A simple random sample of 400 household from a large community was selected to e

Question

A simple random sample of 400 household from a large community was selected to estimate the mean residential electricity usage per household during February last year.Another simple random sample of 450 household was selected independently of the first to estimate mean residential electricity usage during February of this year. the sample result (kilowatts/hour) were:
LAST YEAR n1=400 THIS YEAR n2=450 x1bar=1252 x2bar=1330 s1=257 s2=251
1_ construct 95%C.I for change in mean usage per household between two year.can it be conclude that mean usage per household change between the two year.if so was the change small or large?Explain.write any assumption which you may require to conduct this test.Also conduct an appropriate test of hypothesis for this problem and compare the result with that concluded using confidence interval
2_if 300 of the households selected last year were also included in the sample for this year, would the estimation procedure used in a above be appropriate. criticise. A simple yes or no would not be accepted as an answer

Explanation / Answer

n1=400, n2=450, x1bar=x1-=1252 x2bar=x2-=1330, s1=257, s2=251

standard error of difference of mean =SE(x1- - x2-)=sqrt(s12/n1 + s22/n2)=sqrt(257*257/400 + 251*251/450)=17.47

(1-alpha)*100% confidence interval for difference of  mean= difference of mean± z(alpha/2)*SE(difference of mean)

95% confidence interval for difference of mean=|(1252-1330)|±z(0.05/2)*17.47=

=(1252-1330)±1.96*17.47=-78±34.24=(-112.34, 53.76)

to test significance of change we use z-test and z= (difference of  mean)/SE(difference of mean)=-78/17.47=-4.46

critical z(0.05)=1.96 is less than the absolute value of calculated z=4.46, so we oldconclude that the there is significant change in usage of electiricity.

assumption of the z-test is populatio should come from normal distributed population

(2) yes , since we have to re-calculate means and sample standard deviation and then use z-test(since sample will be more than 30)

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