ࡱ>   y] Sbjbj:: ]Xd(\Xd(\K04V....~~~$6~ }~~~..N~6..~._'mҎ0n݂nn~~~~~~~{~~~~~~~n~~~~~~~~~> :  STAT 200 FINAL PRACTICE TEST (SOLUTIONS) 1.  SHAPE \* MERGEFORMAT  A)  EMBED Equation.3  B)  EMBED Equation.3  C)  EMBED Equation.3  D)  EMBED Equation.3  (note  EMBED Equation.3  means and) 2. SHAPE \* MERGEFORMAT  A)  EMBED Equation.3  B)  EMBED Equation.3  C)  EMBED Equation.3  D)  EMBED Equation.3  E)  EMBED Equation.3  F)  EMBED Equation.3  G)  EMBED Equation.3   EMBED Equation.3  H)  SHAPE \* MERGEFORMAT   EMBED Equation.3   EMBED Equation.3  3A.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  CI for variance or standard deviation use: Confidence interval for population variance: Use  EMBED Equation.3  , get two values from the  EMBED Equation.3 table and solve for  EMBED Equation.3 twice. (Take square root if you want  EMBED Equation.3 ) df = 3 and the two  EMBED Equation.3  table numbers are .216 and 9.348.  EMBED Equation.3  so  EMBED Equation.3  and  EMBED Equation.3  1337.18 to 57870.4 3B. Sample size needed for CI for mean use : sample size needed for CI for mean is  EMBED Equation.3   EMBED Equation.3  3C. Sample size needed for CI for proportion use: sample size for CI for proportion is  EMBED Equation.3 (use  EMBED Equation.3 to guarantee sample size is large enough, use  EMBED Equation.3 in place of  EMBED Equation.3 to get reasonable estimate)  EMBED Equation.3  3D. Difference of three means use ANOVA: ANOVA: Collect data from SRSs from  EMBED Equation.3  different groups. Assume the populations are normal and the data are collected independently. Then the  EMBED Equation.3 -statistic for rejecting EMBED Equation.3 : All population means are equal in favor of EMBED Equation.3 : There is some difference in the population means is found as follows (the sums are over each of the S sources):  EMBED Equation.3  where  EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3 is the mean of all the data,  EMBED Equation.3  is how pieces of data from source i,  EMBED Equation.3 is the sample mean of all the data from source i, and  EMBED Equation.3 is the sample variance of all the data from source i. The Anova Test is always a one-tail test to the right. Company 1Company 2Company 3Number of bulbs studied433Lifetimes990,1010,900,8801000,900,1200850,800,1000 EMBED Equation.3 378031002650 EMBED Equation.3 358460032500002362500Sample mean9451033.33883.33Sample variance4166.6723333.3310833.33  EMBED Equation.3 : All population means are equal in favor  EMBED Equation.3 : There is some difference in the population means  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   NO  EMBED Equation.3  If there is no difference the chance we would find such strong or stronger evidence than we got that there is a difference is over 5%. This is assuming all conditions were met and the data were obtained in a proper fashion. 3E1) Difference of two means from independent samples use: Difference of means from 2 independent samplest, df = min of the sample sizes - 1the difference of the two sample means subtracted in appropriate orderdifference of the population means in Ho (often 0) EMBED Equation.3   EMBED Equation.3  or  EMBED Equation.3   EMBED Equation.3  or  EMBED Equation.3  Picture of how  EMBED Equation.3 would be distributed if  EMBED Equation.3  (using t with df = 9)   EMBED Equation.3  NO If there is the mean of the other company is not higher, the chance we would find such strong or stronger evidence than we got that it is higher is between 5% and 10%. This is assuming all conditions were met and the data were obtained in a proper fashion. 3E2)  EMBED Equation.3  or  EMBED Equation.3   EMBED Equation.3  or  EMBED Equation.3  Picture of how  EMBED Equation.3 would be distributed if  EMBED Equation.3  (using t with df = 9)   EMBED Equation.3  NO If there is the mean of the other company is the same, the chance we would find such strong or stronger evidence than we got that it is different is between 10% and 20%. This is assuming all conditions were met and the data were obtained in a proper fashion. 