High School

A population of values has a normal distribution with [tex]\mu = 167.4[/tex] and [tex]\sigma = 97.7[/tex]. A random sample of size [tex]n = 241[/tex] is drawn.

Find the probability that a sample of size [tex]n = 241[/tex] is randomly selected with a mean greater than [tex]185.7[/tex].

Round your answer to four decimal places.

[tex]P(\overline{M} > 185.7) =[/tex] ____

Answer :

A population of values has a normal distribution with μ = 167.4 and a = 97.7, a random sample of size n = 241, the probability that a sample of size n = 241 is randomly selected with a mean greater than 185.7.

P(M > 185.7) = 2.898.

To find the probability that a sample of size n = 241 has a mean greater than 185.7, we'll use the central limit theorem since the population follows a normal distribution.

Given:

- Population mean = 167.4

- Population standard deviation = 97.7

- Sample size = 241

- Sample mean = 185.7

First, we need to find the standard error of the sample mean (SE):

SE = 97.7 / √241

SE = 6.292

Now, let's calculate the z score:

z = (185.7 - 167.4) / (97.7 / √241)

z = 2.898.

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