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1 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved

2 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Define null and alternative hypothesis and hypothesis testing Define Type I and Type II errors Describe the five-step hypothesis testing procedure Distinguish between a one-tailed and a two-tailed test of hypothesis When you have completed this chapter, you will be able to:

3 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Conduct a test of hypothesis about a population mean Conduct a test of hypothesis about a population proportion Explain the relationship between hypothesis testing and confidence interval estimation Compute the probability of a Type II error, and power of a test

4 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved T erminology Hypothesis …is a statement about a population distribution such that: Examples …the mean monthly income for all systems analysts is $3569. …35% of all customers buying coffee at Tim Hortons return within a week. (i) it is either true or false, but never both, and (ii) with full knowledge of the population data, it is possible to identify, with certainty, whether it is true or false.

5 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved T erminology …is the complement of the alternative hypothesis. We accept the null hypothesis as the default hypothesis. It is not rejected unless there is convincing sample evidence against it. Null Hypothesis H o Alternative Hypothesis H 1 …is the statement that we are interested in proving. It is usually a research hypothesis.

6 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved State the decision rule State the decision rule Identify the test statistic Identify the test statistic Do NOT reject H 0 Reject H 0 and accept H 1 Compute the value of the test statistic and make a decision Compute the value of the test statistic and make a decision Step 1 Select the level of significance Select the level of significance Step 2 Step 3 Step 4 Step 5 Hypothesis Testing State the null and alternate hypotheses State the null and alternate hypotheses

7 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved When a decision is based on analysis of sample data and not the entire population data, it is not possible to make a correct decision all the time. Our objective is to try to keep the probability of making a wrong decision as small as possible!

8 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Lets look at the Canadian legal system for an analogy …the accused person is innocent 2. …the accused person is guilty Two hypotheses: After hearing from both the prosecution and the defence, a decision is made, declaring the accused either: Innocent! But do the courts always make the right decision? Guilty!

9 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Person is innocent Person is guilty Person is declared not guilty Person is declared guilty Correct Decision Error H 0 : person is innocent H 1 : person is guilty H 0 is true H 1 is true Type II Error Type I Error Court Decision Reality

10 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved T erminology Level of Significance …is the probability of rejecting the null hypothesis when it is actually true, i.e. Type I Error …accepting the null hypothesis when it is actually false. Type II Error

11 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved T erminology Test Statistic …is a value, determined from sample information, used to determine whether or not to reject the null hypothesis. Critical Value …is the dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.

12 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Tests

13 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Critical z = rejection region 1- = acceptance region

14 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved = rejection region 1- = acceptance region z /2 -z /2 /2

15 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved A test is one-tailed when the alternate hypothesis, H 1, states a direction. H 1 : The mean yearly commissions earned by full-time realtors is more than $65,000. (µ>$65,000) H 1 : The mean speed of trucks traveling on the 407 in Ontario is less than 120 kilometres per hour. (µ<120) H 1 : Less than 20 percent of the customers pay cash for their gasoline purchase. (p<.20) Examples Tests of Significance

16 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved % Level of Significance =.05 Reject H o when z > = 5% rejection region 1- = 95% acceptance region 1.65

17 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved A test is two-tailed when no direction is specified in the alternate hypothesis, H 1 H 1 : The mean time Canadian families live in a particular home is not equal to 10 years. (µ 10) H 1 : The average speed of trucks travelling on the 407 in Ontario is different than 120 kph. (µ 120) H 1 : The percentage of repeat customers within a week at Tim Horton s is not 50%. (p.50) Examples Tests of Significance

18 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved % Level of Significance Reject H o when z> 1.96 or z< = 5% rejection region = 95% acceptance region & are called critical values

19 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Testing for the Population Mean: Large Sample, Population Standard Deviation Known Testing for the Population Mean: Large Sample, Population Standard Deviation Known Test Statistic to be used: n/ X z

20 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Testing for the Population Mean: Large Sample, Population Standard Deviation Known Testing for the Population Mean: Large Sample, Population Standard Deviation Known The processors of eye drop medication indicate on the label that the bottle contains 16 ml of medication. The standard deviation of the process is 0.5 ml. A sample of 36 bottles from the last hours production revealed a mean weight of ml per bottle At the.05 significance level is the process out of control? That is, can we conclude that the mean amount per bottle is different from 16 ml?

21 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Hypothesis Test State the null and alternate hypotheses Step 1 Select the level of significance Step 2 Identify the test statistic Step 3 State the decision rule Step 4 Compute the test statistic and make a decision Step 5 H 0 : µ = 16 H 1 : µ 16 = 0.05 Because we know the standard deviation, the test statistic is Z Reject H 0 if z > 1.96 or z < n X z Do not reject the null hypothesis. We cannot conclude the mean is different from 16 ml.