3F) Comparing two variances use: Hypothesis test for ratio of two variances: Use  EMBED Equation.3 maker sure the top is bigger than the bottom. Keep track of the two different dfs. Use the F-table for the critical value(s). Note that making F(data)>1 even if you have two critical values as in a two-tail test, only the right hand one matters. Data summary: Before = BAfter = A EMBED Equation.3 240233 EMBED Equation.3 1440213583 EMBED Equation.3 .66673.5833  EMBED Equation.3  or  EMBED Equation.3  (note the A went on the top since As sample standard deviation is bigger)  EMBED Equation.3  or  EMBED Equation.3  Picture of how  EMBED Equation.3 would be distributed assuming  EMBED Equation.3 . Note the tail is .05. The df for the top is 3 and for the bottom is 3.   EMBED Equation.3   EMBED Equation.3  NO If the insert does not raise the variance, the chance we would find such strong or stronger evidence than we got that it does is over 5%. This is assuming all conditions were met and the data were obtained in a proper fashion. G) Sample size needed for CI for proportion use: sample size for CI for proportion is  EMBED Equation.3 (use  EMBED Equation.3 to guarantee sample size is large enough, use  EMBED Equation.3 in place of  EMBED Equation.3 to get reasonable estimate)  EMBED Equation.3  H1) Comparing two means from matched pairs use: Difference of means from 2 dependent samples (a.k.a. matched pairs)t, df = n-1sample mean of the differences subtracted in appropriate orderpopulation difference mean in Ho (often 0) EMBED Equation.3  Where s is the s.d. of the differences.  Bulb 1Bulb 2Bulb 3Bulb 4Power consumption before60615960Power consumption after57585761after - before332-1  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  or  EMBED Equation.3   EMBED Equation.3  or  EMBED Equation.3  Picture of how  EMBED Equation.3 would be distributed if  EMBED Equation.3  (using t with df = 3).   EMBED Equation.3   EMBED Equation.3  NO If the insert does not change power consumption, the chance we would find such strong or stronger evidence than we got that it does is between 10% and 20%. This is assuming all conditions were met and the data were obtained in a proper fashion. H2)  EMBED Equation.3  or  EMBED Equation.3   EMBED Equation.3  or  EMBED Equation.3  Picture of how  EMBED Equation.3 would be distributed if  EMBED Equation.3  (using t with df = 3).   EMBED Equation.3   EMBED Equation.3  NO If the insert does not save power, the chance we would find such strong or stronger evidence than we got that it does is between 5% and 10%. This is assuming all conditions were met and the data were obtained in a proper fashion. I1) HT for standard deviation use: Hypothesis test for population variance: Get critical value(s) from  EMBED Equation.3 table, and use  EMBED Equation.3 where  EMBED Equation.3 is from  EMBED Equation.3 .  EMBED Equation.3   EMBED Equation.3  Picture of how  EMBED Equation.3 would be distributed assuming  EMBED Equation.3 . Note the tails are each .025. df = 3.   EMBED Equation.3   EMBED Equation.3  From A:  EMBED Equation.3   EMBED Equation.3  NO If the standard deviation is 50, the chance we would find such strong or stronger evidence than we got that it is not 50 is between 20% and 100%. This is assuming all conditions were met and the data were obtained in a proper fashion. I2)  EMBED Equation.3   EMBED Equation.3  Picture of how  EMBED Equation.3 would be distributed assuming  EMBED Equation.3 . Note the tail is .05. df = 3.   EMBED Equation.3  From A:  EMBED Equation.3   EMBED Equation.3  NO If the standard deviation is not over 50, the chance we would find such strong or stronger evidence than we got that it is over 50 is between 10% and 90%. This is assuming all conditions were met and the data were obtained in a proper fashion. J) This is not a HT or a CI, it is a probability question, we need to find the area under the curve.  SHAPE \* MERGEFORMAT   EMBED Equation.3   EMBED Equation.3  K) CI for proportion use: 1 sample proportionzsample proportion, p=number of successes / npopulation proportion, p, in HoHT:  EMBED Equation.3  CI:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  L) This is not a HT or a CI, it is a probability question, we need to find the area under the curve.  