Question

Young children need calcium in their diet to support the growth of their bones. The Institute of Medicine provides guidelines for how much calcium should be consumed by people of different ages. One study examined whether or not a sample of children consumed an adequate amount of calcium based on these guidelines. Since there are different guidelines for children aged 5 to 10 years and those aged 11 to 13 years, the children were classified into these two age groups. Each student's calcium intake was classified as meeting or not meeting the guideline. There were 2,028 children in the study. Here are the data.

	Age (years)
Met requirement	5 to 10	11 to 13
No	193	556
Yes	868	411

Use a significance test to make the comparison.

State the null and alternative hypotheses.

H₀: p_{age 5-10} > p_{age 11-13} H_a: p_{age 5-10} ≤ p_{age 11-13}

H₀: p_{age 5-10} = p_{age 11-13} H_a: p_{age 5-10} ≠ p_{age 11-13}    

H₀: p_{age 5-10} ≤ p_{age 11-13} H_a: p_{age 5-10} > p_{age 11-13}

H₀: p_{age 5-10} ≥ p_{age 11-13} H_a: p_{age 5-10} < p_{age 11-13}

Report the test statistic and the P-value. (Use p̂_{age 5-10} − p̂_{age 11-13}. Round your value for z to two decimal places and your P-value to four decimal places.)

z	=
P-value	=

Interpret the result of your test. (Use

α = 0.05.)

Reject the null hypothesis. There is significant evidence that p_{age 5-10} is different from p_{age 11-13}.

Reject the null hypothesis. There is not significant evidence that p_{age 5-10} is different from p_{age 11-13}.    

Fail to reject the null hypothesis. There is not significant evidence that p_{age 5-10} is different from p_{age 11-13}.

Fail to reject the null hypothesis. There is significant evidence that p_{age 5-10} is different from p_{age 11-13}.

Justify for the use of the large-sample procedure for this comparison.

The data are large simple random samples from two dependent populations.

The data are large simple random samples from two independent populations.    

The data are small simple random samples from two dependent populations.

The data are small simple random samples from two independent populations.

Answer 1

H₀: p_{age 5-10} = p_{age 11-13}H_a: p_{age 5-10} ≠ p_{age 11-13}    

	pop 1	pop 2
x=	868	411
n =	1061	967
p̂=x/n=	0.8181	0.4250
estimated prop. diff =p̂1-p̂2    =		0.3931
pooled prop p̂ =(x1+x2)/(n1+n2)=		0.6307
std error Se=√(p̂1*(1-p̂1)*(1/n1+1/n2) =		0.0215
test stat z=(p̂1-p̂2)/Se =		18.32 (please try -18.32 if this comes wrong)
P value   =	0.0000	(from excel:2*normsdist(-18.32)

Reject the null hypothesis. There is significant evidence that p_{age 5-10} is different from p_{age 11-13}.

he data are large simple random samples from two independent populations.