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Chapter 10 Hypothesis Testing Using a Single Sample Sharing prescription drugs with others can be dangerous. Is this a common occurrence among teens? OR How do we answers questions like these using sample data? The National Association of Colleges and In Chapter 9, wethat usedthe sample datastarting to Employers stated average In this chapter, we will use sample data estimate the value of an unknown salary for students graduating with a bachelor’s to test some claim or hypothesis about population characteristic. degree in 2010characteristic is $48,351. Istothis the population seetrue if for your college?it is plausible. To do this, we use a test of hypotheses or test procedure. What is a test of hypotheses? A test of hypotheses is a method that Is it one of the uses sample data to decide between two values of the competing claims (hypotheses) about the sample statistic population characteristic. that are likely to occur? Is theIsvalue of that the sample statistic . . . it one isn’t likely to – a random occurrence due to natural occur? variation? OR – a value that would be considered surprising? Hypothesis statements: YouHare usually The null hypothesis, denoted by 0, is a trying to claim about a population characteristic determine if that is initially assumed to be true. this claim is The hypothesis statements are ALWAYS about the population believable. – NEVER about a sample! The alternative hypothesis, denoted by Ha, is the competing claim. To determine what the alternative hypothesis should be, you need to keep the research objectives in mind. Let’s consider a murder trial . . . To determine which hypothesis is What is the null hypothesis? correct, the jury will listen to the You are trying to is what you if the So we will make one This ofdetermine two evidence. Only if there is “evidence is true Hbeyond is innocent evidence reasonable decisions: doubt” would assume 0: the adefendant before you begin. the null hypothesis be rejected in supports this favor of the alternative hypothesis. claim. • Reject the null hypothesis What is the alternative hypothesis? • Fail to reject null hypothesis If there is not the convincing evidence, then we would “fail to reject” the null Ha: the hypothesis. defendantRemember is guilty that the actually verdict that is returned is “GUILTY” or “NOT GUILTY”. We never end up determining the null hypothesis is true – only that there is not enough evidence to say it’s not true. The Form of Hypotheses: Null hypothesis H0: population characteristic = hypothesized value This one is considered a two-tailed test because you are interested in both Alternative hypothesis The null hypothesis always direction. includes the equal case. Ha: population characteristic > hypothesized value This hypothesized valuevalue is Ha: population characteristic < hypothesized a specific number Ha: population characteristic ≠ hypothesized This signby is the value determined Notice that the alternative These are considered onedetermined by the context of the problem hypothesis usesyou the same tailed tests because context of the population characteristic and the Let’s practice writing are only interested in one problem. same hypothesized value as the hypothesis statements. direction. null hypothesis. Sharing prescription drugs with others can be dangerous. A survey of a representative sample of 592 U.S. teens age 12 to 17 reported that 118 of those surveyed admitted to having shared a prescription drug with a friend. Is this sufficient evidence that more than 10% of teens have shared prescription medication with friends? State the hypotheses : What is the What words indicate What the is the population hypothesized direction of theThe characteristic ofp of value? true proportion H : p = .1 0 alternative hypothesis? teensinterest? who have shared Ha: p > .1 prescription medication with friends Compact florescent (cfl) lightbulbs are much more energy efficient than regular incandescent lightbulbs. Ecobulb brand 60-watt cfl lightbulbs state on the package “Average life 8000 hours”. People who purchase this brand would be unhappy if the bulbs lasted less than 8000 hours. A sample of these bulbs will be selected and tested. What is the StateWhat the words hypotheses : theis the indicate What population hypothesized direction of thecharacteristic of value? H : m alternative = 8000 hypothesis?interest? 