During one particular war, the U.S. military was concerned about the nutrition o

Question

During one particular war, the U.S. military was concerned about the nutrition of its recruits. Many did not have a sufficient number of teeth to chew the food provided to soldiers. As a result, it was likely that they would be undernourished and unable to fulfill their duties as soldiers. The requirements at that time specified that a recruit must have "at least four sound double teeth, one above and one below on each side of the mouth, and so opposed" so that they could chew food. Of the 58,952 recruits who were under the age of 20, 78 were rejected for this reason. For the 43,786 recruits who were 40 or over, 3,601 were rejected.

(a) Find the proportion of rejects for each age group. (Round your answers to four decimal places.)

(b) Find a 99% confidence interval for the difference in the proportions. (Use page40 page<20. Round your answer to four decimal places.)

  ,

(c) Use a significance test to compare the proportions.

State the null and alternative hypotheses.

H0: page40 = page<20
Ha: p40 p<20H0: page40 page<20
Ha: p40 > p<20    H0: page40 page<20
Ha: page40 < page<20H0: page40 > page<20
Ha: page40 page<20

Calculate the test statistic and P-value. (Use page40 page<20. Round your value for z to two decimal places and your P-value to four decimal places.)

Write a short paragraph describing your results and conclusions. (Use

= 0.05.)

Reject the null hypothesis. There is significant evidence that p40 is greater than p<20.Reject the null hypothesis. There is not significant evidence that p40 is greater than p<20.    Fail to reject the null hypothesis. There is significant evidence that p40 is greater than p<20.Fail to reject the null hypothesis. There is not significant evidence that p40 is greater than p<20.

(d) Are the guidelines for the use of the large-sample approach satisfied for your work in parts (b) and (c)? Explain your answers.

No, the conditions are not satisfied. The data are not simple random samples from a larger population.No, the conditions are not satisfied. The data are simple random samples from a larger population.    Yes, the conditions are satisfied. The data are not simple random samples from a larger population.Yes, the conditions are satisfied. The data are simple random samples from a larger population.

page<20 = page40 =

Explanation / Answer

(a)

(b) var(p1)=p1(1-p1)/n1=0.001327*(1-0.001327)/58792=2.25E-08

var(p2)=0.082241*(1-0.082241)/43786=1.72E-06

var(p2-p1)=var(p2)+var(p1)=1.75E-06

SE(p2-v1)=sqrt(var(p2-p1)=0.0013

(1-alpha)*100% confidence interval for difference population proportion=

difference of sample protportion±z(alpha/2)*SE(difference of proportion)

99% confidence interval =(0.0822-0.0013)±z(0.01/2)*0.0013=(0.0822-0.0013)±2.5758*0.0013=

=0.0809±0.0033=(0.0776,0.0832)

(c) H0: page40 page<20
Ha: p40 > p<20

z=(p2-p1)/SE(p2-p1)=0.0809/0.0013=62.23

p-value is <0.001

Reject the null hypothesis. There is significant evidence that p40 is greater than p<20.

(d).Yes, the conditions are satisfied. The data are simple random samples from a larger population.

page<20 =p1=78/58792= 0.001327 page40 =p2=3601/43786= 0.082241

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