Respuesta :
Answer:
(a) Probability that the mean is greater than $38,000 is 0.3446.
(b) Probability that a randomly selected teacher's salary is grater than $45,000 is 0.078.
Step-by-step explanation:
We are given that the average teachers salary in north Dakota is $37,764. Assume a normal distribution with sigma (σ) = 5100.
(a) A sample of 75 teachers is taken.
Let [tex]\bar X[/tex] = sample mean salary
The z score probability distribution for sample mean is given by;
Z = [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
where, [tex]\mu[/tex] = population mean salary = $37,764
[tex]\sigma[/tex] = standard deviation = $5,100
n = sample of teachers = 75
Now, probability that the mean is greater than $38,000 is given by = P([tex]\bar X[/tex] > $38,000)
P([tex]\bar X[/tex] > $38,000) = P( [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] > [tex]\frac{38,000-37,764}{\frac{5,100}{\sqrt{75} } }[/tex] ) = P(Z > 0.40) = 1 - P(Z < 0.40)
= 1 - 0.6554 = 0.3446
The above probability is calculated by looking at the value of x = 0.40 in the z table which has an area of 0.6554.
(b) Let X = a randomly selected teacher's salary
The z score probability distribution for normal distribution is given by;
Z = [tex]\frac{ X-\mu}{\sigma}[/tex] ~ N(0,1)
where, [tex]\mu[/tex] = population mean salary = $37,764
[tex]\sigma[/tex] = standard deviation = $5,100
Now, probability that a randomly selected teacher's salary is grater than $45,000 is given by = P(X > $45,000)
P(X > $45,000) = P( [tex]\frac{ X-\mu}{\sigma}[/tex] > [tex]\frac{45,000-37,764}{5,100} }[/tex] ) = P(Z > 1.42) = 1 - P(Z < 1.42)
= 1 - 0.9222 = 0.078
The above probability is calculated by looking at the value of x = 1.42 in the z table which has an area of 0.9222.
