Answer to Question #187637 in Statistics and Probability for Ella Ssebulime

Question

The National Center for Education Statistics surveyed 4400 college graduates about the lengths of time

required to earn their bachelor’s degrees. The mean was 5.15 years and the standard deviation was 1.68

years. Based on the above information, construct a 95% confidence interval for the mean time required to

earn a bachelor’s degree by all college students.

The sample size (n) is _____________________

 

Answer 1

We are given that the National Center for Education Statistics surveyed a random sample of 4400 college graduates about the lengths of time required to earn their bachelor’s degrees. The mean was 5.15 years and the standard deviation was 1.68 years respectively.

Firstly, the pivotal quantity for 95% confidence interval for the population mean is given by:

$$P.Q. =\dfrac{\bar X-\mu}{\frac{\sigma}{\sqrt n}}\approx N(0,1)$$

where,   $\bar X$ = sample mean time = 5.15 years

         $\sigma$ = sample standard deviation = 1.68 years

         n = sample of college graduates = 4400

       $\mu$  = population mean time

Here for constructing 95% confidence interval we have used One-sample z test statistics although we are given sample standard deviation because the sample size is very large so at large sample values t distribution also follows normal.

So, 95% confidence interval for the population mean,  $\mu$ is ;

$P(-1.96 < N(0,1) < 1.96)$ = 0.95 {As the critical value of z at 2.5%

                         level of significance are -1.96 & 1.96}  

$P(-1.96<\dfrac{\bar X-\mu}{\frac{\sigma}{\sqrt n}}<1.96)=0.95\\\Rightarrow P(-1.96\times \frac{\sigma}{\sqrt n}<\bar X-\mu<1.96\times \frac{\sigma}{\sqrt n})=0.95\\\Rightarrow P(\bar X-1.96\times\frac{\sigma}{\sqrt n}<\mu<\bar X+1.96\times\frac{\sigma}{\sqrt n})=0.95$

95% confidence interval for $\mu=[\bar X-1.96\times\frac{\sigma}{\sqrt n},\bar X+1.96\times\frac{\sigma}{\sqrt n}]$

$=[5.15-1.96\times\frac{1.68}{\sqrt{4400}},5.15+1.96\times\frac{1.68}{\sqrt{4400}}]\\=[5.100,5.199]\approx[5.10,5.20]$

Therefore, 95​% confidence interval for the mean time required to earn a bachelor’s degree by all college students is [5.10 years , 5.20 years].