Chapter 8. Tests of Hypotheses Based on a Single Sample. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.1

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1 Chapter 8 Tests of Hypotheses Based on a Single Sample Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.1

2 Nonstatistical Hypothesis Testing A criminal trial is an example of hypothesis testing without the statistics. In a trial a jury must decide between two hypotheses. The null hypothesis is H 0 : The defendant is innocent The alternative hypothesis or research hypothesis is H 1 : The defendant is guilty The jury does not know which hypothesis is true. They must make a decision on the basis of evidence presented. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.2

3 Nonstatistical Hypothesis Testing In the language of statistics convicting the defendant is called rejecting the null hypothesis in favor of the alternative hypothesis. That is, the jury is saying that there is enough evidence to conclude that the defendant is guilty (i.e., there is enough evidence to support the alternative hypothesis). If the jury acquits it is stating that there is not enough evidence to support the alternative hypothesis. Notice that the jury is not saying that the defendant is innocent, only that there is not enough evidence to support the alternative hypothesis. That is why we never say that we accept the null hypothesis. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.3

4 Nonstatistical Hypothesis Testing There are two possible errors. A Type I error occurs when we reject a true null hypothesis. That is, a Type I error occurs when the jury convicts an innocent person. A Type II error occurs when we don t reject a false null hypothesis. That occurs when a guilty defendant is acquitted. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.4

5 Nonstatistical Hypothesis Testing The probability of a Type I error is denoted as α (Greek letter alpha). The probability of a type II error is β (Greek letter beta). The two probabilities are inversely related. Decreasing one increases the other. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.5

6 Nonstatistical Hypothesis Testing In our judicial system Type I errors are regarded as more serious. We try to avoid convicting innocent people. We are more willing to acquit guilty people. We arrange to make α small by requiring the prosecution to prove its case and instructing the jury to find the defendant guilty only if there is evidence beyond a reasonable doubt. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.6

7 Hypothesis Testing The critical concepts are these: 1. There are two hypotheses, the null and the alternative hypotheses. 2. The procedure begins with the assumption that the null hypothesis is true. 3. The goal is to determine whether there is enough evidence to reject the truth of the null hypothesis. 4. There are two possible decisions: Conclude there is enough evidence to reject the null hypothesis. [Conclude alternative hypothesis is true] Conclude there is not enough evidence to reject the null hypothesis. [Conclude null hypothesis is true] Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.7

8 Nonstatistical Hypothesis Testing 5. Two possible errors can be made. Type I error: Reject a true null hypothesis Type II error: Do not reject a false null hypothesis. P(Type I error) = α P(Type II error) = β In practice, Type II errors are by far the most serious mistake because you go away believing the null is true and it is not true. Type I errors would allow you to conduct additional testing to verify conclusion. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.8

9 Introduction In addition to estimation, hypothesis testing is a procedure for making inferences about a population. Population Sample Inference Parameter Statistic Hypothesis testing allows us to determine whether enough statistical evidence exists to conclude that a belief (i.e. hypothesis) about a parameter is supported by the data. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.9

10 Statistical Hypotheses Testing A statistical hypothesis, or just hypothesis, is a claim either about the value of a single parameter, or values of several parameters, or the form of an entire probability. Example: Test if population mean µ = 350 Test if population mean µ is between 340 and 360 Test if the samples come from normal distribution etc Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

11 Exercise 1 For each of the following assertion, state whether it is a legitimate statistical hypothesis and why: 1. H: σ > 100 x~ 2. H: = H: s H: σ 1 /σ 2 < 1 5. H: X Y = 5 6. H: λ.01, where λ is the parameter of an exponential distribution used to model component lifetime. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

12 Concepts of Hypothesis Testing There are two hypotheses. One is called the null hypothesis and the other the alternative or research hypothesis. The usual notation is: pronounced H nought H 0 : the null hypothesis H 1 : the alternative or research hypothesis The null hypothesis (H 0 ) will always include = [two tail test], < [one tail test], and > [one tail test] in this course Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