SHAPE \* MERGEFORMAT   EMBED Equation.3   EMBED Equation.3  M1) Difference of two proportions, use: Difference of proportions (percentages, or probabilities of success) from 2 sampleszthe difference of the two sample proportions subtracted in appropriate orderdifference in population proportions in Ho (often 0) (0 and non 0 differences have different standard deviations see ( HT(0 case):  EMBED Equation.3 where  EMBED Equation.3  EMBED Equation.3  CI & HT(non 0 case):  EMBED Equation.3   EMBED Equation.3  or  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  or  EMBED Equation.3  Picture of how all  EMBED Equation.3  would be distributed if  EMBED Equation.3 . The best evidence that Ha is true is in the shaded part that is in both tails. The total shaded areais.05.   EMBED Equation.3   EMBED Equation.3  NO If the percentages are the same, the chance we would find such strong or stronger evidence than we got they differ is .7188. This is assuming all conditions were met and the data were obtained in a proper fashion. M2)  EMBED Equation.3  or  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  or  EMBED Equation.3  Picture of how all  EMBED Equation.3  would be distributed if  EMBED Equation.3 . The best evidence that Ha is true is in the right tail of .05.  EMBED Equation.3   EMBED Equation.3  NO If the percentage is not greater for the other company, the chance we would find such strong or stronger evidence than we got that it is higher is .3594. This is assuming all conditions were met and the data were obtained in a proper fashion. N) Matched pairs, use: Difference of means from 2 dependent samples (a.k.a. matched pairs)t, df = n-1sample mean of the differences subtracted in appropriate orderpopulation difference mean in Ho (often 0) EMBED Equation.3  Where s is the s.d. of the differences. From earlier:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  O) Difference of two proportions, use: Difference of proportions (percentages, or probabilities of success) from 2 sampleszthe difference of the two sample proportions subtracted in appropriate orderdifference in population proportions in Ho (often 0) (0 and non 0 differences have different standard deviations see ( HT(0 case):  EMBED Equation.3 where  EMBED Equation.3  EMBED Equation.3  CI & HT(non 0 case):  EMBED Equation.3  other original  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  P) One mean, use: 1 sample meant, df = n-1sample meanpopulation mean in Ho EMBED Equation.3  From A)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  945 EMBED Equation.3  945 EMBED Equation.3  Q1) HT for proportion use: 1 sample proportionzsample proportion, p=number of successes / npopulation proportion, p, in HoHT:  EMBED Equation.3  CI:  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  Picture of how all p s would be distributed if  EMBED Equation.3 . The best evidence for  EMBED Equation.3 is in both tails. The total area of both tails is .05.   EMBED Equation.3  NO If the percentage is 1/3, the chance we would find such strong or stronger evidence than we got that it is not 1/3 is .5842. This is assuming all conditions were met and the data were obtained in a proper fashion. Q2)  EMBED Equation.3   EMBED Equation.3  Picture of how all p s would be distributed if  EMBED Equation.3 . The best evidence for  EMBED Equation.3 is in the right tail of .05.   EMBED Equation.3  NO If the percentage is not over 1/3, the chance we would find such strong or stronger evidence than we got that it is over 1/3 is .2912. This is assuming all conditions were met and the data were obtained in a proper fashion. R1) One mean, use: 1 sample meant, df = n-1sample meanpopulation mean in Ho EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  Picture of how all  EMBED Equation.3 s would be distributed if  EMBED Equation.3 . The best evidence that  EMBED Equation.3  is in both tails. Each tail is .025.  