0 Ha: m < 8000 The true mean (m) life of the cfl lightbulbs Because in variation of the manufacturing process, tennis balls produced by a particular machine do not have the same diameters. Suppose the machine was initially calibrated to achieve the specification of m = 3 inches. However, the manager is now concerned that the diameters no longer conform to this specification. If the mean diameter is not 3 inches, production will have to be halted. State the hypotheses : What words indicateWhat the is the population direction of The true mean m H0: m =of3 the characteristic alternative hypothesis? diameter interest? of tennis Ha: m ≠ 3 balls For each pair of hypotheses, indicate which are Must useand a population characteristic - x not legitimate explain why is a statistics (sample) a) H0 : m 15 ; Ha : m 15 greater than! b) H0 : xMust 4be ; Honly : x 15 a Must use same c) H0number : p .1as; H : p a in H0! .1 d) H0 :Hm0 MUST 2.3 ;be Ha“=“ : m! 3.2 e) H0 : p .5 ; Ha : p .5 When you perform a hypothesis test you make a decision: reject H0 or fail to reject H0 When one of Each could possibly beyoua make wrong these decisions, there is decision; therefore, there are a possibility thattwo you types of errors. could be wrong! That you made an error! Type I error • The error of rejecting H0 when H0 is true • The probability of a Type I error is denoted by a. a is called the significance level of the This is the lower-case test Greek letter “alpha”. Type II error • The error of failing to reject H0 when H0 is false • The probability of a Type II error is denoted by b This is the lower-case Greek letter “beta”. Here is another way to look at the types of errors: Suppose H0H is is true Suppose Suppose H is true is Suppose 0H 00 and wereject fail to and false weand wewefail it, false and reject it, what toreject what rejecttype it,what what of it, type of decision decision type decision of decision was type of was made? was made? made? was made? H0 is true H0 is false Reject H0 Type I error Correct Fail to reject H0 Type II Correct error The U.S. Bureau of Transportation Statistics reports that for 2009 72% of all domestic passenger flights arrived on time (meaning within 15 minutes of its scheduled arrival time). Suppose that an airline with a poor on-time record decides to offer its employees a bonus if, in an upcoming month, the airline’s proportion of on-time flights exceeds the overall 2009 industry rate of .72. State the hypotheses. Type I error – the airline reward the H0: p = .72 State a decides Type Ito error employees when the in context. H : p > .72 a proportion of on-time flights doesn’t exceeds .72 State a Type II error Type II error – the airline in context. employees do not receive the bonus when they deserve it. In 2004, Vertex Pharmaceuticals, a biotechnology company, issued a press release announcing that it had filed an application with the FDA to begin clinical trials on an experimental drug VX-680 that had been found to reduce State the growth rate of pancreatic a Type I error in the and colon cancer tumors in animal context of studies. this problem. A potential consequence of making a Type I What is athe potential error would be that the company would Data resulting from planned clinical trials can be tothis devote resources to the error? used toconsequence test: continueof development of the drug when it really is Let m = the true mean growth rate of tumors for patients not effective. taking the experimental drug H0: m = mean growth rate of tumors for patients not taking the experimental drug Ha: m < mean growth rate of tumors for patients not taking the experimental drug A Type I error would be to incorrectly conclude that the experimental drug is effective in slowing the growth rate of tumors In 2004, Vertex Pharmaceuticals, a biotechnology company, issued a press release announcing that it had filed an application with the FDA to begin clinical trials on an experimental drug VX-680 Statetoa reduce Type IIthe error in the that had been found growth rate of context this problem. pancreaticAand colon canceroftumors in animal potential consequence of making a Type What a potential erroriswould be that the company might studies. II abandon development of a drug that was consequence of this error? Data resulting from theeffective. planned clinical trials can be used to test: H0: m = mean growth rate of tumors for patients not taking the experimental drug Ha: m < mean growth rate of tumors for patients not taking the experimental drug A Type II error would be to conclude that the drug is ineffective when in fact