Respuesta :
Answer:
(a) 95% confidence interval for the average of the age of people denied is [43.9 , 50.1].
(b) Based on our confidence interval, we would disagree with the manager's claim.
Step-by-step explanation:
We are given that a survey of the age of people denied promotion was conducted. In a random sample of 23 people the average age was 47.0 with a sample standard deviation of 7.2.
Assume this sample comes from a population that is normally distributed.
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 average age = 47.0
[tex]\sigma[/tex] = sample standard deviation = 7.2
n = sample of people = 23
[tex]\mu[/tex] = true average of the age of people denied
Here for constructing 95% confidence interval we have used One-sample t test statistics as we don't know about population standard deviation.
So, 95% confidence interval for the population mean, [tex]\mu[/tex] is ;
P(-2.074 < [tex]t_1_0_0[/tex] < 2.074) = 0.95 {As the critical value of t at 22 degree
of freedom are -2.074 & 2.074 with P = 2.5%}
P(-2.074 < [tex]\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }[/tex] < 2.074) = 0.95
P( [tex]-2.074 \times {\frac{s}{\sqrt{n} } }[/tex] < [tex]{\bar X-\mu}[/tex] < [tex]2.074 \times {\frac{s}{\sqrt{n} } }[/tex] ) = 0.95
P( [tex]\bar X-2.074 \times {\frac{s}{\sqrt{n} } }[/tex] < [tex]\mu[/tex] < [tex]\bar X+2.074 \times {\frac{s}{\sqrt{n} } }[/tex] ) = 0.95
(a) 95% confidence interval for [tex]\mu[/tex] = [ [tex]\bar X-2.074 \times {\frac{s}{\sqrt{n} } }[/tex] , [tex]\bar X+2.074 \times {\frac{s}{\sqrt{n} } }[/tex] ]
= [ [tex]47-2.074 \times {\frac{7.2}{\sqrt{23} } }[/tex] , [tex]47+2.074 \times {\frac{7.2}{\sqrt{23} } }[/tex] ]
= [43.9 , 50.1]
Therefore, 95% confidence interval for the average of the age of people denied is [43.9 , 50.1].
(b) Based on our confidence interval, we would disagree with the manager's claim that the average age of people denied promotion was 51 because the interval doesn't contain the value 51.