From earlier:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  NO If the mean was 1000, the chance we would find such strong or stronger evidence than we got that it is not 1000 is between 10% and 20%. This is assuming all conditions were met and the data were obtained in a proper fashion. R2)  EMBED Equation.3   EMBED Equation.3  Picture of how all  EMBED Equation.3 s would be distributed if  EMBED Equation.3 . The best evidence that  EMBED Equation.3  is in the left tail of .05.  From earlier:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  NO If the mean was at least 1000, the chance we would find such strong or stronger evidence than we got that it is less than 1000 is between 5% and 10%. This is assuming all conditions were met and the data were obtained in a proper fashion. S) Difference of two means from independent samples use: Difference of means from 2 independent samplest, df = min of the sample sizes - 1the difference of the two sample means subtracted in appropriate orderdifference of the population means in Ho (often 0) EMBED Equation.3 Other Original  EMBED Equation.3  or  EMBED Equation.3  T) z 4. Show two characteristics are related, use: O and E stuff : Right tails only. Use  EMBED Equation.3 = EMBED Equation.3   EMBED Equation.3 :two characteristics are independent  EMBED Equation.3 : they are related df=(r-1)(c-1) Es are found by (row total)(column total)/(grand total)  EMBED Equation.3  color and year independent  EMBED Equation.3  color and year related OsblackwhitebluetanTotals200025351010802001606016241602002323011780totals1171253741320 Esblackwhitebluetan2000 EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 2001 EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 2002 EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  Picture of how all  EMBED Equation.3 s would be distributed if color and year were independent. df =(4-1)(3-1) = 6   EMBED Equation.3   EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  NO If there was no relationship, the chance we would find such strong or stronger evidence than we got that there is between 10% and 90%. This is assuming all conditions were met and the data were obtained in a proper fashion. 5. Showing data is not distributed a certain way, use: O and E stuff : Right tails only. Use  EMBED Equation.3 = EMBED Equation.3   EMBED Equation.3 : data distributed a certain way  EMBED Equation.3 : its not distributed that way df=number of categories 1 Es are found using np where p is the probability of being in a category in Ho  EMBED Equation.3   EMBED Equation.3 not as above blackwhitebluetanTotalOs2535101080Es80(.35) = 2880(.35) = 2880(.15) = 1280(.15) = 1280 Picture of how all  EMBED Equation.3 s would be distributed if color and year were independent. df =(4-1) = 3   EMBED Equation.3   EMBED Equation.3 = EMBED Equation.3 = EMBED Equation.3  NO If the colors were distributed 35-35-15-15, the chance we would find such strong or stronger evidence than we got that the data is not distributed that way is between 10% and 90%. This is assuming all conditions were met and the data were obtained in a proper fashion. 6. A)   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  .  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  B)  EMBED Equation.3 =.955 C)  EMBED Equation.3 : EMBED Equation.3   EMBED Equation.3 :  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  YES D) EMBED Equation.3  = EMBED Equation.3   EMBED Equation.3 = EMBED Equation.3   EMBED Equation.3  or  EMBED Equation.3  Plugging in x = 0 we get y = 19.2 and plugging in x = 14 we get y = 61.2. Next we plot the points and draw the line in the graph. E) EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  or 61.2 EMBED Equation.3  F)  EMBED Equation.3  or 61.2 EMBED Equation.3  +-01HIJLPSTghijkopֿ֣ևzk^jh[jh `[EHUjvL h `[CJUVaJjDh[jh[jEHUjvL h[jCJUVaJjh[jh[jEHUjMvL h[jCJUVaJjh[jh[jEHUjuL h[jCJUVaJjh[jUh[jjhV.UmHnHuhV.jhV.Uhrhg./h$,-MNOPkl  / J        + , - . 2 3 F G H I M N a b c d h i | ̾ԧԋ~objh `[h `[EHUjxL h `[CJUVaJjh `[h `[EHUjxL h `[CJUVaJjK h `[h `[EHUj"xL h `[CJUVaJj h `[Ujh `[UmHnHujh `[Uh `[j h[jh `[EHUjgwL h `[CJUVaJjh[jUh[j#J e  V W o p gd? 0^`0gd?| } ~  ˼ˠˈreVIjh]yhh EHUj|L hh CJUVaJjgh]yhh EHUj{L hh CJUVaJ h]yhh jh]yhh Ujhh h JMEHUjL h JMCJUVaJjhh h JMEHUjL h JMCJUVaJhh jhh Uh `[jh `[UjOh[jhXEHUjpW hXCJUVaJ      > ? 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