the mean growth rate of tumors is reduced The relationship between a and b The ideal test procedure would result in both Selecting a of significance level aand = .05 a = 0 (probability a Type I error) b=0 results inof a test procedure that, used over (probability a Type II error). and over with different samples, rejects a This isSo impossible to achieve since we must why choose ainsmall a base truenot H0always about 5 times 100. a = .05data. or a = .01)? our decision(like on sample Standard test procedures allow us to select a, the significance level of the test, but we have no direct control over b. The relationship between a and b Suppose thisisnormal curve If the null hypothesis false and the Let’s consider the represents the sampling alternative hypothesis is true, then the tail distribution would represent the the following This hypotheses: pb,when This is thefor part of true proportion is believed to be the greater probabilitycurve of failing to reject a a null hypothesis is true. that represents than .5 – so the curve should really be false H0Type . or the I error. shifted to the right. H0: p = .5 Ha: p > .5 Let a = .05 .5 The relationship between a and b If the null hypothesis is false and the Let’s consider the hypothesis is true, then the alternative This tailthat would b, the following hypotheses: true proportion to be bgreater Notice asisrepresent abelieved gets smaller, probability of failing to reject a than .5 – sogets the curve should really be larger! false to H0the . shifted right. H0: p = .5 Ha: p > .5 Let a = .01 How does one decide what a level to use? After assessing the consequences of type I and type II errors, identify the largest a that is tolerable for the problem. Then employ a test procedure that uses this maximum acceptable value –rather than anything smaller – as the level of significance. Remember, using a smaller a increases b. The EPA has adopted what is known as the Lead and Copper Rule, which defines drinking water as unsafe if the concentration of lead is 15 parts per billion (ppb) or greater or if the concentration of copper is 1.3 ppb or greater. aa Type III error inmight Which type of error has a more serious The manager of aState community water system use lead State Type error incontext. context. level measurements from aconsequence? sample water specimens Since most people wouldof consider theI? to What is a consequence of a Type What is a consequence ofIaerror Type II? test the following hypotheses: consequence of the Type H0: m = 15we versus Hawant : m < 15to keep more serious, would A Type I error theselect conclusion that a water source a leads smallto – so a smaller meets EPA standards when the water significance level of ais=really .01. unsafe. There are possible health risks to the community A Type II error leads to the conclusion that a water source does NOT meet EPA standards when the water is really safe. The community might lose a good water source. Large-Sample Hypothesis Test for a Population Proportion The fundamental idea behind hypothesis testing is: We reject H0 if the observed sample is very unlikely to occur if H0 is true. Recall the General Properties for Sampling Distributions of p These three properties imply that the standardized variable pˆ p z 1. ˆ p 1 p p n As long normal as the distribution sample size is has an approximately p (1 p )standard less than 10% of the population pˆn is large. when 2. m p n 3. When n is large, the sampling distribution of p is approximately normal. In June 2006, an Associated Press survey was conducted to investigate how people use the nutritional information provided on food packages. Interviews were conducted with 1003 randomly selected adult Americans, and each participant was asked a series of questions, including the following two: Based on these data, is it reasonable to Question 1: When purchasing packaged food,Americans how often do conclude that a majority of adult you check thecheck nutritional labeling on the package? frequently nutritional labels when purchasing packaged Question 2: How often do you purchasefoods? food that is bad for you, even after you’ve checked the nutrition labels? It was reported that 582 responded “frequently” to the question about checking labels and 441 responded “very often” or “somewhat often” to the question about purchasing bad foods even after checking the labels. Nutritional Labels Continued . . . H0: p = .5 Ha:We p > .5 will create a test statistic using: p = true proportion of adult pˆ p