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[null,"Exercises on Hypothesis Testing"]
[]
[1,"A sample of lightbulbs is studied, to test the hypothesis that the mean lifetime of the bulbs is 200"]
[null,"hours.  The sample data has a significance level of 1%.  True or false:  If the mean lifetime in the"]
[null,"population is indeed 200 hours, then a second sample (of the same size, analyzed similarly) has"]
[null,"only one chance in a hundred of yielding a sample mean as far from 200 as the first sample mean."]
[]
[2,"When test results are significant at the 5-percent level, this means that there is at least a"]
[null,"95-percent chance of being correct if you reject the null hypothesis.  True or false?"]
[]
[3,"True or false:  If the significance level of sample data, with respect to some null hypothesis, is 7%,"]
[null,"then we could say, \"The data is significant at the 10% level, but not at the 5% level.\""]
[]
[4,"A company wonders if a new marketing campaign has increased demand for their product.  They"]
[null,"examine weekly demand prior to the campaign, and afterwards.  If  m1  is the old (true, long-term"]
[null,"average weekly demand, and  m2  is the new (true, long-term) average weekly demand, and if the"]
[null,"decision they must make is whether or not to allocate substantial further resources to the"]
[null,"campaign, they should take as their null hypothesis that …"]
[]
[5,"A large distributor of cosmetics has kept his outstanding accounts receivable to a mean age of 18"]
[null,"days, over the past year.  This average is considered a standard by which to measure the efficiency"]
[null,"of the credit and collections department.  During the current month, however, a random check of"]
[null,"100 accounts yields an average of 20 days, with a sample standard deviation of 9 days."]
[]
[null,"(a)","How significant is the difference from the standard?"]
[]
[null,"(b)","If management has reason to believe that the collection of accounts is becoming slower,"]
[null,null,"and is interested only in the possibility that the average has increased, how significant"]
[null,null,"is the observed difference?"]
[]
[6,"In a brand-preference survey of 1,600 consumers in a given area, 760 express a preference for"]
[null,"brand A, and 840 for all other brands combined."]
[]
[null,"(a)","Construct a 95%-confidence interval for the percentage of consumers preferring brand A."]
[]
[null,"(b)","Is the proportion of consumers who prefer brand A significantly less than 50%?"]

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[null,"Exercises on Hypothesis Testing"]
[]
[1,"A sample of lightbulbs is studied, to test the hypothesis that the mean lifetime of the bulbs is 200"]
[null,"hours.  The sample data has a significance level of 1%.  True or false:  If the mean lifetime in the"]
[null,"population is indeed 200 hours, then a second sample (of the same size, analyzed similarly) has"]
[null,"only one chance in a hundred of yielding a sample mean as far from 200 as the first sample mean."]
[]
[null,"True. This is a restatement of the definition of \"significance level,\" illustrating that there's a tacit"]
[null,"\"second\" comparative study in the definition. The significance level of the data (with respect to the"]
[null,"null hypothesis) is the probability that, in a world where the null hypothesis is true, if we were to"]
[null,"carry out the procedure we just carried out, we would see data at least as contradictory to the null"]
[null,"hypothesis as the data we are, in fact, seeing."]
[]
[null,"We can restate the true-or-false sentence as: \"If the null hypothesis were true, then performing the"]
[null,"procedure we just performed we would have only a 1%-chance of obtaining data at least as far from"]
[null,"200 as the data we just did see.\""]
[]
[]
[2,"When test results are significant at the 5-percent level, this means that there is at least a"]
[null,"95-percent chance of being correct if you reject the null hypothesis.  True or false?"]
[]
[null,"False. The second half of the sentence is equivalent to saying, \"… there's at least a 95%-chance"]
[null,"that the null hypothesis is true.\" Significance level doesn't tell us the precise probability that the"]
[null,"hypothesis is true:  That probability depends on more than just the data - it depends, for example,"]
[null,"on the reputation of the person making the statement, and what our general background knowledge"]
[null,"suggests."]
