Respuesta :
Answer:
95% confidence interval for the mean credit hours taken by a student each quarter is [14.915 hours , 15.485 hours].
Step-by-step explanation:
We are given that a random sample of 250 students at a university finds that these students take a mean of 15.2 credit hours per quarter with a standard deviation of 2.3 credit hours.
Firstly, the pivotal quantity for 95% confidence interval for the population mean is given by;
P.Q. = [tex]\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }[/tex] ~ [tex]t_n_-_1[/tex]
where, [tex]\bar X[/tex] = sample credit hours per quarter = 15.2 credit hours
s = sample standard deviation = 2.3 credit hours
n = sample of students = 250
[tex]\mu[/tex] = population mean credit hours per quarter
Here for constructing 95% confidence interval we have used One-sample t test statistics as we know don't about population standard deviation.
So, 95% confidence interval for the population mean, [tex]\mu[/tex] is ;
P(-1.96 < [tex]t_2_4_9[/tex] < 1.96) = 0.95 {As the critical value of t at 249 degree of
freedom are -1.96 & 1.96 with P = 2.5%}
P(-1.96 < [tex]\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }[/tex] < 1.96) = 0.95
P( [tex]-1.96 \times {\frac{s}{\sqrt{n} } }[/tex] < [tex]{\bar X-\mu}[/tex] < [tex]1.96 \times {\frac{s}{\sqrt{n} } }[/tex] ) = 0.95
P( [tex]\bar X-1.96 \times {\frac{s}{\sqrt{n} } }[/tex] < [tex]\mu[/tex] < [tex]\bar X+1.96 \times {\frac{s}{\sqrt{n} } }[/tex] ) = 0.95
95% confidence interval for [tex]\mu[/tex] = [ [tex]\bar X-1.96 \times {\frac{s}{\sqrt{n} } }[/tex] , [tex]\bar X+1.96 \times {\frac{s}{\sqrt{n} } }[/tex] ]
= [ [tex]15.2-1.96 \times {\frac{2.3}{\sqrt{250} } }[/tex] , [tex]15.2+1.96 \times {\frac{2.3}{\sqrt{250} } }[/tex] ]
= [14.915 hours , 15.485 hours]
Therefore, 95% confidence interval for the mean credit hours taken by a student each quarter is [14.915 hours , 15.485 hours].
The interpretation of the above confidence interval is that we are 95% confident that the true mean credit hours taken by a student each quarter will be between 14.915 credit hours and 15.485 credit hours.
