Answer to High school statistics example. Here is a photo

Houda Merfouk

Answered question

2022-03-30

Answer & Explanation

star233

Skilled2023-04-26Added 403 answers

We are given that the population of values has a normal distribution with a mean of

μ = 244.7

and a standard deviation of

σ = 12

. We intend to draw a random sample of size

n = 201

and find the probability that the sample mean falls between 243.6 and 245.7.
We can start by standardizing the distribution of the sample mean using the standard error of the mean, which is given by:

S E = \frac{σ}{\sqrt{n}}

where

σ

is the population standard deviation and

n

is the sample size. Substituting the given values, we get:

S E = \frac{12}{\sqrt{201}} \approx 0.8485

Next, we can standardize the sample mean using the standard normal distribution, which has a mean of 0 and a standard deviation of 1. This gives us:

Z = \frac{\bar{X} - μ}{S E}

where

\bar{X}

is the sample mean. Substituting the given values, we get:

Z = \frac{243.6 - 244.7}{0.8485} \approx - 1.299

and

Z = \frac{245.7 - 244.7}{0.8485} \approx 1.178

Using a standard normal distribution table or calculator, we can find the probabilities associated with these two values of

Z

. The probability of the sample mean falling between 243.6 and 245.7 is then given by the difference between these two probabilities:

P (243.6 < \bar{X} < 245.7) = P (- 1.299 < Z < 1.178) \approx 0.8366 - 0.1020 \approx 0.7346

Therefore, the probability that a sample of size

n = 201

is randomly selected with a mean between 243.6 and 245.7 is approximately 0.7346.

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