[]
[null,"The significance level of the data tells us how likely it would be for us to see such data, if we live in"]
[null,"a world where the null hypothesis is definitely true. The lower the significance level, the less likely it"]
[null,"is that the null hypothesis is true. But there's not a direct correspondence between the two specific"]
[null,"probabilities."]
[]
[null,"From Bayes' Rule, we know that"]
[]
[null,"Pr( statement is true | we see this data ) ="]
[null,null,"Pr( we see this data | statement is true ) x Pr( statement is true ) / Pr (we see this data )"]
[]
[null,"The final two probabilities both depend on our prior beliefs."]
[]
[]
[3,"True or false:  If the significance level of sample data, with respect to some null hypothesis, is 7%,"]
[null,"then we could say, \"The data is significant at the 10% level, but not at the 5% level.\""]
[]
[null,"True. This is purely terminology. You will sometimes hear people bracket the precise significance"]
[null,"level instead of telling you the specific value. \"Significant at x%\" means the precise significance"]
[null,"level is less than x%. \"Not significant at y%\" means the precise significance level is greater than y%."]
[]
[]
[4,"A company wonders if a new marketing campaign has increased demand for their product.  They"]
[null,"examine weekly demand prior to the campaign, and afterwards.  If  m1  is the old (true, long-term"]
[null,"average weekly demand, and  m2  is the new (true, long-term) average weekly demand, and if the"]
[null,"decision they must make is whether or not to allocate substantial further resources to the"]
[null,"campaign, they should take as their null hypothesis that …"]
[]
[null,"m1 ³ m2. This will force the campaign to \"prove itself\" before they throw new resources into it."]
[]
[]
[5,"A large distributor of cosmetics has kept his outstanding accounts receivable to a mean age of 18"]
[null,"days, over the past year.  This average is considered a standard by which to measure the efficiency"]
[null,"of the credit and collections department.  During the current month, however, a random check of"]
[null,"100 accounts yields an average of 20 days, with a sample standard deviation of 9 days."]
[]
[null,"(a)","How significant is the difference from the standard?",null,null,null,null,null,null,null,null,null,"From the \"Hypothesis Testing Tool\""]
[]
[null,"The standard error of the mean in our study is  9 / Ö100 = 0.9, and so the actual data (20) is 2.22",null,null,null,null,null,null,null,null,null,null,20,"estimate/prediction of unknown quantity"]
[null,"standard deviations above what we'd expect to see if the standard (18) still held.",null,null,null,null,null,null,null,null,null,null,0.9,"measure of uncertainty"]
[null,null,null,null,null,null,null,null,null,null,null,null,100,"sample size"]
[null,"If the null hypothesis is that the truth is still 18, then a sample mean above 20 or below 16 would be",null,null,null,null,null,null,null,null,null,null,0,"number of explanatory variables in regression, or"]
[null,"at least as contradictory to the null hypothesis as is the data we're seeing here. So we do a",null,null,null,null,null,null,null,null,null,null,null,"0 if dealing with a population mean"]
[null,"two-tailed test, and find that the significance level of the data is ",null,null,null,null,null,{"$f":"NORMDIST(18-ABS(20-18),18,9/SQRT(100),TRUE)+(1-NORMDIST(20,18,9/SQRT(100),TRUE))","v":2.626829138204223E-2},{"$f":"NORMDIST(16,18,9/SQRT(100),TRUE)+(1-NORMDIST(20,18,9/SQRT(100),TRUE))","v":2.626829138204223E-2},{"$f":"2*(1-NORMDIST(20,18,9/SQRT(100),TRUE))","v":2.626829138204223E-2},{"$f":"2*(1-NORMDIST((20-18)/0.9,0,1,TRUE))","v":2.626829138204223E-2}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"significance level of data with respect to null hypothesis"]
[null,"(b)","If management has reason to believe that the collection of accounts is becoming slower,",null,null,null,null,null,null,null,null,null,"Null hypothesis:"]
