STATISTICS 333-Discussion 1Practice Problems1. It seems that crop rotation could affect the yield in certain situations. In one experiment, potatoeswere gathered from 2 fields. One filed(X) has been planted with only potatoes for years, and theother field(Y) has previously been planted with corn. The weight of collected potatoes are givenbelow.X: 6.5 5.5 5.0 7.0Y: 7.5 6.5 8.0 9.0 8.5Test whether the mean weight of potatoes grown on the “non-rotated” yield(X) equals to the meanweight of potatoes on the “rotated” field(Y), assuming that the variances of weights of potatoesfrom 2 fields are equal (“pooled” method). Find the p-value for the test and interpret the results.State the assumption(s) you make.2. volunteer working at an animal shelter conducted a study of effect of catnip on cats at the shelter.She recorded the number of “negative interaction” each of 15 cat made in 15 minute periods beforeand after being given a teaspoon of catnip. The paired measurements were collected on the sameday within 30 minutes of one another; the data are given in the accompanying table.Cat Before(Y1) After(Y2) Difference1 0 0 02 3 6 -33 3 4 -14 0 1 -15 0 0 06 4 5 -17 1 3 -28 2 1 19 3 5 -210 5 7 -211 1 0 112 0 1 -113 3 5 -214 0 2 -215 2 2 0Mean 1.8 2.8 -1SD 1.66 2.37 1.20(a) Construct a 95% confidence interval for the difference in mean difference.(b) Construct a 95% confidence interval the wrong way, using the independent sample methods.How does this interval differ from the one obtained in (a)?(c) Compare the before and after population using a t test at α = 0.05. Use a nondirectionalalternative.3. Suppose scientists attempt to measure the distance to the Moon by bouncing a laser beam off amirror left behind by the Apollo astronauts. Two different sets of equipments can be used for suchexperiments. One is quite precise but very expensive for each measurement, and the other one isless expensive but less precise. In either way, we record the time (in seconds) for a laser beam totravel to the Moon and back to the Earth, repeat it several times, and then use the average lasertravel time to compute the Moon-Earth distance.You have a choice between one of the following two alternatives, each of which has roughly thesame cost:1I. Use the more precise equipments (σ = 0.1). Take 10 measurements.II. Use the less precise equipments (σ = 0.3). Take 30 measurements.(a) Which alternative would you select to meet the stated objective? (Give your reason).(b) What if we take ONE measurements using the more precise equipments, and ONE measure-ment using the less precise equipments, and then average those TWO measurements? Whatwould be the standard deviation of that estimate?2Solution1. H0: µX= µYv.s. H1: µX6= µY¯x = 6, sX= 0.913, ¯y = 7.9, sY= 0.962.s2p=(nX− 1)s2X+ (nY− 1)s2YnX+ nY− 2= 0.886, so sp= 0.941.t =¯x − ¯yspp1/nX+ 1/nY=−1.90.941 ×√0.2 + 0.25= −3.011p-value> 2*(1-pt(3.011, 7))[1] 0.01963437There is strong evidence against that the mean weight of potatoes grown on the “non-rotated”yield equals to the mean weight of potatoes on the “rotated” field. In other words, the datasuggest that there is significant differences between the mean weight of potatoes from 2 fields.2. (a) t0.025(14) = 2.14 and (¯D ± t0.025SD√n) = (−1.6645378, −0.3354622).(b)¯Y1−¯Y2= 1.8 − 2.8 = −1, t0.025(28) = 2.05 and Spool =q14∗1.662+14∗2.37315+15−2= 2.046. Thusthe C.I. for µ1− µ2is (−1 ± 2.05 ∗ 2.046 ∗p1/15 + 1/15) = (−2.5315418, 0.5315418).(c) T =¯DsD/√n= −3.23 and the interval P value is 2 ∗(0.0005, 0.005) = (0.001, 0.01)3. I. σ = 0.1, n = 10 and II. σ = 0.3, n = 30.(a) For I, V ar(¯X) =σ2n=0.1210= 0.001. For II, V ar(¯X) =σ2n=0.3230= 0.003.Since I has smaller variance, it should be more precise. Choose I.(b) Let X1∼ N(µ, 0.12), X2∼ N(µ, 0.32).V arX1+ X22=V ar(X1+ X2)4=V ar(X1) + V ar(X2)4(by independence)=0.12+ 0.324=0.14= 0.025the standard deviation