Mathematics, 15.07.2020 02:01 cireland

Suppose that a population is known to be normally distributed with £ = 2400 and € = 210. Of a random sample of size n = 8 is selected, calculate the probability that the sample mean will exceed 2,500.

Suppose that a population is known to be normally distributed with £ = 2400 and € = 210. Of a rando

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Answer from: samieskyy290

0.0885 = 8.85% probability that the sample mean will exceed 2,500.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

When the distribution is normal, we use the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean $\mu = p$ and standard deviation $s = \sqrt{\frac{p(1-p)}{n}}$

In this question:

$\mu = 2400, \sigma = 210, n = 8, s = \frac{210}{\sqrt{8}} = 74.25$

Calculate the probability that the sample mean will exceed 2,500.

This is 1 subtracted by the pvalue of Z when X = 2500. So

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem:

$Z = \frac{X - \mu}{s}$

$Z = \frac{2500 - 2400}{74.25}$

Z = 1.35

Z = 1.35 has a pvalue of 0.9115

1 - 0.9115 = 0.0885

0.0885 = 8.85% probability that the sample mean will exceed 2,500.

Answer from: Quest

the answer is (-8,-5)

step-by-step explanation:

the endpoint to midpoint is half the length. the length is (-2,-1) so when you are at the midpoint just add it to it to find your answer -6+(-2)= -8 and -4+(-1)=-5 so (-8,-5)

Answer from: Quest

your answer is -9

step-by-step explanation:

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Suppose that a population is known to be normally distributed with £ = 2400 and € = 210. Of a random...