Question

An analysis of the Business School graduates found that 211 out of 310 randomly selected graduates used a statistical inference technique during their first year of emplyment. (a) Calculate a 90% confidence interval for the proportion of graduates who used a statistical inference technique within the first year of their employment. Present your work. (b) Interpret the confidence interval. (c) If a new follow-up study is to be undertaken, what sample size should be taken in order to estimate the true proportion of graduates who used a statistical inference technique in the first year of employment within 2%, with a confidence interval of 90%? (d) If a new follow-up study is to be undertaken, what sample size should be taken in order to estimate the true proportion of graduates who used a statistical inference technique in the first year of employment within 2%, with a confidence interval of 99%?

Answer 1

Solution:-

a) 90% confidence interval for the proportion of graduates who used statistical inference technique within first year of employment is C.I = ( 0.6371, 0.7242).

$C.I = \hat{p}\pm z_{\alpha /2}\times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

C.I = 0.68065 + 1.645*0.02648

C.I = 0.68065 + 0.04356

C.I = ( 0.6371, 0.7242)

b) If repeated samples were taken and the 90% confidence interval was computed for each sample, 90% of the intervals would contain the population proportion.

c) The required sample size is 1471.

M.E = 0.02, C.L = 90%

$M.E = z_{\alpha /2}\times\sqrt{ \frac{p(1-p)}{n}}$

$n =\left ( \frac{z_{\alpha /2}}{M.E} \right )^{2}\times p(1-p)$

$n =\left ( \frac{1.645}{0.02} \right )^{2}\times 0.68065(1-0.68065)$

n = 1470.51

n = 1471

d) The required sample size is 3607.

M.E = 0.02, C.L = 99%

$M.E = z_{\alpha /2}\times\sqrt{ \frac{p(1-p)}{n}}$

$n =\left ( \frac{z_{\alpha /2}}{M.E} \right )^{2}\times p(1-p)$

$n =\left ( \frac{2.576}{0.02} \right )^{2}\times 0.68065(1-0.68065)$

n = 3606.03

n = 3607