22 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Testing for the Population Mean: Large Sample, Population Standard Deviation Unknown Testing for the Population Mean: Large Sample, Population Standard Deviation Unknown Rocks Discount Store chain issues its own credit card. Lisa, the credit manager, wants to find out if the mean monthly unpaid balance is more than $400. Should Lisa conclude that the population mean is greater than $400, or is it reasonable to assume that the difference of $7 ($407-$400) is due to chance? A random check of 172 unpaid balances revealed the sample mean to be $407 and the sample standard deviation to be $38. The level of significance is set at.05.

23 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved When the sample is large, i.e. over 30, you can use the z-distribution as your test statistic. Remember, use the best that you have! (Just replace the sample standard deviation for the population standard deviation)

24 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Hypothesis Test State the null and alternate hypotheses Step 1 Select the level of significance Step 2 Identify the test statistic Step 3 State the decision rule Step 4 Compute the test statistic and make a decision Step 5 H 0 : µ = 400 H 1 : µ > 400 = 0.05 Because the sample is large, we use the test statistic Z Reject H 0 if z > n X z Reject the hypothesis. H 0. Lisa can conclude that the mean unpaid balance is greater than $400! 17238$ 400$407$

25 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Test Statistic to be used: Testing for the Population Mean: Small Sample, Population Standard Deviation Unknown Testing for the Population Mean: Small Sample, Population Standard Deviation Unknown ns X t /

26 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved The current production rate for producing 5 amp fuses at Neds Electric Co. is 250 per hour. Testing for the Population Mean: Small Sample, Population Standard Deviation Unknown Testing for the Population Mean: Small Sample, Population Standard Deviation Unknown A new machine has been purchased and installed that, according to the supplier, will increase the production rate! A sample of 10 randomly selected hours from last month revealed the mean hourly production on the new machine was 256 units, with a sample standard deviation of 6 per hour. At the.05 significance level, can Ned conclude that the new machine is faster?

27 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Hypothesis Test State the null and alternate hypotheses Step 1 Select the level of significance Step 2 Identify the test statistic Step 3 State the decision rule Step 4 Compute the test statistic and make a decision Step 5 H 0 : µ = 250 H 1 : µ > 250 = 0.05 Because the sample is small and is unknown, we use the t -test Reject H 0 if t > n X t Reject the hypothesis. H 0. Ned can conclude that the new machine will increase the production rate! … = 9 degrees of freedom

28 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved A P -Value is the probability, (assuming that the null hypothesis is true) of finding a value of the test statistic at least as extreme as the computed value for the test! If the P-Value is smaller than the significance level, H 0 is rejected. If the P-Value is larger than the significance level, H 0 is not rejected.

29 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Since P-value is smaller than of 0.05, reject H 0. The population mean is greater than $400. Rocks Discount Store chain issues its own credit card. Lisa, the credit manager, wants to find out if the mean monthly unpaid balance is more than $400. The level of significance is set at.05. A random check of 172 unpaid balances revealed the sample mean to be $407 and the sample standard deviation to be $38. Should Lisa conclude that the population mean is greater than $400? = 0.05 P(z 2.42) = Previously determined… = ns X z

30 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved P-Value = p(z |computed value|) P-Value = p(z |computed value|) P-Value = 2p(z |computed value|) P-Value = 2p(z |computed value|) |....| means absolute value of…

31 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved The processors of eye drop medication indicate on the label that the bottle contains 16 ml of medication. The standard deviation of the process is 0.5 ml. A sample of 36 bottles from last hours production revealed a mean weight of ml per bottle. At the.05 significance level is the process out of control? That is, can we conclude that the mean amount per bottle is different from 16 ml? = 0.05 = 0.05 Previously determined… P-Value = 2p(z |computed value|) P-Value = 2p(z |computed value|) = 2p(z |1.44|) = 2( ) = 2(.0749) =.1498 = 2p(z |1.44|) = 2( ) = 2(.0749) =.1498 Since.1498 >.05, do not reject H n X z

32 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Interpreting the Weight of Evidence against H o If the P-value is less than ….10 we have some evidence that H o is not true.05 we have strong evidence that H o is not true.01 we have very strong evidence that H o is not true.001 we have extremely strong evidence that H o is not true

33 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved If the P-value is less than….10 we have some evidence.05 we have strong evidence.01 we have very strong evidence.001 we have extremely strong evidence that H o is not true Since P-value is.0078 … we have very strong evidence to conclude that the population mean is greater than $400! … we have very strong evidence to conclude that the population mean is greater than $400!