Americans who z nutritional labels frequently check p 1 p n We use p > .5 to test for a majority 582 pˆ who .58 frequently of adult Americans For this sample: 1003 check nutritional labels. A test statistic indicates how many standard This observed is is greater than deviations thesample sampleproportion statistic (p) from the .5..58 Isit.5plausible a characteristic sample proportion of p = .58 population (p). z occurred 5a.08 as result of chance variation, or is it .5.5 unusual to observe a sample proportion this large 1003 when p = .5? Nutritional Labels Continued . . . H0: p = .5 Ha: p > .5 p = true proportion of adult Americans who frequently check nutritional labels the normal NextIn we findstandard the P-value for curve, this seeing a value 5.08 or larger is unlikely. testofstatistic. It’s probability 582 is approximately 0. pˆ .58 For this sample: 1003 Since the P-value is so small, .58P-value .5 The is the probability of obtaining a we z 5.08 reject H0. There is convincing test.5statistic at least as inconsistent with H0 .5 evidence to suggest that the as was observed, assuming H is true. 0 1003 P-value ≈ 0 majority of adult Americans frequently check the nutritional labels on packaged0foods. Computing P-values The calculation of the P-value depends on the form of the inequality in the alternative hypothesis. • Ha: p > hypothesize value z curve P-value = area in upper tail Calculated z Computing P-values The calculation of the P-value depends on the form of the inequality in the alternative hypothesis. • Ha: p < hypothesize value z curve P-value = area in lower tail Calculated z Computing P-values The calculation of the P-value depends on the form of the inequality in the alternative hypothesis. • Ha: p ≠ hypothesize value P-value = sum of area in two tails z curve Calculated z and –z Using P-values to make a decision: To decide whether or not to reject H0, we compare the P-value to the significance level a If the P-value > a, we “fail to reject” the null hypothesis. If the P-value < a, we “reject” the null hypothesis. Summary of the Large-Sample z Test for p Null hypothesis: Test Statistic: H0: p = hypothesized value pˆ p z p (1 P ) n Alternative Hypothesis: P-value: Ha: p > hypothesized value Area to the right of calculated z Ha: p < hypothesized value Area to the left of calculated z Ha: p ≠ hypothesized value 2(Area to the right of z) of +z or 2(Area to the left of z) of -z Summary of the Large-Sample z Test for p Continued . . . Assumptions: 1. p is a sample proportion from a random sample 2. The sample size n is large. (np > 10 and n(1 - p) > 10) 3. If sampling is without replacement, the sample size is no more than 10% of the population size A report states that nationwide, 61% of high school graduates go on to attend a two-year or four-year college the year after graduation. Suppose a random sample of 1500 high school graduates in 2009 from a particular state estimated the proportion of high school graduates that attend college the year after graduation to be 58%. Can we reasonably conclude that the proportion of this state’s high school graduates in 2009 who attended college the year after graduation is different from the national figure? Use a = .01. H0: p = .61 Ha: p ≠ .61 Where p is the proportion of all State hypotheses. 2009 highthe school graduates in this state who attended college the year after graduation College Attendance Continued . . . H0: p = .61 Ha: p ≠ .61 Where p is the proportion of all 2009 high school graduates in this state who attended college the year after graduation Assumptions: • Given a random sample of 1500 high school graduates • Since 1500(.61) > 10 and 1500(.39) > 10, sample size is large enough. • Population size is much larger than the sample size. College Attendance Continued . . . H0: p = .61 Ha: p ≠ .61 Where p is the proportion of all 2009 high school graduates in this state who attended college the year after graduation Test statistic: z .58 .61 2.38 .61(.39) What potential error could you 1500 have made? Type II The= area P-value = 2(.0087) .0174to the left Useof a -2.38 = .01 is approximately .0087 Since P-value > a, we fail to reject H0. The evidence does not suggest that the proportion of 2009 high school graduates in this state who attended college the year after graduation differs from the national value. In December 2009, a county-wide water conservation campaign was conducted in a particular county. In 2010, a random sample of 500 homes was selected and water usage was