13 Concepts of Hypothesis Testing Consider a problem dealing with the mean demand for computers during assembly lead time. Rather than estimate the mean demand, our operations manager wants to know whether the mean is different from 350 units. We can rephrase this request into a test of the hypothesis: H 0 : = 350 Thus, our research hypothesis becomes: H 1 : 350 This is what we are interested in determining Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

14 Concepts of Hypothesis Testing The testing procedure begins with the assumption that the null hypothesis is true. Thus, until we have further statistical evidence, we will assume: H 0 : = 350 (assumed to be TRUE) Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

15 Concepts of Hypothesis Testing The goal of the process is to determine whether there is enough evidence to infer that the null hypothesis is most likely not true and consequently the alternative hypothesis then must be true. That is, is there sufficient statistical information to determine if this statement: H 1 : 350, is true? Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

16 Concepts of Hypothesis Testing There are two possible decisions that can be made: Conclude that there is enough evidence to reject the null hypothesis Conclude that there is not enough evidence to reject the null hypothesis NOTE: we do not say that we accept the null hypothesis, although most people in industry will say this. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

17 Concepts of Hypothesis Testing Once the null and alternative hypotheses are stated, the next step is to randomly sample the population and calculate a test statistic (in this example, the sample mean). If the test statistic s value is inconsistent with the null hypothesis we reject the null hypothesis and infer that the alternative hypothesis is true. For example, if we re trying to decide whether the mean is not equal to 350, a large value of (say, 600) would provide enough evidence. If is close to 350 (say, 355) we could not say that this provides a great deal of evidence to infer that the population mean is different than 350. Is 355 really close?? Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

18 Concepts of Hypothesis Testing Two possible errors can be made in any test: A Type I error occurs when we reject a true null hypothesis and A Type II error occurs when we don t reject a false null hypothesis. There are probabilities associated with each type of error: P(Type I error) = P(Type II error ) = Maximum value of α is called the significance level. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

19 Types of Errors A Type I error occurs when we reject a true null hypothesis (i.e. Reject H 0 when it is TRUE) H 0 T F Reject I Reject II A Type II error occurs when we don t reject a false null hypothesis (i.e. Do NOT reject H 0 when it is FALSE) Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

20 Types of Errors Back to our example, we would commit a Type I error if: Reject H 0 when it is TRUE We reject H 0 ( = 350) in favor of H 1 ( 350) when in fact the real value of is 350. We would commit a Type II error in the case where: Do NOT reject H 0 when it is FALSE We believe H 0 is correct ( = 350), when in fact the real value of is something other than 350. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

21 Exercise 2 For the following pairs of assertions, indicate which do not comply with our rules for setting up hypotheses and why (the subscripts 1 and 2 differentiate between quantities for two different populations or samples) 1. H 0 : µ = 100, H a : µ > H 0 : σ = 20, H a : σ H 0 : p.25, H a : p = H 0 : µ 1 µ 2 = 25, H a : µ 1 µ 2 > H 0 : S, H a : 1 = S2 S1 S2 6. H 0 : µ = 120, H a : µ = H 0 : σ 1 /σ 2 = 1, H a : σ 1 /σ H 0 : p 1 p 2 =.1, H a : p 1 p 2 <.1 Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

22 Exercise 3 To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of welds is selected, and tests are conducted on each weld in the sample. Weld strength is measured as the force required to break the weld. Suppose the specifications state that the mean strength of welds should exceed 100 lb/in 2 ; the inspection team decides to test H 0 : µ = 100 versus H a : µ > 100. Explain why it might be preferable to use this H a rather than µ < 100. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

23 Exercise 4 Let µ denote the true average radioactivity level (picocuries per litter). The value 5pCi/L is considered the dividing line between safe and unsafe water. Would you recommend testing H 0 : µ = 5 vs. H a : µ > 5 or H 0 : µ = 5 vs. H a : µ < 5? Explain your reasoning. [Hint: Think about the consequnces of a type I and type II error for each possibility.] Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

24 Recap I 1) Two hypotheses: H 0 & H 1 2) ASSUME H 0 is TRUE 3) GOAL: determine if there is enough evidence to infer that H 1 is TRUE 4) Two possible decisions: Reject H 0 in favor of H 1 NOT Reject H 0 in favor of H 1 5) Two possible types of errors: Type I: reject a true H 0 [P(Type I)= ] Type II: not reject a false H 0 [P(Type II)= ] Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

