Question

The student affairs office of a public university wishes to study absenteeism among students at the School of Business during the semester. A random sample of 100 business students reveals the following:

•Absenteeism: mean = 9.7 classes & standard deviation = 8.0 classes

•48 business studentswere absent more than 5 classes

a.Construct a 95% confidence interval estimate of the population mean number of absences for business students during the semester.

b.Construct a 90% confidence interval estimate of the population proportion of business students absent more than 5 classes during the semester.

Suppose that the student affairs office also wishes to take a survey in another school.

c.What sample size is needed to have 95% confidence in estimating the population mean absenteeism to within 1.5 classes if the population standard deviation is estimated to be 9.0 classes?

d.How many students need to be selected to have 90% confidence in estimating the population proportion to within 0.045 if no previous estimate is available?

e.Based on your answers to part (c) and part (d), what sample size is needed if a single survey is being conducted? Why?

Answer 1

Answer:-

Given That:-

The student affairs office of a public university wishes to study absenteeism among students at the School of Business during the semester. A random sample of 100 business students reveals the following:

•Absenteeism: mean = 9.7 classes & standard deviation = 8.0 classes

•48 business studentswere absent more than 5 classes

a.Construct a 95% confidence interval estimate of the population mean number of absences for business students during the semester.

Standard error of mean , SE = 8 / $\sqrt{100}$

= 0.8

Degree of freedom = n-1

= 100-1

= 99

Critical t value for 95% confidence interval and df = 99 is 1.98

95% confidence interval estimate of the population mean number of absences is,

(9.7 - 1.98 0.8 , 9.7 + 1.98 0.8)

(8.116 , 11.284)

b.Construct a 90% confidence interval estimate of the population proportion of business students absent more than 5 classes during the semester. Suppose that the student affairs office also wishes to take a survey in another school.

Sample proportion, p = 48/100

= 0.48

Standard error of proportion, SE = $\sqrt{0.48*(1-0.48)/100}$

= 0.05

Z value for 90% confidence interval is 1.645

90% confidence interval estimate of the population mean number of absences is,

(0.48 - 1.645 0.05 , 0.48 + 1.645 0.05)

(0.39775 , 0.56225)
c.What sample size is needed to have 95% confidence in estimating the population mean absenteeism to within 1.5 classes if the population standard deviation is estimated to be 9.0 classes?

Margin of Error , E = 1.5

Z value for 95% confidence interval is 1.96

Sample size = $(Z \sigma/E)^2$

= 138 (Rounding to nearest integer)

d.How many students need to be selected to have 90% confidence in estimating the population proportion to within 0.045 if no previous estimate is available?

Margin of Error , E = 0.045

Z value for 90% confidence interval is 1.645

Sample size =

= (1.645 /0.045)^2 0.48 * (1-0.48)

= 334 (Rounding to nearest integer)

e.Based on your answers to part (c) and part (d), what sample size is needed if a single survey is being conducted? Why?

The sample size is needed is the maximum of two sample size which can satisfies both the requirements in part c and d. Sample size needed is 334.

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