unknown) The average cost of a hotel room in New York is said to be $168 per night. To determine if this is true, a random sample of 25 hotels is taken and resulted in an X of $172.50 and an S of $15.40. Test the appropriate hypotheses at = 0.05. (Assume the population distribution is normal) Chap 9-40 H0: ______ H1: ______
- 41. Chap 9-41 = 0.05 n = 25, df = 25- 1=24 is unknown, so use a t statistic Critical Value: ±t24,0.025 = ± 2.0634 Example Solution: Two-Tail t Test Chap 9-41 Do not reject H0: insufficient evidence that true mean cost is different from $168 Reject H0 Reject H0 /2=.025 -t 24,0.025 Do not reject H0 0 /2=.025 -2.0639 2.0639 1.46 25 15.40 168 172.50 n S μ X STAT t 1.46 H0: μ = 168 H1: μ 168 t 24,0.025
- 42. Chap 9-42 Example Two-Tail t Test Using A p- value from Excel Since this is a t-test we cannot calculate the p- value without some calculation aid. The Excel output below does this: t Test for the Hypothesis of the Mean Null Hypothesis µ= 168.00 $ Level of Significance 0.05 Sample Size 25 Sample Mean 172.50 $ Sample Standard Deviation 15.40 $ Standard Error of the Mean 3.08 $ =B8/SQRT(B6) Degrees of Freedom 24 =B6-1 t test statistic 1.46 =(B7-B4)/B11 Lower Critical Value -2.0639 =-TINV(B5,B12) Upper Critical Value 2.0639 =TINV(B5,B12) p-value 0.157 =TDIST(ABS(B13),B12,2) =IF(B18<B5, "Reject null hypothesis", "Do not reject null hypothesis") Data Intermediate Calculations Two-Tail Test Do Not Reject Null Hypothesis Chap 9-42 p-value > α So do not reject H0
- 45. Exercise 3.
- 47. Chap 9-47 5.
- 48. Hypothesis Tests for Proportions Involves categorical variables Two possible outcomes Possesses characteristic of interest (probability p) Does not possess characteristic of interest (1-p) The hypothesis testing for population proportion based on one sample is also known as one-sample test for proportion Fraction or proportion of the population in the category of interest is denoted by π Chap 9-48
- 49. Hypothesis Tests for Proportions According to the central limit theorem of proportions, the sampling distribution of proportions 𝑝 for a large sample follows an approximate normal distribution with mean π (the population proportion) and standard deviation π (1−π) 𝑛 . Thus , the Z-statistic is defined as : 𝑍 = 𝑝 − π π (1 − π) 𝑛 It will follow a standard normal distribution. Chap 9-49
- 50. Proportions Sample proportion in the category of interest is denoted by p When both X and n – X are at least 5, p can be approximated by a normal distribution with mean and standard deviation Chap 9-50 size sample sample in interest of category in number n X p̂ p μ n ) (1 σ p
- 51. The sampling distribution of p is approximately normal, so the test statistic is a ZSTAT value: Hypothesis Tests for Proportions Chap 9-51 n ) (1 p̂ ZSTAT π π π X 5 and n – X 5 Hypothesis Tests for p X < 5 or n – X < 5 Not discussed in this chapter
- 52. An equivalent form to the last slide, but in terms of the number in the category of interest, X: Z Test for Proportion in Terms of Number in Category of Interest Chap 9-52 ) (1 n n X ZSTAT X 5 and n-X 5 Hypothesis Tests for X X < 5 or n-X < 5 Not discussed in this chapter
- 53. Example: Z Test for Proportion A marketing company claims that it receives responses from 8% of those surveyed. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the = 0.05 significance level. Chap 9-53 Check: X = 25 n-X = 475
- 54. Chap 9-54 Z Test for Proportion: Solution = 0.05 n = 500, p = 0.05 Chap 9-54 Reject H0 at = 0.05 H0: π = 0.08 H1: π 0.08 Critical Values: ± 1.96 Test Statistic: Decision: Conclusion: z 0 Reject Reject .025 .025 1.96 -2.47 There is sufficient evidence to reject the company’s claim of 8% response rate. 2.47 500 .08) .08(1 .08 .05 n ) (1 p̂ ZSTAT -1.96
- 55. Calculate the p-value and compare to (For a two-tail test the p-value is always two-tail) p-Value Solution Do not reject H0 Reject H0 Reject H0 /2 = .025 1.96 0 Z = -2.47 (continued) 0.0136 2(0.0068) 2.47) P(Z 2.47) P(Z p-value = 0.0136: Reject H0 since p-value = 0.0136 < = 0.05 Z = 2.47 -1.96 /2 = .025 0.0068 0.0068 Calculation of the corresponding p-value, In Excel, NORM.S.DIST(+-Zstat)
