46 The aluminum bottle first introduced in 1991 for mainly p

4-6. The aluminum bottle first introduced in 1991 for mainly personal and household items like lotions has become popular with beverage companies. Besides being light weight, the aluminum bottle is reported to cool faster and stay cold longer than glass bottles. The data below is used to analyze the bottle choices.

Bottle type      Sample Size                 Mean               St Dev

Glass               42                                133.8 min        9.9

Aluminum       35                                92.4 min          7.3

a. Use the data above and construct a 95% confidence interval for the difference between the mean scores for glass and aluminum bottles.

b. Use the data above and construct a 95% confidence interval estimate for the mean time for aluminum bottle

c.Construct a 90% confidence interval estimate for the variance in the mean flexibility score for all aluminum bottles.

Solution

(a) Given a=1-0.95=0.05, Z(0.025) = 1.96 (from standard normal table)

So the lower bound is

(xbar1-xbar2) - Z*sqrt(s1^2/n1+s2^2/n2)

=(133.8-92.4)- 1.96*sqrt(9.9^2/42+7.3^2/35)

=37.55114

So the upper bound is

(xbar1-xbar2) + Z*sqrt(s1^2/n1+s2^2/n2)

=(133.8-92.4)+ 1.96*sqrt(9.9^2/42+7.3^2/35)

=45.24886

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(b)So the lower bound is

xbar - Z*s/vn =92.4-1.96*7.3/sqrt(35) = 89.98151

So the upper bound is

xbar + Z*s/vn =92.4+1.96*7.3/sqrt(35) =94.81849

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(c) The degree of freedom =n-1=35-1=34

Given a=1-0.9=0.1, chisquare with 0.05 and df=34 is 21.66 (from chisquare table)

chisquare with 0.95 and df=34 is 48.60(from chisquare table)

So the lower bound is

(n-1)*s^2/48.60 = 34*7.3^2/48.60 =37.28107

So the upper bound is

(n-1)*s^2/21.66 =34*7.3^2/21.66 =83.65005