[null,null,"and is interested only in the possibility that the average has increased, how significant",null,null,null,null,null,null,null,null,null,null,null,null,"≥",null,null,null,{"$f":"IF(M65>=Q74,1,TDIST((Q74-M65)/M66,M67-1-M68,TRUE))","v":1}]
[null,null,"is the observed difference?",null,null,null,null,null,null,null,null,null,null,null,"true value   ","=",18,null,null,{"$f":"2*TDIST(ABS(Q74-M65)/M66,M67-1-M68,TRUE)","v":2.8544485605744328E-2}]
[null,null,null,null,null,null,null,null,null,null,null,"okay, using normal distribution",null,null,null,"≤",null,null,null,{"$f":"IF(Q74>=M65,1,TDIST((M65-Q74)/M66,M67-1-M68,TRUE))","v":1.4272242802872164E-2}]
[null,"Here we'd do a one-tailed test, so the significance level of the data is ",null,null,null,null,null,null,{"$f":"(1-NORMDIST(20,18,9/SQRT(100),TRUE))","v":1.3134145691021115E-2},null,null,"better, using t-distribution",null,null,null,null,null,null,null,{"$f":"\"(from t-distribution with \"&TEXT(M67-1-M68,\"0\")&\" degrees of freedom)\"","v":"(from t-distribution with 99 degrees of freedom)"}]
[]
[null,"We get a smaller number (I.e., stronger evidence against the null hypothesis) because, when the"]
[null,"null hypothesis is that the true mean is less than or equal to 18, only data above 18 would"]
[null,"contradict the null hypothesis … and that's exactly what we're seeing."]
[]
[]
[6,"In a brand-preference survey of 1,600 consumers in a given area, 760 express a preference for"]
[null,"brand A, and 840 for all other brands combined."]
[]
[null,"(a)","Construct a 95%-confidence interval for the percentage of consumers preferring brand A."]
[]
[null,null,{"$f":"760/1600","v":0.47499999999999998},"±",{"$f":"1.96*SQRT(C88*(1-C88)/1600)","v":2.4469355835411766E-2}]
[]
[null,"(b)","Is the proportion of consumers who prefer brand A significantly less than 50%?"]
[]
[null,"We take as our null hypothesis that the proportion preferring brand A is at least 50%.",null,null,null,null,null,null,null,null,null,null,0.47499999999999998,"estimate/prediction of unknown quantity"]
[null,"Assuming that the truth is precisely 50% (the closest the null hypothesis can come to fitting the",null,null,null,null,null,null,null,null,null,null,{"$f":"SQRT(M92*(1-M92))/SQRT(M94)","v":1.2484365222148863E-2},"measure of uncertainty"]
[null,"data), the standard error of the proportion in our study is",null,null,null,null,null,{"$f":"SQRT(0.5*(1-0.5)/1600)","v":1.2500000000000001E-2},null,null,null,null,1600,"sample size"]
[null,null,null,null,null,null,null,null,null,null,null,null,0,"number of explanatory variables in regression, or"]
[null,"Our estimated proportion is precisely two standard deviations below the hypothesized proportion,",null,null,null,null,null,null,null,null,null,null,null,"0 if dealing with a population mean"]
[null,"so the significance level of our data is",null,null,null,{"$f":"NORMDIST((0.475-0.5)/H94,0,1,TRUE)","v":2.2750131948179098E-2},{"$f":"NORMDIST(0.475,0.5,H94,TRUE)","v":2.2750131948179098E-2}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"significance level of data with respect to null hypothesis"]
[null,null,null,null,null,null,null,null,null,"okay, using normal distribution",null,null,"Null hypothesis:"]
[null,null,null,null,null,null,null,null,null,"better, using t-distribution",null,null,null,null,null,"≥",null,null,null,{"$f":"IF(M92>=Q101,1,TDIST((Q101-M92)/M93,M94-1-M95,TRUE))","v":2.2699529878411503E-2}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"true value   ","=",0.5,null,null,{"$f":"2*TDIST(ABS(Q101-M92)/M93,M94-1-M95,TRUE)","v":4.5399059756823006E-2}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"≤",null,null,null,{"$f":"IF(Q101>=M92,1,TDIST((M92-Q101)/M93,M94-1-M95,TRUE))","v":1}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"\"(from t-distribution with \"&TEXT(M94-1-M95,\"0\")&\" degrees of freedom)\"","v":"(from t-distribution with 1599 degrees of freedom)"}]