Statistics for Business and Economics 6 th Edition
Statistics for Business and Economics 6 th Edition Chapter 10 Hypothesis Testing Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -1
What is a Hypothesis? n A hypothesis is a claim (assumption) about a population parameter: n population mean Example: The mean monthly cell phone bill of this city is μ = $42 n population proportion Example: The proportion of adults in this city with cell phones is p =. 68 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -2
The Null Hypothesis, H 0 n States the assumption (numerical) to be tested Example: The average number of TV sets in U. S. Homes is equal to three ( ) n A hypothesis about a parameter that will be maintained unless there is strong evidence against the null hypothesis. Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -3
The Alternative Hypothesis, H 1 n Is the opposite of the null hypothesis n n e. g. , The average number of TV sets in U. S. homes is not equal to 3 ( H 1: μ ≠ 3 ) If we reject the null hypothesis, then the second hypothesis, named the “alternative hypothesis” will be accepted. Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -4
Level of Significance, n n Defines rejection region of the sampling distribution Is designated by , (level of significance) n Typical values are. 01, . 05, or. 10 n Is selected by the researcher at the beginning n Provides the critical value(s) of the test Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -5
Errors in Making Decisions n Type I Error n The rejection of a true null hypothesis n Considered a serious type of error The probability of Type I Error is n Called level of significance of the test n Set by researcher in advance Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -6
Errors in Making Decisions (continued) n Type II Error n The failure to reject a false null hypothesis The probability of Type II Error is β Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -7
Type I & II Error Relationship § Type I and Type II errors can not happen at the same time § Type I error can only occur if H 0 is true § Type II error can only occur if H 0 is false If Type I error probability ( ) , then Type II error probability ( β ) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -8
Hypothesis Tests for the Mean Hypothesis Tests for Known Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Unknown Chap 10 -9
Test of Hypothesis for the Mean (σ Known) n Convert sample result ( ) to a z value Hypothesis Tests for σ Known σ Unknown Consider the test The decision rule is: (Assume the population is normal) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -10
Decision Rule H 0: μ = μ 0 H 1: μ > μ 0 Alternate rule: Z Do not reject H 0 0 zα Reject H 0 μ 0 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Critical value. Chap 10 -11
p-Value Approach to Testing n n p-value: Probability of obtaining a test statistic more extreme ( ≤ or ) than the observed sample value given H 0 is true n Also called observed level of significance n Smallest value of for which H 0 can be rejected Decision rule: compare the p-value to n If p-value < , reject H 0 n If p-value , do not reject H 0 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -12
Example 10. 1 Evaluating a new production process (hypothesis test: Upper tail Z test) n n n The production manager of Northern Windows Inc. has asked you to evaluate a proposed new procedure for producing its Regal line of double-hung windows. The present process has a mean production of 80 units per hour with a population standard deviation of σ = 8. the manager indicates that she does not want to change a new procedure unless there is strong evidence that the mean production level is higher with the new process. Assume n=25 and α=0. 05. Also assume that the sample mean was 83. What decision would you recommend based on hypothesis testing? Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -13
One-Tail Tests n In many cases, the alternative hypothesis focuses on one particular direction H 0: μ ≤ 3 H 1: μ > 3 H 0: μ ≥ 3 H 1: μ < 3 This is an upper-tail test since the alternative hypothesis is focused on the upper tail above the mean of 3 This is a lower-tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -14
Upper-Tail Tests n There is only one critical value, since the rejection area is in only one tail H 0: μ ≤ 3 H 1: μ > 3 Do not reject H 0 Z 0 zα Reject H 0 μ Critical value Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -15
Lower-Tail Tests n There is only one critical value, since the rejection area is in only one tail H 0: μ ≥ 3 H 1: μ < 3 Reject H 0 -z Do not reject H 0 0 Z μ Critical value Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -16
Two-Tail Tests n n In some settings, the alternative hypothesis does not specify a unique direction There are two critical values, defining the two regions of rejection H 0: μ = 3 H 1: μ ¹ 3 /2 x 3 Reject H 0 Do not reject H 0 -z /2 Lower critical value Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 0 Reject H 0 +z /2 z Upper critical value Chap 10 -17
Example 10. 2 Lower Tail Test n n n The production manager of Twin Forks ball bearing has asked your assistance in evaluating a modified ball bearing production process. When the process is operating properly the process produces ball bearings whose weights are normally distributed with a population mean of 5 ounces and a population standard deviation of 0. 1 ounce. The sample mean was 4. 962 and n= 16. A new raw material supplier was used for a recent production run, and the manager wants to know if that change has resulted in lowering of the mean weight of bearings. What will be your conclusion for a lower tail test? Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -18
Example 10. 3 Two tailed test n n The production manager of Circuits Unlimited has asked for your assistance in analyzing a production process. The process involves drilling holes whose diameters are normally distributed with population mean 2 inches and population standard deviation 0. 06 inch. A random sample of nine measurements had a sample mean of 1. 95 inches. Use a significance level of 0. 05 to determine if the observed sample mean is unusual and suggests that the machine should be adjusted. Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -19
t Test of Hypothesis for the Mean (σ Unknown) n Convert sample result ( ) to a t test statistic Hypothesis Tests for σ Known σ Unknown Consider the test The decision rule is: (Assume the population is normal) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -20
t Test of Hypothesis for the Mean (σ Unknown) (continued) n For a two-tailed test: Consider the test (Assume the population is normal, and the population variance is unknown) The decision rule is: Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -21
Example: Two-Tail Test ( Unknown) The average cost of a hotel room in New York is said to be $168 per night. A random sample of 25 hotels resulted in x = $172. 50 and s = $15. 40. Test at the = 0. 05 level. H 0: μ = 168 H 1: μ ¹ 168 (Assume the population distribution is normal) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -22
Example Solution: Two-Tail Test H 0: μ = 168 H 1: μ ¹ 168 n = 0. 05 /2=. 025 Reject H 0 -t n-1, α/2 -2. 0639 n n = 25 n is unknown, so use a t statistic /2=. 025 Do not reject H 0 0 1. 46 Reject H 0 t n-1, α/2 2. 0639 n Critical Value: t 24 , . 025 = ± 2. 0639 Do not reject H 0: not sufficient evidence that true mean cost is different than $168 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -23
Tests of the Population Proportion n Involves categorical variables n Two possible outcomes n n n “Success” (a certain characteristic is present) n “Failure” (the characteristic is not present) Fraction or proportion of the population in the “success” category is denoted by P Assume sample size is large Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -24
Proportions (continued) n Sample proportion in the success category is denoted by n n When n. P(1 – P) > 9, can be approximated by a normal distribution with mean and standard deviation n Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -25
Hypothesis Tests for Proportions n The sampling distribution of is Hypothesis approximately Tests for P normal, so the test statistic is a z n. P(1 – P) < 9 n. P(1 – P) > 9 value: Not discussed in this chapter Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -26
Example: Z Test for Proportion A marketing company claims that it receives 8% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the =. 05 significance level. Check: Our approximation for P is = 25/500 =. 05 n. P(1 - P) = (500)(. 05)(. 95) = 23. 75 > 9 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 10 -27
Z Test for Proportion: Solution Test Statistic: H 0: P =. 08 H 1: P ¹. 08 =. 05 n = 500, =. 05 Decision: Critical Values: ± 1. 96 Reject . 025 -1. 96 0 1. 96 z -2. 47 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Reject H 0 at =. 05 Conclusion: There is sufficient evidence to reject the company’s claim of 8% response rate. Chap 10 -28
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