Question

Can someone please help me with this.

Thank you much!

1. A population has parameters μ=36.3 and σ=57. You intend to draw a random sample of size n=139.
What is the mean of the distribution of sample means?
μ¯x=________________

b. A population of values has a normal distribution with μ=201.8 and σ=90.9. You intend to draw a random sample of size n=20.

Find the probability that a single randomly selected value is less than 266.8.
P(X < 266.8) _________________

Answer 1

Solution :

Given that,

1.a.

mean = $\mu$ = 36.3

standard deviation = $\sigma$ = 57

n = 139

The sampling distribution of mean and standard deviation is ,

$\mu$_{$\bar x$} = 36.3

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 57 / $\sqrt$ 139 = 4.8347

b.

mean = $\mu$ = 201.8

standard deviation = $\sigma$ = 90.9

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 90.9 / $\sqrt$ 20 = 20.3259

P($\bar x$ < 2.66.8) = P(($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (266.8 - 201.8) / 20.3259)

= P(z < 3.20)

= 0.9993

P($\bar x$ < 2.66.8) = 0.9993