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Hypothesis Testing Variance known?

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Sampling Distribution n Over-the-counter stock selling prices calculate average price of all stocks listed [ ]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]calculate average price of the 25 stocks [x-bar] n take all possible samples of size 25 would all x-bars be equal?would all x-bars be equal? n average all the possible x-bars …equals

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Levine, Prentice-Hall Sampling Distribution 20 H0H0

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Levine, Prentice-Hall Sampling Distribution It is unlikely that we would get a sample mean of this value... 20 H0H0

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Levine, Prentice-Hall Sampling Distribution It is unlikely that we would get a sample mean of this value...... if in fact this were the population mean 20 H0H0

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Levine, Prentice-Hall Sampling Distribution It is unlikely that we would get a sample mean of this value...... if in fact this were the population mean... therefore, we reject the hypothesis that = 50. 20 H0H0

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Levine, Prentice-Hall Null Hypothesis n What is tested Always has equality sign: , or Always has equality sign: , or n Designated H 0 Example ………... H 0 : 3 Example ………... H 0 : 3

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Levine, Prentice-Hall Alternative Hypothesis Opposite of null hypothesis Opposite of null hypothesis Always has inequality sign: , , or Always has inequality sign: , , or n Designated H 1 n Example H 1 : < 3 H 1 : < 3

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Levine, Prentice-Hall Decision Reject null hypothesis Reject null hypothesis n Retain, or, fail to reject, null hypothesis n Do not use the term “accept”

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Levine, Prentice-Hall p-value Probability of obtaining a test statistic more extreme ( or than actual sample value given H 0 is true Probability of obtaining a test statistic more extreme ( or than actual sample value given H 0 is true n Called observed level of significance Smallest value of H 0 can be rejected Smallest value of H 0 can be rejected n Used to make rejection decision If p-value , reject H 0 If p-value , reject H 0

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Levine, Prentice-Hall Level of Significance n Defines unlikely values of sample statistic if null hypothesis is true Called rejection region of sampling distribution Called rejection region of sampling distribution Designated (alpha) Designated (alpha) Typical values are.01,.05,.10 Typical values are.01,.05,.10 n Selected by researcher at start

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Levine, Prentice-Hall Rejection Region (one- tail test) Sampling Distribution 1 - Level of Confidence

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Levine, Prentice-Hall Rejection Region (one- tail test) Sampling Distribution 1 - Level of Confidence Observed sample statistic

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Levine, Prentice-Hall Rejection Region (one- tail test) Sampling Distribution 1 - Level of Confidence Observed sample statistic

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Levine, Prentice-Hall Rejection Regions (two- tailed test) Sampling Distribution 1 - Level of Confidence

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Levine, Prentice-Hall Rejection Regions (two- tailed test) Sampling Distribution 1 - Level of Confidence Observed sample statistic

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Levine, Prentice-Hall Rejection Regions (two- tailed test) Sampling Distribution 1 - Level of Confidence Observed sample statistic

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Levine, Prentice-Hall Rejection Regions (two- tailed test) Sampling Distribution 1 - Level of Confidence Observed sample statistic

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Levine, Prentice-Hall Risk of Errors in Making Decision n Type I error Reject true null hypothesis Reject true null hypothesis Has serious consequences Has serious consequences Probability of Type I error is alpha [ Probability of Type I error is alpha [ – Called level of significance n Type II error Do not reject false null hypothesis Do not reject false null hypothesis Probability of Type II error is beta [ Probability of Type II error is beta [

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Levine, Prentice-Hall Decision Results H 0 : Innocent

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Levine, Prentice-Hall Hypothesis Testing n State H 0 n State H 1 Choose Choose n Choose n n Choose test

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Levine, Prentice-Hall Hypothesis Testing n Set up critical values n Collect data n Compute test statistic n Make statistical decision n Express decision n State H 0 n State H 1 Choose Choose n Choose n n Choose test

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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.

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Levine, Prentice-Hall Two-tailed z-test H 0 : H 1 : n Critical Value(s): Test Statistic: Decision:Conclusion:

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Levine, Prentice-Hall Two-tailed z-test H 0 : = 368 H 1 : 368 n Critical Value(s): Test Statistic: Decision:Conclusion:

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Levine, Prentice-Hall Two-tailed z-test H 0 : = 368 H 1 : 368 .05 n 25 Critical Value(s): Test Statistic: Decision:Conclusion:

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Levine, Prentice-Hall Two-tailed z-test H 0 : = 368 H 1 : 368 .05 n 25 Critical Value(s): Test Statistic: Decision:Conclusion:

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Levine, Prentice-Hall Two-tailed z-test H 0 : = 368 H 1 : 368 .05 n 25 Critical Value(s): Test Statistic: Decision:Conclusion:

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Levine, Prentice-Hall Two-tailed z-test H 0 : = 368 H 1 : 368 .05 n 25 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at =.05

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Levine, Prentice-Hall Two-tailed z-test H 0 : = 368 H 1 : 368 .05 n 25 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at =.05 No evidence average is not 368

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Levine, Prentice-Hall Two-tailed z-test [p-value]] Z value of sample statistic

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Levine, Prentice-Hall Two-tailed z-test [p-value] p-value is P(z -1.50 or z 1.50) Z value of sample statistic

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Levine, Prentice-Hall Two-tailed z-test [p-value] p-value is P(z -1.50 or z 1.50) Z value of sample statistic

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Levine, Prentice-Hall Two-tailed z-test [p-value] p-value is P(z -1.50 or z 1.50) Z value of sample statistic From Z table: lookup 1.50.4332

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Levine, Prentice-Hall Two-tailed z-test [p-value] p-value is P(z -1.50 or z 1.50) Z value of sample statistic From Z table: lookup 1.50.4332.5000 -.4332.0668

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Levine, Prentice-Hall Two-tailed z-test [p-value] p-value is P(z -1.50 or z 1.50) =.1336 Z value of sample statistic From Z table: lookup 1.50.4332.5000 -.4332.0668

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Levine, Prentice-Hall Two-tailed z-test [p-value] 1/2 p-value =.0668 1/2 =.025

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Levine, Prentice-Hall Two-tailed z-test [p-value] (p-Value =.1336) ( =.05) Do not reject. 1/2 p-Value =.0668 1/2 =.025 Test statistic is in ‘Do not reject’ region

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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?

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Levine, Prentice-Hall solution template ( known) H 0 : H 1 : = n = Critical Value(s): Test Statistic: Decision: Conclusion:

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Levine, Prentice-Hall Two-tailed z-test ( known) H 0 : H 1 : = n = Critical Value(s): Test Statistic: Decision: Conclusion:

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Levine, Prentice-Hall Two-tailed z-test ( known) H 0 : = 70 H 1 : 70 = n = Critical Value(s): Test Statistic: Decision: Conclusion:

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Levine, Prentice-Hall Two-tailed z-test ( known) H 0 : = 70 H 1 : 70 =.05 n = 36 Critical Value(s): Test Statistic: Decision: Conclusion:

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Levine, Prentice-Hall Two-tailed z-test ( known) H 0 : = 70 H 1 : 70 =.05 n = 36 Critical Value(s): Test Statistic: Decision: Conclusion:

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Levine, Prentice-Hall Two-tailed z-test ( known) H 0 : = 70 H 1 : 70 =.05 n = 36 Critical Value(s): Test Statistic: Decision: Conclusion:

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Levine, Prentice-Hall Two-tailed z-test ( known) H 0 : = 70 H 1 : 70 =.05 n = 36 Critical Value(s): Test Statistic: Decision: Conclusion: Do not reject at =.05

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Levine, Prentice-Hall Two-tailed z-test ( known) H 0 : = 70 H 1 : 70 =.05 n = 36 Critical Value(s): Test Statistic: Decision: Conclusion: Do not reject at =.05 No evidence average is not 70

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Levine, Prentice-Hall One-tailed z-test ( known) n Assumptions Population is normally distributed Population is normally distributed If not normal, can be approximated by normal distribution for large samples If not normal, can be approximated by normal distribution for large samples

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Levine, Prentice-Hall One-tailed z-test ( known) n Assumptions Population is normally distributed Population is normally distributed If not normal, can be approximated by normal distribution for large samples If not normal, can be approximated by normal distribution for large samples Null hypothesis has or sign only Null hypothesis has or sign only

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Levine, Prentice-Hall One-tailed z-test ( known) n Assumptions Population is normally distributed Population is normally distributed If not normal, can be approximated by normal distribution for large samples If not normal, can be approximated by normal distribution for large samples Null hypothesis has or sign only Null hypothesis has or sign only n Z-test statistic

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Levine, Prentice-Hall One-tailed z-test ( known) H 0 : 0 H 1 : < 0 Must be significantly below

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Levine, Prentice-Hall One-tailed z-test ( known) H 0 : 0 H 1 : < 0H 0 : 0 H 1 : > 0 Must be significantly below Small values satisfy H 0. Do not reject!

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Levine, Prentice-Hall One-tailed z-test ( known) What is “z” given =.025? =.025

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Levine, Prentice-Hall One-tailed z-test ( known).500 -.025.475 What Is Z given =.025? =.025

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Levine, Prentice-Hall One-tailed z-test ( known).500 -.025.475.06 1.9.4750 Standardized Normal Probability Table (Portion) What is “z” given =.025? =.025

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Levine, Prentice-Hall One-tailed z-test ( known).500 -.025.475.06 1.9.4750 Standardized Normal Probability Table (Portion) What Is Z given =.025? =.025

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Levine, Prentice-Hall 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. 368 gm.