25 Recap II The null hypothesis must specify a single value of the parameter (e.g. = ) Assume the null hypothesis is TRUE. Sample from the population, and build a statistic related to the parameter hypothesized (e.g. the sample mean, ) Compare the statistic with the value specified in the first step Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

26 Example A department store manager determines that a new billing system will be cost-effective only if the mean monthly account is more than $170. [One tail test] A random sample of 400 [n] monthly accounts is drawn, for which the sample mean is $178 [X-bar]. The accounts are approximately normally distributed with a standard deviation of $65 [Sigma]. Can we conclude that the new system will be cost-effective? Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

27 Example The system will be cost effective if the mean account balance for all customers is greater than $170. We express this belief as a our research hypothesis, that is: H 1 : > 170 (this is what we want to determine) Thus, our null hypothesis becomes: H 0 : < 170 (we use the = part to conduct the test) Some people will leave the less than part off of the null hypothesis. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

28 Example What we want to show: H 1 : > 170 H 0 : < 170 (we ll assume the = is true) We know: n = 400, = 178, and = 65 What to do next?! Determine sampling distribution of X-bar. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

29 Example To test our hypotheses, we can use two different approaches: The rejection region approach based on sampling distribution of X-bar (typically used when computing statistics manually), and The p-value approach (which is generally used with a computer and statistical software and also based on the sampling distribution of X-bar). We will explore both Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

30 Example Rejection Region [Based on sampling distribution of X-bar] The rejection region is a range of values such that if the test statistic falls into that range, we decide to reject the null hypothesis in favor of the alternative hypothesis. is the critical value of to reject H 0. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

31 Example It seems reasonable to reject the null hypothesis in favor of the alternative if the value of the sample mean is large relative to 170, that is if >. Normally stated as if X- bar is greater than the critical value. = P( > ) is also = P(rejecting H 0 given that H 0 is true) = P(Type I error) Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

32 Example All that s left to do is calculate and compare it to 170. we can calculate this based on any level of significance ( ) we want Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

33 Example At a 5% significance level (i.e. =0.05), we get Solving we compute = Since our sample mean (178) is greater than the critical value we calculated (175.34), we reject the null hypothesis in favor of H 1, i.e. that: > 170 and that it is cost effective to install the new billing system Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

34 Example: The Big Picture H 1 : > 170 H 0 : = 170 Reject H 0 in favor of = =178 Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

35 Standardized Test Statistic An easier method is to use the standardized test statistic: and compare its result to : (rejection region: z > ) Since z = 2.46 > (z.05 ), we reject H 0 in favor of H 1 Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

36 Conclusions of a Test of Hypothesis If we reject the null hypothesis, we conclude that there is enough evidence to infer that the alternative hypothesis is true. If we do not reject the null hypothesis, we conclude that there is not enough statistical evidence to infer that the alternative hypothesis is true. Remember: The alternative hypothesis is the more important one. It represents what we are investigating. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

37 One and Two Tail Testing The department store example was a one tail test, because the rejection region is located in only one tail of the sampling distribution: More correctly, this was an example of a right tail test. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

38 Right-Tail Testing Calculate the critical value of the mean ( ) and compare against the observed value of the sample mean ( ) Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

39 Left-Tail Testing Calculate the critical value of the mean ( ) and compare against the observed value of the sample mean ( ) Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

40 Two Tail Testing Two tail testing is used when we want to test a research hypothesis that a parameter is not equal ( ) to some value Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

41 Example AT&T s argues that its rates are such that customers won t see a difference in their phone bills between them and their competitors. They calculate the mean and standard deviation for all their customers at $17.09 and $3.87 (respectively). These are assumed to be the values of the population parameters [mean and standard deviation] You don t believe them, so you sample 100 customers at random and calculate a monthly phone bill based on competitor s rates. What we want to show is whether or not: H 1 : We do this by assuming that: H 0 : = Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

42 Example At a 5% significance level (i.e. =.05), we have /2 =.025. Thus, z.025 = 1.96 and our rejection region is: z < or- z > z z.025 z Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

