Respuesta :
Answer:
95% confidence interval for the mean of the 800 instructors is [126.80 , 133.20].
Step-by-step explanation:
We are given that there were 800 math instructors at a mathematics convention.
Forty instructors were randomly selected and given an IQ test. The scores produced a mean of 130 with a standard deviation of 10.
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 mean score = 130
s = sample standard deviation = 10
n = sample of instructors = 40
[tex]\mu[/tex] = population mean of 800 instructors
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(-2.0225 < [tex]t_3_9[/tex] < 2.0225) = 0.95 {As the critical value of t at 39 degree of
freedom are -2.0225 & 2.0225 with P = 2.5%}
P(-2.0225 < [tex]\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }[/tex] < 2.0225) = 0.95
P( [tex]-2.0225 \times {\frac{s}{\sqrt{n} } }[/tex] < [tex]{\bar X-\mu}[/tex] < [tex]2.0225 \times {\frac{s}{\sqrt{n} } }[/tex] ) = 0.95
P( [tex]\bar X-2.0225 \times {\frac{s}{\sqrt{n} } }[/tex] < [tex]\mu[/tex] < [tex]\bar X+2.0225 \times {\frac{s}{\sqrt{n} } }[/tex] ) = 0.95
95% confidence interval for [tex]\mu[/tex] = [ [tex]\bar X-2.0225 \times {\frac{s}{\sqrt{n} } }[/tex] , [tex]\bar X+2.0225 \times {\frac{s}{\sqrt{n} } }[/tex] ]
= [ [tex]130-2.0225 \times {\frac{10}{\sqrt{40} } }[/tex] , [tex]130+2.0225 \times {\frac{10}{\sqrt{40} } }[/tex] ]
= [126.80 , 133.20]
Therefore, 95% confidence interval for the mean of the 800 instructors is [126.80 , 133.20].
Answer: The required interval would be (127.711, 132.289).
Step-by-step explanation:
Since we have given that
N = 800
n = 40
Mean = 130
Standard deviation = 10
degree of freedom df = n-1 = 40 -1 = 39
At 95% confidence level, [tex]t_{39}=2.023[/tex]
So, 95% confidence interval would be
[tex]\bar{x}\pm t_{n-1}\dfrac{s}{\sqrt{n}}\sqrt{\dfrac{N-1}{n-1}}\\\\=130\pm 2.023\times \dfrac{10}{\sqt{40}}\sqrt{\dfrac{800-1}{40-1}}\\\\=130\pm 2.023\times \dfrac{1}{4}\sqrt{\dfrac{799}{39}}\\\\=130\pm 2.289\\\\=(130-2.289,130+2.289)\\\\=(127.711,132.289)[/tex]
Hence, the required interval would be (127.711, 132.289).
