Question

I need help with this chapter 13 statistics problem! Can someone find the answer and/or show me how to solve for it

Answer 1

Using the normal distribution, it is found that the probabilities are given as follows:

• $P(\overline{x} < 29) = 0.5596$.

• $P(\overline{x} > 26) = 0.6879)$.

Normal Probability Distribution

The z-score of a measure X of a normally distributed variable with mean $\mu$ and standard deviation $\sigma$ is given by:

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

• The z-score measures how many standard deviations the measure is above or below the mean.

• Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.

• By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

In this problem, the parameters are given as follows:

$\mu = 28.29, \sigma = 33.493, n = 52, s = \frac{33.493}{\sqrt{52}} = 4.6446$

The probability that the mean is less than 29 million is the p-value of Z when X = 29, hence:

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

By the Central Limit Theorem:

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

$Z = \frac{29 - 28.29}{4.6446}$

Z = 0.15

Z = 0.15 has a p-value of 0.5596.

Hence, $P(\overline{x} < 29) = 0.5596$.

The probability that the mean is more than 26 million is the 1 subtracted by the p-value of Z when X = 26, hence:

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

By the Central Limit Theorem:

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

$Z = \frac{26 - 28.29}{4.6446}$

Z = -0.49

Z = -0.49 has a p-value of 0.3121.

1 - 0.3121 = 0.6879.

Then, $P(\overline{x} > 26) = 0.6879)$.

More can be learned about the normal distribution at https://brainly.com/question/27919134

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