43 Example From the data, we calculate = Using our standardized test statistic: We find that: Since z = 1.19 is not greater than 1.96, nor less than 1.96 we cannot reject the null hypothesis in favor of H 1. That is there is insufficient evidence to infer that there is a difference between the bills of AT&T and the competitor. Does not mean that you have proven that the true mean is $ Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

44 Probability of a Type II Error It is important that that we understand the relationship between Type I and Type II errors; that is, how the probability of a Type II error is calculated and its interpretation. Recall example about department store H 0 : = 170 H 1 : > 170 At a significance level of 5% we rejected H 0 in favor of H 1 since our sample mean (178) was greater than the critical value of (175.34) Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

45 Probability of a Type II Error A Type II error occurs when a false null hypothesis is not rejected. In example, this means that if is less than (our critical value) we will not reject our null hypothesis, which means that we will not install the new billing system. Thus, we can see that: = P( < given that the null hypothesis is false) Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

46 Example - Calculating P(Type II errors) Our original hypothesis our new assumption Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

47 Effects on of Changing Decreasing the significance level, increases the value of and vice versa. Consider this diagram again. Shifting the critical value line to the right (to decrease ) will mean a larger area under the lower curve for (and vice versa) Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

48 Another example A certain type of automobile is known to sustain no visible damage 25% of time in 10-mph crash tests. A modified bumper design has been proposed in an effort to increase this percentage. The experiment involves n = 20 independent crashes with prototypes of the new design. Let p denote the proportion of all 10-mph crashes with new bumper that result in no visible damage. Hypothesis: H 0 : p =.25 H a : p >.25 No Improvement Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

49 Another example Intuitively, H 0 should be rejected if a substantial number of crashes show no damage. Test statistic: X = the number of crashes with no visible damages. How large is enough? Rejection region: R 8 = {8, 9, 10,, 19, 20} i.e, we will reject H 0 if x 8. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

50 Calculating type I error When H 0 is true, X has a binomial probability distribution with n = 20 and p =.25 α = P( type I error) = P( H 0 is rejected when it is true) = P( X 8 when X ~ Bin(20,.25)) = 1 B(7;20,.25) = =.102 So, when H 0 is actually true, roughly 10% of all experiment consisting of 20 crashes would result in H 0 being incorrectly rejected. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

51 Calculating type II error In contrast to α, there is no single β. Instead, there is a different β for each different p that exceeds.25. Let s pick p =.3 (i.e, we assume that the true p =.3) β (.3) = P( type II error when p =.3) = = P( H 0 is not rejected when it is P( X 7 when X ~ Bin(20,.3)) false because = B(7;20,.3) = p =.3).772 When p is actually.3 rather than.25 (a small departure from H 0 ), roughly 77% of all experiments of this type would result in H 0 being incorrectly not rejected! Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

52 Calculating type II error Picking different values of p results different type II error p β(p) Clearly, β decreases as the value of p moves farther to the right of the null value.25 Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

53 Example What if we want a smaller α? Reduce the rejection region to R 9 ={9,10,,19,20} α = P( type I error) = P( H 0 is rejected when it is true) = P( X 9 when X ~ Bin(20,.25)) = 1 B(8;20,.25) = β.041 (. 3) = P( H 0 is not rejected when it is false because p =.3) = P( X 8 when X ~ Bin(20,.3)) = B(8;20,.3) =.887 β (.5) = B(8;20,.5) =.252 Compare these with the previous β s for R and.132 respectively Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

54 Judging the Test A statistical test of hypothesis is effectively defined by the significance level ( ) and the sample size (n), both of which are selected by the statistics practitioner. Therefore, if the probability of a Type II error ( ) is judged to be too large, we can reduce it by increasing, and/or increasing the sample size, n. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc

55 Judging the Test The power of a test is defined as 1. It represents the probability of rejecting the null hypothesis when it is false [the true mean is some value other than that specified by the null hypothesis]. I.e. when more than one test can be performed in a given situation, its is preferable to use the test that is correct more often. If one test has a higher power than a second test, the first test is said to be more powerful and the preferred test. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc