A population of values has a normal distribution with \mu=204.3 and \sigma=43. You intend to draw a random sample of size n=111.Find the probability that

Question

A population of values has a normal distribution with $μ = 204.3$ and $σ = 43$ . You intend to draw a random sample of size $n = 111$ .
Find the probability that a single randomly selected value is less than 191.2.
$P (X < 191.2) =$ ?
Find the probability that a sample of size $n = 111$ is randomly selected with a mean less than 191.2.
$P (M < 191.2) =$ ?
Write your answers as numbers accurate to 4 decimal places.

Answer 1

Step 1
Solution:
Let X be random variable.
From the given information, X follows normal distribution with mean $μ = 204.3$ and a standard deviation is $σ = 43$ .
Step 2
Then, the probability that a single randomly selected value is less than 191.2 is
$P (X < 191.2) = P (\frac{X - μ}{σ} < \frac{191.2 - 204.3}{43})$
$= P (Z < - 0.305)$
$= 0.3802 [U s i n g t h e e x c e l f u n c t i o n = N O R M . D I S T (- 0.305, 0, 1, T R U E)]$
Thus, the probability that a single randomly selected value is less than 191.2 is 0.3802.
Step 3
From the given information, a sample size is $n = 111$ .
Then, the probability that a randomly selected with a mean less than 191.2 is
$P (M < 191.2) = P (\frac{M - μ}{\frac{σ}{\sqrt{n}}} < \frac{191.2 - 204.3}{\frac{43}{\sqrt{111}}})$
$= P (Z < \frac{\sqrt{111} (- 13.1)}{43})$
$= P (Z < - 3.210)$
$= 0.0007 [U s i n g t h e e x c e l f u n c t i o n = N O R M . D I S T (- 3.210, 0, 1, T R U E)]$
Thus, the probability that a randomly selected with a mean less than 191.2 is 0.0007.

A population of values has a normal distribution with \mu=204.3 and \sigma=43. You intend to draw a random sample of size n=111.Find the probability that

Expert Answer

Not exactly what you’re looking for?

Relevant Questions