- 56. Chap 9-56
- 57. Chap 9-57 One-Tail Tests In many cases, the alternative hypothesis focuses on a particular direction Chap 9-57 H0: μ ≥ 3 H1: μ < 3 H0: μ ≤ 3 H1: μ > 3 This is a lower-tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 This is an upper-tail test since the alternative hypothesis is focused on the upper tail above the mean of 3
- 58. Chap 9-58 Lower-Tail Tests Chap 9-58 Reject H0 Do not reject H0 There is only one critical value, since the rejection area is in only one tail -Zα or -tα 0 μ H0: μ ≥ 3 H1: μ < 3 Z or t X Critical value
- 59. Chap 9-59 Upper-Tail Tests Chap 9-59 Reject H0 Do not reject H0 Zα or tα 0 μ H0: μ ≤ 3 H1: μ > 3 There is only one critical value, since the rejection area is in only one tail Critical value Z or t X _
- 60. Chap 9-60 Example: Upper-Tail t Test for Mean ( unknown) A phone industry manager thinks that customer monthly cell phone bills have increased, and now average over $52 per month. The company wishes to test this claim. (Assume a normal population) Chap 9-60 H0: μ ≤ 52 the average is not over $52 per month H1: μ > 52 the average is greater than $52 per month (i.e., sufficient evidence exists to support the manager’s claim) Form hypothesis test:
- 61. Suppose that = 0.10 is chosen for this test and n = 25. Find the rejection region: Chap 9-61 Reject H0 Do not reject H0 = 0.10 1.318 0 Reject H0 Reject H0 if tSTAT > 1.318 Example: Find Rejection Region (continued)
- 62. Obtain sample and compute the test statistic Suppose a sample is taken with the following results: n = 25, X = 53.1, and S = 10 Then the test statistic is: Example: Test Statistic 0.55 25 10 52 53.1 n S μ X tSTAT (continued)
- 63. Chap 9-63 Example: Decision Reach a decision and interpret the result: Chap 9-63 Reject H0 Do not reject H0 = 0.10 1.318 0 Reject H0 Do not reject H0 since tSTAT = 0.55 ≤ 1.318 There is insufficient evidence that the mean bill is over $52. tSTAT = 0.55 (continued)
- 64. Example: Utilizing The p- value for The Test Calculate the p-value and compare to (p-value below calculated using Excel spreadsheet on next page) Chap 9-64 Reject H0 = .10 Do not reject H0 1.318 0 Reject H0 tSTAT = .55 p-value = .2937 Do not reject H0 since p-value = .2937 > = .10
- 65. Excel Spreadsheet Calculating The p- value for The Upper Tail t Test t Test for the Hypothesis of the Mean Null Hypothesis µ= 52.00 Level of Significance 0.1 Sample Size 25 Sample Mean 53.10 Sample Standard Deviation 10.00 Standard Error of the Mean 2.00 =B8/SQRT(B6) Degrees of Freedom 24 =B6-1 t test statistic 0.55 =(B7-B4)/B11 Upper Critical Value 1.318 =TINV(2*B5,B12) p-value 0.2937 =TDIST(ABS(B13),B12,1) =IF(B18<B5, "Reject null hypothesis", "Do not reject null hypothesis") Data Intermediate Calculations Upper Tail Test Do Not Reject Null Hypothesis Chap 9-65
- 66. Chap 9-66 Possible Errors in Hypothesis Test Decision Making Type I Error Reject a true null hypothesis Considered a serious type of error The probability of a Type I Error is Called level of significance of the test Set by researcher in advance Type II Error Failure to reject a false null hypothesis The probability of a Type II Error is β Chap 9-66
- 67. Chap 9-67 Possible Errors in Hypothesis Test Decision Making Possible Hypothesis Test Outcomes Actual Situation Decision H0 True H0 False Do Not Reject H0 No Error Probability 1 - α Type II Error Probability β Reject H0 Type I Error Probability α No Error Probability 1 - β Chap 9-67 (continued)
- 68. Chap 9-68 Chapter Summary Addressed hypothesis testing methodology Performed Z Test for the mean (σ known) Discussed critical value and p–value approaches to hypothesis testing Performed one-tail and two-tail tests Chap 9-68
- 69. Chap 9-69 Chapter Summary Performed t test for the mean (σ unknown) Performed Z test for the proportion Chap 9-69 (continued)