recorded for each home in the sample. Suppose the sample results were that 220 households had reduced water consumption. The county supervisors wanted to know if their data supported the claim that fewer than half the households in the county reduced water consumption. H0: p = .5 Ha: p < .5 State the hypotheses. where p is the proportion of all households in Calculate the county with reduced water p. usage pˆ 220 .44 500 Water Usage Continued . . . H0: p = .5 Ha: p < .5 where p is the proportion of all households in the county with reduced water usage Verify assumptions 1. p is from a random sample of households 2. Sample size n is large because np = 250 >10 and n(1-p) = 250 > 10 3. It is reasonable that there are more than 5000 (10n) households in the county. Water Usage Continued . . . H0: p = .5 Ha: p < .5 where p is the proportion of all households in the county with reduced water usage Calculate the up test Look this value in the .44 .5 statistic P-value z 2.What 68 potential error could you table of z and curve areas .5(.5) Type I have made? 500 P-value = .0037 Use a = .01 Since P-value < a, we reject H0. There is convincing evidence that the proportion of households with reduced water usage is less than half. Water Usage Continued . . . H0: p = .5 Ha: p < .5 where p is the proportion of all households in the county with reduced water usage Since P-value < a, we reject H0. Confidence intervals are twoUsed a = .01 Compute 98% confidence tailed, soain we need to put in With .01 each tail, that.01 puts interval: the upper tail (since the –curve is .98 in the middle Notice that the Let’s create a confidence symmetrical). are is testing Ha: p < .05, a hypothesizedSince valueweinterval this the appropriate with this data. . 44 (. 56 ) would also be in the lower tail. level of .5 is NOT in the . 44 2 . 326 confidence What is the appropriate 500 .98 98% confidence to use? confidence level interval and that we (. 388 , . 492 ) “rejected” H0! .5 College Attendance Revisited . . . H0: p = .61 Ha: p ≠ .61 Where p is the proportion of all 2009 high school graduates in this state who attended college the year after graduation Since P-value > a, we fail to reject H0. Use a = .01 Let’s compute a a This is a two-tailed test so for Notice that the gets splitconfidence evenly intointerval both tails, this problem. .58 (.42) hypothesized value leaving 99% in the middle. .58 2.576 of .5 IS in the 98% .99 1500 confidence interval and that we “failed (.547, .613) to reject” H0! Hypothesis Tests for a Population Mean Let’s review the assumptions for a confidence interval for a population mean The assumptions are the same for a 1) x is the large-sample sample mean from a random hypothesis testsample, for a 2) the sample size n population is large (n mean. > 30), and 3) , the population standard deviation, is known or unknown This Thisisisthe thetest teststatistic statistic when when isisunknown. known. x m z n P-value is area under the z curve x m t s n P-value is area under the t curve with df=n-1 The One-Sample t-test for a Population Mean Null hypothesis: Test Statistic: H0: m = hypothesized value x m t s n Alternative Hypothesis: P-value: Ha: m > hypothesized value Area to the right of calculated t with df = n-1 Ha: m < hypothesized value Area to the left of calculated t with df = n-1 Ha: m ≠ hypothesized value 2(Area to the right of t) of +t or 2(Area to the left of t) of -t The One-Sample t-test for a Population Mean Continued . . . Assumptions: 1. x and s are the sample mean and sample standard deviation from a random sample 2. The sample size n is large (n > 30) or the population distribution is at least approximately normal. A study conducted by researchers at Pennsylvania State University investigated whether time perception, an indication of a person’s ability to concentrate, is impaired during nicotine withdrawal. After a 24-hour smoking abstinence, 20 smokers were asked to estimate how much time had elapsed during a 45-second period. Researchers wanted to see whether smoking abstinence had a negative impact on time perception, causing elapsed time to be overestimated. Suppose the resulting data on perceived elapsed time (in seconds) were as follows: 69 65 72 73 59 55 39 52 67 57 56 50 70 47 56 45 70 64 67 53 What is the mean and standard x = 59.30 s = 9.84 n = 20 deviation of the sample? Smoking Abstinence Continued . . . 