Hypothesis Testing Variance known Sampling Distribution n Overthecounter
Hypothesis Testing Variance known?
Sampling Distribution n Over-the-counter stock selling prices • calculate average price of all stocks listed [ ] n take a sample of 25 stocks and record price • calculate average price of the 25 stocks [x-bar] n take all possible samples of size 25 • would all x-bars be equal? n average all the possible x-bars …equals
Sampling Distribution 20 H Levine, Prentice-Hall 0
Sampling Distribution It is unlikely that we would get a sample mean of this value. . . 20 H Levine, Prentice-Hall 0
Sampling Distribution It is unlikely that we would get a sample mean of this value. . . if in fact this were the population mean 20 H Levine, Prentice-Hall 0
Sampling Distribution It is unlikely that we would get a sample mean of this value. . . therefore, we reject the hypothesis that = 50. . if in fact this were the population mean 20 H Levine, Prentice-Hall 0
Null Hypothesis n What is tested n Always has equality sign: , or n Designated H 0 • Example ………. . . H 0: 3 Levine, Prentice-Hall
Alternative Hypothesis n Opposite of null hypothesis n Always has inequality sign: , , or n Designated H 1 n Example • H 1: < 3 Levine, Prentice-Hall
Decision n Reject null hypothesis n Retain, or, fail to reject, null hypothesis n Do not use the term “accept” Levine, Prentice-Hall
p-value n Probability of obtaining a test statistic more extreme ( or than actual sample value given H 0 is true n Called observed level of significance • n Smallest value of H 0 can be rejected Used to make rejection decision • If p-value , reject H 0 Levine, Prentice-Hall
Level of Significance n Defines unlikely values of sample statistic if null hypothesis is true • n Designated (alpha) • n Called rejection region of sampling distribution Typical values are. 01, . 05, . 10 Selected by researcher at start Levine, Prentice-Hall
Rejection Region (one-tail test) Sampling Distribution Level of Confidence 1 - Levine, Prentice-Hall
Rejection Region (one-tail test) Sampling Distribution Level of Confidence 1 - Observed sample statistic Levine, Prentice-Hall
Rejection Region (one-tail test) Sampling Distribution Level of Confidence 1 - Observed sample statistic Levine, Prentice-Hall
Rejection Regions (two-tailed test) Sampling Distribution Level of Confidence 1 - Levine, Prentice-Hall
Rejection Regions (two-tailed test) Sampling Distribution Level of Confidence 1 - Levine, Prentice-Hall Observed sample statistic
Rejection Regions (two-tailed test) Sampling Distribution Level of Confidence 1 - Observed Levine, Prentice-Hall sample statistic
Rejection Regions (two-tailed test) Sampling Distribution Level of Confidence 1 - Observed sample statistic Levine, Prentice-Hall
Risk of Errors in Making Decision n Type I error • • • n Reject true null hypothesis Has serious consequences Probability of Type I error is alpha [ ] – Called level of significance Type II error • • Do not reject false null hypothesis Probability of Type II error is beta [ ] Levine, Prentice-Hall
Decision Results H 0: Innocent Levine, Prentice-Hall
Hypothesis Testing n State H 0 n State H 1 n Choose n n Choose test Levine, Prentice-Hall
Hypothesis Testing n State H 0 n Set up critical values n State H 1 n Collect data n Choose n Compute test statistic n Choose n n Make statistical decision n Choose test n Express decision Levine, Prentice-Hall
Two-tailed z-test Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes has an average weight = 372. 5 grams. The company has specified to be 15 grams. Test at the. 05 level. 368 gm. Levine, Prentice-Hall
Two-tailed z-test Test Statistic: H 0: H 1: n Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
Two-tailed z-test Test Statistic: H 0: = 368 H 1: 368 n Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
Two-tailed z-test H 0: = 368 H 1: 368 . 05 n 25 Critical Value(s): Test Statistic: Decision: Conclusion: Levine, Prentice-Hall
Two-tailed z-test H 0: = 368 H 1: 368 . 05 n 25 Critical Value(s): Test Statistic: Decision: Conclusion: Levine, Prentice-Hall
Two-tailed z-test H 0: = 368 H 1: 368 . 05 n 25 Critical Value(s): Test Statistic: Decision: Conclusion: Levine, Prentice-Hall
