1) Let ??1, ??2, ??3, . . . , ??25 be random sample of size 25 from a normally distributed population with the mean of 4 and variance of 16. Find the point estimate of the population mean if ??(??¯ >...

1) Let ??1, ??2, ??3, . . . , ??25 be random sample of size 25 from a normally distributed population with the mean of 4 and variance of 16. Find the point estimate of the population mean if ??(??¯ > ??¯) = 0.9798. 2) Pine Corporation manufactures automotive car batteries. The company claims that 30% of its 652?? batteries are good for 36 months or longer. Assume that the claim is true. Let ??^be the proportion in a sample of 100 such batteries that are good for 36 months or longer. Calculate the probability that the value of the sample proportion will be between 22% and 30%. 3) The 90% confidence interval for the true mean of national statistics examination scores is 81 <>< 87.="" assume="" that="" the="" population="" standard="" deviation="" was="" not="" known="" and="">< 30.="" i)="" find="" the="" standard="" deviation="" of="" the="" sample="" mean="" for="" the="" data="" used="" to="" obtain="" the="" confidence="" interval.="" ii)="" construct="" and="" interpret="" the="" 95%="" confidence="" interval="" for="" the="" true="" mean="" of="" national="" statistics="" examination="" scores.="" ssts012="" assignment="" 1="" 2021="" 2="" |="" p="" a="" g="" e="" 4)="" the="" percentage="" rates="" of="" home="" ownership="" for="" eight="" randomly="" selected="" states="" are="" listed="" below.="" estimate="" the="" population="" variance="" for="" the="" percentage="" rate="" if="" home="" ownership="" with="" 99%="" confidence="" level.="" 66.0="" |="" 75.8="" |="" 70.9="" |="" 73.9="" |="" 63.4="" |="" 68.5="" |="" 73.3="" |="" 65.9="" 5)="" the="" following="" information="" was="" obtained="" from="" two="" independent="" samples="" selected="" from="" two="" normally="" distributed="" populations="" with="" unknown="" but="" equal="" standard="" deviations.="" sample="" 1="" ¯1="13.97" 1="3.78" 1="21" sample="" 2="" ¯2="15.55" 2="3.26" 2="20" i)="" what="" is="" the="" point="" estimate="" of="" the="" population="" standard="" deviation?="" ii)="" find="" and="" interpret="" a="" 95%="" interval="" estimate="" for="" 2="" -="" 1.="" iii)="" based="" on="" the="" answer="" in="" ii)="" would="" you="" conclude="" that="" 1="??2?" substantiate.="" 6)="" a="" sample="" of="" 1000="" observations="" taken="" from="" population="" gave="" =="" 290.="" another="" sample="" of="" size="" 1200="" taken="" from="" population="" gave="" =="" 396.="" i)="" what="" would="" be="" the="" value="" of="" the="" test="" statistics="" for="" testing="" 0:="" =="" against="" 1:=""> ??????. ii) Find the critical value and draw conclusion about the hypothesis stated in i). Use 1% level of significance