69 65 72 73 59 55 39 52 67 57 56 50 70 47 56 45 70 64 67 53 x = 59.30 s = 9.84 n = 20 Where m is the true mean perceived elapsed H0: m = 45 time for smokers who haveState abstained from the Ha: m > 45 smoking for 24-hours Since the boxplot is approximately hypotheses. Assumptions: symmetrical, it is plausible that the 1) It is reasonable todistribution believe thatisthe sampleVerify of smokers population To do this, we need to graph the assumptions. is representative of all smokers. approximately normal. data using a boxplot or normal 2) Since the sample size is probability not plot at least 30, we must determine if it is plausible that the population distribution is approximately 40 50 60 70 normal. Smoking Abstinence Continued . . . 69 65 72 73 59 55 39 52 67 57 56 50 70 47 56 45 70 64 67 53 x = 59.30 s = 9.84 n = 20 H0: m = 45 Ha: m > 45 Where m is the true mean perceived elapsed time for smokers whoCompute have abstained from the test smoking for 24-hours Test statistic: P-value ≈ 0 statistic and P-value. 59.30 45 t 6.50 9.84 20 a = .05 Since P-value < a, we reject H0. There is convincing evidence that the mean perceived elapsed time is greater than the actual elapsed time of 45 seconds. Smoking Abstinence Continued . . . 69 65 72 73 59 55 39 52 67 57 56 50 70 47 56 45 70 64 67 53 x = 59.30 s = 9.84 n = 20 Compute the appropriate Where m is the true mean perceived elapsed H0: m = 45 confidence time for smokers who have interval. abstained from Ha: m > 45 smoking for 24-hours Since P-value < a, we reject H0. Notice that the a = .05 hypothesized value 9.84 59 . 30 1 . 729 of 45 is NOT in the 20 90% confidence Since this is a one-tailed test, a ) (55.497 , 63.103 interval wetail. .05 goes in goes inand thethat upper “rejected” H0! leaving .90 in the the lower tail, middle. A growing concern of employers is time spent in activities like surfing the Internet and emailing friends during work hours. The San Luis Obispo Tribune summarized the findings of a large survey of workers in an article that ran under the headline “Who Goofs Off More than 2 Hours a Day? Most Workers, Survey Says” (August 3, 2006). Suppose that the CEO of a large company wants to determine whether the average amount of wasted time during an 8-hour day for employees of her company is less than the reported 120 minutes. Each person in a random sample of 10 employees was contrasted and asked about daily wasted time at work. The resulting data are the following: 108 112 117 130 111 131 113 113 105 128 What is the mean and standard x = 116.80 s = 9.45 n = 10 deviation of the sample? Surfing Internet Continued . . . 108 112 117 130 111 131 113 113 105 128 x = 116.80 s = 9.45 n = 10 H0: m = 120 Ha: m < 120 Where m is the true mean daily wasted time for employees of this company The boxplot reveals some skewness, State the Verify the but there is no outliers. It is plausible Assumptions: hypotheses. assumptions. that the population distribution is 1) The given sample was a random sample of employees approximately normal. 2) Since the sample size is not at least 30, we must determine if it is plausible that the population 110 120 130 distribution is approximately normal. Surfing Internet Continued . . . 108 112 117 130 111 131 113 113 105 128 x = 116.80 s = 9.45 n = 10 H0: m = 120 Ha: m < 120 Where m is the true mean daily wasted time for employees of this company the testwe 116potential .80 Compute 120 error What could and 1.07 Test Statistic: thave statistic TypeP-value. II 9made? .45 P-value =.150 10 a = .05 Since p-value > a, we fail to reject H0. There is not sufficient evidence to conclude that the mean daily wasted time for employees of this company is less than 120 minutes. Power and the Probability of Type II Error Recall Type I and Type II Errors: Suppose H0H is is true Suppose Suppose H is true is Suppose 0H 00 and wereject fail to and false weand wewefail it, false and reject it, what toreject what rejecttype it,what what of it, type of decision decision type decision of decision was type of was made? was made? made? was made? Reject H0 H0 is true H0 is false Type I error Correct Power Fail to Type II Correct reject that The probability we correctly reject H0 is error called the power of the test. H 0 Suppose that the student body president at a university is interested in studying the amount of money that students spend on textbooks each semester. The director of financial aid services Thethat power of a test depends true value of the believes average amount spentononthe textbooks mean! Becauseand the uses actual value m is unknown, is $500 each semester, this to of However, if the true mean was $525, it is less likely If the true mean is greater than $500, then we of we cannot know the power for the actual value determine the amount of financial aid for which a that the reject sampleHwould be mistaken for a sample from should true mean is ONLY a m . 