A random sample of 78 students was interviewed, and 59 students said that they w

Question

A random sample of 78 students was interviewed, and 59 students said that they would vote for Mandy Milcovich as student body president. (a) Let p represent the proportion of all students at this college who will vote for Mandy. Find a point estimate for p and for q. (b) Find a 90% confidence interval for p and q. Use 3 decimal places for each and the E term. (c) What assumptions are required for the calculation of part (b)? Are these conditions satisfied? (d) How many more students should be included in the sample to be 90% sure that a point estimate will be within a distance of 0.05 from p? A random sample of 53 students was asked for the number of semester hours they are taking this semester. The sample standard deviation was found to be s = 4.7 semester hours. How many more students should be included in the sample to be 99% sure that the sample mean x is within 1 semester hour of the population mean mu for all students at this college?

Explanation / Answer

(3a) p=59/78=0.756 and q=1-p=1-0.756=0.244

(3b)SE(p)=sqrt(p(1-p)/n)=sqrt(0.756*(1-0.756)/78)=0.049

SE(q)=sqrt(q(1-q)/n)=sqrt(0.244*(1-0.244)/78)=0.049

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

90% confidence interval for p=0.756±z(0.1/2)*0.049=0.756±1.96*0.049=0.756±0.096=(0.707,0.805)

90% confidence interval for q=0.756±z(0.1/2)*0.049=0.244±1.96*0.049=0.244±0.096=(0.195,0.293)

(c) np>10 and nq>10 and it is satisfied

(d) here margin of error=0.05

with 90% confidence margin of error=z(0.1/2)*SE(p)

or, 0.05=1.6449*sqrt(p(1-p)/n)

or,0.05/1.6449=sqrt(p(1-p)/n)

or,p(1-p)/n=0.0009

or, n=p(1-p)/0.0009=0.756*0.244/0.0009=204.96 (next whole number is 205
answer is 205

(4) with (1-alpha)*100% confidence margin of error for mean= z(alpha/2)*sd/sqrt(n)

with 99% confidence margin of error for mean= z(0.01/2)*4.7/sqrt(n)

or, 1=2.5758*4.7/sqrt(n)

or, sqrt(n)=12.106

or, n=146.56 ( next whole number is 147)

n=147

more students required=147-53=94

requied answer is 94

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