34 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved … is the fraction or percentage that indicates the part of the population or sample having a particular trait of interest … is the fraction or percentage that indicates the part of the population or sample having a particular trait of interest A Proportion … is denoted by p … is found by: Sample Proportion sampledNumber sample in the successes ofNumber p

35 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Testing a Single Population Proportion: Test Statistic to be used: … is the symbol for sample proportion … is the symbol for population proportion p p ^ p0p0 … represents a population proportion of interest n pp pp z )1( ˆ 00 0

36 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved In the past, 15% of the mail order solicitations for a certain charity resulted in a financial contribution. At the.05 significance level can it be concluded that the new letter is more effective? A new solicitation letter that has been drafted is sent to a sample of 200 people and 45 responded with a contribution.

37 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Hypothesis Test State the null and alternate hypotheses Step 1 Select the level of significance Step 2 Identify the test statistic Step 3 State the decision rule Step 4 Compute the test statistic and make a decision Step 5 = 0.05 We will use the z -test Reject the hypothesis. More than 15% are responding with a pledge, therefore, the new letter is more effective! H 1 : p >.15 H 0 : p =.15 Reject H 0 if z > pp z ˆ n pp)1( ˆ 200 )15.1(

38 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Relationship Between Hypothesis Testing Procedure and Confidence Interval Estimation Case 1: Our decision rule can be restated as: Do not reject H 0 if 0 lies in the (1- ) confidence interval estimate of the population mean, computed from the sample data TEST

39 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved = rejection region 1- = Confidence Interval region Do not reject H o when z falls in the confidence interval estimate

40 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Relationship Between Hypothesis Testing Procedure and Confidence Interval Estimation Case 2: Lower-tailed test Our decision rule can be restated as: Do not reject H 0 if 0 is less than or equal to the (1- ) upper confidence bound for, computed from the sample data.

41 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved = rejection region 1- = confidence level region Do not reject Relationship Between Hypothesis Testing Procedure and Confidence Interval Estimation

42 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Relationship Between Hypothesis Testing Procedure and Confidence Interval Estimation Case 3: Upper-tailed test Our decision rule can be restated as: Do not reject H 0 if 0 is greater than or equal to the (1- ) lower confidence bound for, computed from the sample data.

43 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved = rejection region 1- = acceptance region

44 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Level of Significance …is the probability of rejecting the null hypothesis when it is actually true, i.e. Type I Error …accepting the null hypothesis when it is actually false. Type II Error

45 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Calculating the Probability of a Type II Error A batch of 5000 light bulbs either belong to a superior type, with a mean life of 2400 hours, or to an inferior type, with a mean life of 2000 hours. (By default, the bulbs will be sold as the inferior type.) Suppose we select a sample of 4 bulbs. Find the probability of a Type II error. Both bulb distributions are normal, with a standard deviation of 300 hours. =

46 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved State the null and alternate hypotheses Step 1 Select the level of significance Step 2 Identify the test statistic Step 3 State the decision rule Step 4 H 0 : µ = 2000 H 1 : µ = 2400 = As populations are normal, is known, we use the z-test Reject H 0 if the computed z > 1.96, or stated another way, If the computed value x bar is greater than x u = (300/ n), REJECT H 0 in favour of H 1 Superior: =2400 Inferior: =2000 =300 =0.025 Superior: =2400 Inferior: =2000 =300 =0.025

47 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Suppose H 0 is false and H 1 is true. i.e. the true value of µ is 2400, then x bar is approximately normally distributed with a mean of 2400 and a standard deviation of / n = 300/ n Suppose H 0 is false and H 1 is true. i.e. the true value of µ is 2400, then x bar is approximately normally distributed with a mean of 2400 and a standard deviation of / n = 300/ n …is the probability of not rejecting H o …is the probability that the value of x bar obtained will be less than or equal to x u The probability of a Type II Error XuXu X

48 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Suppose we select a sample of 4 bulbs. Then x bar has a mean of 2400 and a sd of 300/ 4 = 150 Xu = (300/ 4) = 2294 A 1 = , giving us a left tail area of n X z

49 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved The probability of a Type II error is 0.24 i.e. =0.24

50 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved If we decrease the value of (alpha), the value z increases and the critical value x u moves to the right, and therefore the value of (beta) increases. Conversely, if we increase the value of (alpha), x u moves to the left, thereby decreasing the value of (beta) For a given value of (alpha), the value of (beta) can be decreased by increasing the sample size.

51 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Power of a Test … is defined as the probability of rejecting H 0 when H 0 is false, or … the probability of correctly identifying a true alternative hypothesis … it is equal to (1- ) In previous example, = 0.24 Therefore, the tests power is = 0.76 In previous example, = 0.24 Therefore, the tests power is = 0.76

52 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved Test your learning … Click on… Online Learning Centre for quizzes extra content data sets searchable glossary access to Statistics Canadas E-Stat data …and much more!

53 Copyright © 2003 McGraw-Hill Ryerson Limited. All rights reserved This completes Chapter 10


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