Two-tailed z-test H 0: = 368 H 1: 368 . 05 n 25 Critical Value(s): Test Statistic: Decision: Do not reject at =. 05 Conclusion: Levine, Prentice-Hall
Two-tailed z-test H 0: = 368 H 1: 368 . 05 n 25 Critical Value(s): Test Statistic: Decision: Do not reject at =. 05 Conclusion: No evidence average is not 368 Levine, Prentice-Hall
Two-tailed z-test [p-value]] Levine, Prentice-Hall Z value of sample statistic
Two-tailed z-test [p-value] p-value is P(z -1. 50 or z 1. 50) Levine, Prentice-Hall Z value of sample statistic
Two-tailed z-test [p-value] p-value is P(z -1. 50 or z 1. 50) Levine, Prentice-Hall Z value of sample statistic
Two-tailed z-test [p-value] p-value is P(z -1. 50 or z 1. 50) . 4332 From Z table: lookup 1. 50 Levine, Prentice-Hall Z value of sample statistic
Two-tailed z-test [p-value] p-value is P(z -1. 50 or z 1. 50) . 4332 From Z table: lookup 1. 50 Levine, Prentice-Hall . 5000 -. 4332. 0668 Z value of sample statistic
Two-tailed z-test [p-value] p-value is P(z -1. 50 or z 1. 50) =. 1336 . 4332 From Z table: lookup 1. 50 Levine, Prentice-Hall . 5000 -. 4332. 0668 Z value of sample statistic
Two-tailed z-test [p-value] 1/2 p-value =. 0668 1/2 =. 025 Levine, Prentice-Hall
Two-tailed z-test [p-value] (p-Value =. 1336) ( =. 05) 1/2 p-Value =. 0668 Do not reject. 1/2 p-Value =. 0668 1/2 =. 025 Test statistic is in ‘Do not reject’ region Levine, Prentice-Hall
Two-tailed z-test ( known) challenge You are a Q/C inspector. You want to find out if a new machine is making electrical cords to customer specification: average breaking strength of 70 lb. with = 3. 5 lb. You take a sample of 36 cords & compute a sample mean of 69. 7 lb. At the. 05 level, is there evidence that the machine is not meeting the average breaking strength? Levine, Prentice-Hall
Two-tailed z-test ( known) Test Statistic: H 0: H 1: = n= Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
Two-tailed z-test ( known) Test Statistic: H 0: = 70 H 1: 70 = n= Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
Two-tailed z-test ( known) Test Statistic: H 0: = 70 H 1: 70 =. 05 n = 36 Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
Two-tailed z-test ( known) Test Statistic: H 0: = 70 H 1: 70 =. 05 n = 36 Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
Two-tailed z-test ( known) Test Statistic: H 0: = 70 H 1: 70 =. 05 n = 36 Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
Two-tailed z-test ( known) Test Statistic: H 0: = 70 H 1: 70 =. 05 n = 36 Critical Value(s): Decision: Do not reject at =. 05 Conclusion: Levine, Prentice-Hall
Two-tailed z-test ( known) Test Statistic: H 0: = 70 H 1: 70 =. 05 n = 36 Critical Value(s): Decision: Do not reject at =. 05 Conclusion: No evidence average is not 70 Levine, Prentice-Hall
One-tailed z-test ( known) n Assumptions • • Population is normally distributed If not normal, can be approximated by normal distribution for large samples Levine, Prentice-Hall
One-tailed z-test ( known) n Assumptions • • n Population is normally distributed If not normal, can be approximated by normal distribution for large samples Null hypothesis has or sign only Levine, Prentice-Hall
One-tailed z-test ( known) n Assumptions • • n n Population is normally distributed If not normal, can be approximated by normal distribution for large samples Null hypothesis has or sign only Z-test statistic Levine, Prentice-Hall
One-tailed z-test ( known) H 0: 0 H 1: < 0 Must be significantly below Levine, Prentice-Hall
One-tailed z-test ( known) H 0: 0 H 1: < 0 H 0: 0 H 1: > 0 Must be significantly below Levine, Prentice-Hall Small values satisfy H 0. Do not reject!
One-tailed z-test ( known) What is “z” given =. 025? =. 025 Levine, Prentice-Hall
One-tailed z-test ( known) What Is Z given =. 025? . 500 -. 025. 475 =. 025 Levine, Prentice-Hall
One-tailed z-test ( known) What is “z” given =. 025? . 500 -. 025. 475 Standardized Normal Probability Table (Portion) . 06 =. 025 1. 9 Levine, Prentice-Hall . 4750
One-tailed z-test ( known) What Is Z given =. 025? . 500 -. 025. 475 Standardized Normal Probability Table (Portion) . 06 =. 025 1. 9 Levine, Prentice-Hall . 4750
One-tailed z-test ( known) Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed X = 372. 5. The company has specified to be 15 grams. Test at the. 05 level. Levine, Prentice-Hall 368 gm.