0. BUT, if the student isgreater, eligible.if The student body president the population the meanthen were $500. So,mean it is might more little say $505, the sample BUT, wethat can we gain insight to the power m by plans to ask each student in acorrectly random sample how likely will reject H0of .$500. look like we expect if the true mean were investigating somethis what if scenarios ... muchThus he or she spent on books semester and we wouldn’t have convincing evidence to reject use the data to test (using a H =0.05) the following . hypotheses: H0: m = 500 versus Ha: m > 500 Let’s consider a one-sided, upper tail test. Fail to Reject H0 Reject H0 Power = 1 - b b m0 a ma If the null hypothesis is false, then m > hypothesized value Textbooks Continued . . . H0: m = 500 Ha: m > 500 Suppose that = $85 and n = 100. (Since n is large, the sampling distribution of x is approximately normal.) What is the probability of committing a Type I error? a = .05 This is the z value with area to its of left. = 500 is true, for .95 what values the sample If m mean would you reject the null hypothesis? Rejection Region x 500 Use: 1.645 What is the value of this x? .95 We would reject H0 for x > 513.98. 85 100 Textbooks Continued . . . Suppose that = $85 and n = 100. Ha: m > 500 We would reject H 0 for x > 513.98. If the null hypothesis is false, then m > 500. What is theThis probability of a Type II error (isb)? area (to the left of x = 513.98) What if m =520?b. H0: m = 500 513.98 520 z .708 85 100 Rejection Region b = .239Look this up in the table of areas for z curves Textbooks Continued . . . H0: m = 500 Ha: m > 500 Suppose that = $85 and n = 100. We would reject H0 for x > 513.98. What is the power of the test if m = 520? b = .239 Power is the probability of power is in the Power = Notice 1 -correctly .239that = .761 rejecting H . SAME curve as b 0 Rejection Region Power = 1 – b Textbooks Continued . . . Suppose that = $85 and n = 100. Ha: m > 500 We would reject H 0 for x > 513.98. If we reject H0, then m Notice that, as the distance between > 500. Find b and power. the null hypothesized valueiffor and What m =m530? our alternative value for m increases, b decreases AND power increases. b = .03 power = .97 H0: m = 500 Rejection Region Textbooks Continued . . . Suppose that = $85 and n = 100. Ha: m > 500 We would reject H 0 for x > 513.98. If the null hypothesis Notice that, as theisdistance between false, then m > 500. Find b and power. the null hypothesizedWhat valueiffor and m =m 510? our alternative value for m decreases, b increases AND power decreases. b = .68 power = .32 H0: m = 500 Rejection Region Textbooks Continued . . . Suppose that = $85 and n = 100. H0: m = 500 Ha: m > 500 b will increase and power will decrease. What happens if we use a = .01? Rejection Region b Rejection Region Power b Power What happens to a, b, & power when the sample size is increased? Fail to Reject H0 a The standard b decreases and The significance deviation will power increases! Reject levelH(a) remains 0 decrease making thecurve same taller – so the the valueskinnier. where the and rejection region begins must move. m0 b Power ma Effects of Various Factors on the Power of a Test • The larger the size of the discrepancy between the hypothesized value and the actual value of the population characteristic, the higher the power. • The larger the significance level a, the higher the power of a test. • The larger the sample size, the higher the power of a test b and Power for a t Test When using a t-test, the population standard deviation is unknown. b not only depends on a, n, and the actual value of m, but b also depends so we must have an estimate of . The b curves (on the next slide) can be used to estimate b and the power of a test based upon the value of d. d alternativ e value - hypothesiz ed value σ b curves Consider testing H0: m = 100 versus Ha: m > 100 and focus onpopulation the alternative value mis= normal, 110. When the distribution Suppose = 10, n = 7,hypotheses and a = .01.about m the t that test for testing has smaller b than does any other test 110 100 d procedure that 1has the same significance level a. 10 Calculate d. Use the df = 6 Power ≈ .4 curve to estimate b b ≈ .6