Question

Without using a software

As part of the Women's Health Trial, one group of women were encouraged to adopt a low-fat diet while a second group received no dietary counseling. A year later, the women in the intervention group had successfully maintained their diets. At that time, a study was undertaken to determine whether their husbands also had a reduced level of fat intake. (a) In the intervention group, a sample of 156 husbands has mean daily fat intake (x_1 ) ̅ = 54.8 grams and standard deviation s_1 = 28.1 grams. In the control group, a sample of 148 husbands has mean intake (x_2 ) ̅ = 69.5 grams and standard deviation s_2 = 34.7 grams. Calculate separate 95% confidence intervals for the true mean fat intakes of men in each group. (b) Formally test the null hypothesis that the two groups of men have the same mean dietary fat intake using a two-sided test. What do you conclude? (c) Construct a 95% confidence interval for the true difference in population means. (d) A researcher might also be interested in knowing whether the men differ with respect to the intake of other types of food, such as protein or carbohydrates. In the intervention group, the husbands have mean daily carbohydrate intake (x_1 ) ̅ = 172.5 grams and standard deviation s_1 = 68.8 grams. In the control group, the men have mean carbohydrate intake (x_2 ) ̅ = 185.5 grams and standard deviation s_2 = 69.0 grams. Test the null hypothesis that the two populations have the same mean carbohydrate intake. What do you conclude?

Answer 1

Answer:
one group of women were encouraged to adopt a low-fat diet while a second group received no dietary counseling. A year later, the women in the intervention group had successfully maintained their diets.
a)
group1

54.8-1.96*28.1/sqrt156<mean<54.8+1.96*28.1/sqrt156

50.390<mean<59.210

group2

69.5-1.96*34.7/sqrt148<mean<69.5+1.96*34.7/sqrt148

63.909<mean<75.091

b)H0: both mean equal

H1: mean 1 not equal to mean2

testvalue=54.8-69.5/sqrt(28.1^2/156+34.7^2/148)=-4.046

critical value=+/- 1.96

as test value<-1.96

wecan rejerct null hypothesis

so two groups of men have the different mean dietary fat intake

c)54.8-69.5-1.96*sqrt(28.1^2/156+34.7^2/148) <mean1-mean2< 54.8-69.5+1.96*sqrt(28.1^2/156+34.7^2/148)

-21.82<mean1-mean2<-7.579

d)H0: both mean equal

H1: mean 1 not equal to mean2

test value=172.5-185.5/sqrt(68.8^2/156+69^2/148)=-1.644

critical value=-/+1.96

as -1.96<test value<1.96

we fail to reject null hypothesis

so two populations have the same mean carbohydrate intake