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Levine, Prentice-Hall One-tailed z-test ( known) H 0 : H 1 : = n = Critical Value(s): Test Statistic: Decision: Conclusion:

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Levine, Prentice-Hall One-tailed z-test ( known) H 0 : 368 H 1 : > 368 = n = Critical Value(s): Test Statistic: Decision: Conclusion:

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Levine, Prentice-Hall One-tailed z-test ( known) H 0 : 368 H 1 : > 368 =.05 n = 25 Critical Value(s): Test Statistic: Decision: Conclusion:

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Levine, Prentice-Hall One-tailed z-test ( known) H 0 : 368 H 1 : > 368 =.05 n = 25 Critical Value(s): Test Statistic: Decision: Conclusion:

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Levine, Prentice-Hall One-tailed z-test ( known) H 0 : 368 H 1 : > 368 =.05 n = 25 Critical Value(s): Test Statistic: Decision: Conclusion:

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Levine, Prentice-Hall One-tailed z-test ( known) H 0 : 368 H 1 : > 368 =.05 n = 25 Critical Value(s): Test Statistic: Decision: Conclusion: Do not reject at =.05

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Levine, Prentice-Hall One-tailed z-test ( known) H 0 : 368 H 1 : > 368 =.05 n = 25 Critical Value(s): Test Statistic: Decision: Conclusion: Do not reject at =.05 No evidence average is more than 368

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Levine, Prentice-Hall One-tailed z-test ( known) p-value Solution Z value of sample statistic Use alternative hypothesis to find direction

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Levine, Prentice-Hall One-tailed z-test ( known) p-value Z value of sample statistic Use alternative hypothesis to find direction

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Levine, Prentice-Hall One-tailed z-test ( known) p-value Z value of sample statistic From Z table: lookup 1.50..4332 Use alternative hypothesis to find direction

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Levine, Prentice-Hall One-tailed z-test ( known) p-value Z value of sample statistic From Z table: lookup 1.50..4332 Use alternative hypothesis to find direction.5000 -.4332.0668

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Levine, Prentice-Hall One-tailed z-test ( known) p-value Z value of sample statistic From Z table: lookup 1.50..4332 Use alternative hypothesis to find direction.5000 -.4332.0668 p-value =.0668

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Levine, Prentice-Hall One-tailed z-test ( known) p-value p-value =.0668 =.05

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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

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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)?

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Levine, Prentice-Hall p-value Z value of sample statistic From Z table: lookup 2.645.4960 Use alternative hypothesis to find direction.5000 -.4960.0040 p-value =.004 p-value < ( =.01) Reject H 0.

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Levine, Prentice-Hall p-value Probability of obtaining a test statistic more extreme ( or than actual sample value given H 0 is true Probability of obtaining a test statistic more extreme ( or than actual sample value given H 0 is true n Called observed level of significance Smallest value of H 0 can be rejected Smallest value of H 0 can be rejected n Used to make rejection decision If p-value , reject H 0 If p-value , reject H 0

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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.

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Levine, Prentice-Hall H 0 : H 1 : = n = Critical Value(s): Test Statistic: Decision: Conclusion: One-tailed t-test ( unknown)

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Levine, Prentice-Hall H 0 : 368 H 1 : < 368 = n = Critical Value(s): Test Statistic: Decision: Conclusion: One-tailed t-test ( unknown)

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Levine, Prentice-Hall H 0 : 368 H 1 : < 368 =.05 n = 25, d.f. = 24 Critical Value(s): Test Statistic: Decision: Conclusion: One-tailed t-test ( unknown)

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Levine, Prentice-Hall H 0 : 368 H 1 : < 368 =.05 n = 25, d.f. = 24 Critical Value(s): Test Statistic: Decision: Conclusion: One-tailed t-test ( unknown)

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Levine, Prentice-Hall H 0 : 368 H 1 : < 368 =.05 n = 25, d.f. = 24 Critical Value(s): Test Statistic: Decision: Conclusion: One-tailed t-test ( unknown)

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Levine, Prentice-Hall H 0 : 368 H 1 : < 368 =.05 n = 25, d.f. = 24 Critical Value(s): Test Statistic: Decision: Conclusion: Do not reject at =.05 One-tailed t-test ( unknown)

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Levine, Prentice-Hall H 0 : 368 H 1 : < 368 =.05 n = 25, d.f. = 24 Critical Value(s): Test Statistic: Decision: Conclusion: Do not reject at =.05 No evidence average is less than 368 One-tailed t-test ( unknown)

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Levine, Prentice-Hall One-tailed t-test ( unknown) p-value Solution t value of sample statistic Use alternative hypothesis to find direction

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Levine, Prentice-Hall One-tailed t-test ( unknown) p-value Use alternative hypothesis to find direction t value of sample statistic From t table: lookup -1.50 for 24 d.f. P-value = 0.075

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Levine, Prentice-Hall p-value =.075 =.05 One-tailed t-test ( unknown) p-value

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Levine, Prentice-Hall p-value =.075 =.05 One-tailed t-test ( unknown) p-value Test statistic is in ‘Fail to reject’ region (p-value =.075) ( =.05). Do not reject. Reject

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Questions?

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ANOVA

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© 2001 Prentice-Hall, Inc.Chap 9-1 BA 201 Lecture 15 Test for Population Mean Known.

© 2001 Prentice-Hall, Inc.Chap 9-1 BA 201 Lecture 15 Test for Population Mean Known.

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