One-tailed z-test ( known) Test Statistic: H 0: H 1: = n= Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
One-tailed z-test ( known) Test Statistic: H 0: 368 H 1: > 368 = n= Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
One-tailed z-test ( known) Test Statistic: H 0: 368 H 1: > 368 =. 05 n = 25 Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
One-tailed z-test ( known) Test Statistic: H 0: 368 H 1: > 368 =. 05 n = 25 Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
One-tailed z-test ( known) Test Statistic: H 0: 368 H 1: > 368 =. 05 n = 25 Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
One-tailed z-test ( known) Test Statistic: H 0: 368 H 1: > 368 =. 05 n = 25 Critical Value(s): Decision: Do not reject at =. 05 Conclusion: Levine, Prentice-Hall
One-tailed z-test ( known) Test Statistic: H 0: 368 H 1: > 368 =. 05 n = 25 Critical Value(s): Decision: Do not reject at =. 05 Conclusion: No evidence average is more than 368 Levine, Prentice-Hall
One-tailed z-test ( known) p-value Solution Use alternative hypothesis to find direction Levine, Prentice-Hall Z value of sample statistic
One-tailed z-test ( known) p-value Use alternative hypothesis to find direction Levine, Prentice-Hall Z value of sample statistic
One-tailed z-test ( known) p-value Use alternative hypothesis to find direction . 4332 From Z table: lookup 1. 50 Levine, Prentice-Hall Z value of sample statistic
One-tailed z-test ( known) p-value Use alternative hypothesis to find direction . 4332 From Z table: lookup 1. 50 Levine, Prentice-Hall . 5000 -. 4332. 0668 Z value of sample statistic
One-tailed z-test ( known) p-value =. 0668 Use alternative hypothesis to find direction . 4332 From Z table: lookup 1. 50 Levine, Prentice-Hall . 5000 -. 4332. 0668 Z value of sample statistic
One-tailed z-test ( known) p-value =. 0668 =. 05 Levine, Prentice-Hall
One-tailed z-test ( known) p-value (p-value =. 0668) ( =. 05). Do not reject. p-Value =. 0668 =. 05 Test statistic is in ‘Fail to reject’ region Levine, Prentice-Hall
p-value Challenge You’re an analyst for Ford. You want to find out if the average miles per gallon of Escorts is at least 32 mpg. Similar models have a standard deviation of 3. 8 mpg. You take a sample of 60 Escorts & compute a sample mean of 30. 7 mpg. What is the value of the observed level of significance (p-Value)? Levine, Prentice-Hall
p-value =. 004 p-value < ( =. 01) Reject H 0. Use alternative hypothesis to find direction . 5000 -. 4960. 0040 . 4960 Z value of sample statistic Levine, Prentice-Hall From Z table: lookup 2. 645
p-value n Probability of obtaining a test statistic more extreme ( or than actual sample value given H 0 is true n Called observed level of significance • n Smallest value of H 0 can be rejected Used to make rejection decision • If p-value , reject H 0 Levine, Prentice-Hall
One-tailed t-test ( unknown) Does an average box of cereal contain less than the 368 grams indicated on the package? A random sample of 25 boxes showed X = 363. 5 and s=15. Test at the. 05 level. 368 gr. Levine, Prentice-Hall
One-tailed t-test ( unknown) Test Statistic: H 0: H 1: = n= Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
One-tailed t-test ( unknown) Test Statistic: H 0: 368 H 1: < 368 = n= Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
One-tailed t-test ( unknown) Test Statistic: H 0: 368 H 1: < 368 =. 05 n = 25, d. f. = 24 Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
One-tailed t-test ( unknown) Test Statistic: H 0: 368 H 1: < 368 =. 05 n = 25, d. f. = 24 Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
One-tailed t-test ( unknown) Test Statistic: H 0: 368 H 1: < 368 =. 05 n = 25, d. f. = 24 Critical Value(s): Decision: Conclusion: Levine, Prentice-Hall
One-tailed t-test ( unknown) Test Statistic: H 0: 368 H 1: < 368 =. 05 n = 25, d. f. = 24 Critical Value(s): Decision: Do not reject at =. 05 Conclusion: Levine, Prentice-Hall
One-tailed t-test ( unknown) Test Statistic: H 0: 368 H 1: < 368 =. 05 n = 25, d. f. = 24 Critical Value(s): Decision: Do not reject at =. 05 Conclusion: No evidence average is less than 368 Levine, Prentice-Hall
One-tailed t-test ( unknown) p-value Solution t value of sample statistic Levine, Prentice-Hall Use alternative hypothesis to find direction
One-tailed t-test ( unknown) p-value From t table: lookup -1. 50 for 24 d. f. P-value = 0. 075 t value of sample statistic Levine, Prentice-Hall Use alternative hypothesis to find direction
One-tailed t-test ( unknown) p-value =. 075 =. 05 Levine, Prentice-Hall
One-tailed t-test ( unknown) p-value (p-value =. 075) ( =. 05). p-value =. 075 Do not reject. Reject =. 05 Test statistic is in ‘Fail to reject’ region Levine, Prentice-Hall
Questions?
ANOVA
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