Question

A distribution of values is normal with a mean of 60 and a standard deviation of 16. From this distribution, you are drawing samples of size 25.

Find the interval containing the middle-most 76% of sample means:

-Enter your answer using interval notation.

-In this context, either inclusive or exclusive intervals would be acceptable.

-Round your numbers to one decimal place.

-Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer 1

Answer:

$\left[56.2, 63.8]\right$

Step-by-step explanation:

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

Normal probability distribution

Problems of normally distributed samples are solved using 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 random variable X, with mean $\mu$ and standard deviation $\sigma$, the sample means with size n of at least 30 can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 60, \sigma = 16, n = 25, s = \frac{16}{\sqrt{25}} = 3.2$

Find the interval containing the middle-most 76% of sample means:

50 - 76/2 = 12th percentile to the 50 + 76/2 = 88th percentile.

12th percentile

value of X when Z has a pvalue of 0.12. So X when Z = -1.175

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

By the Central Limit Theorem

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

$-1.175 = \frac{X - 60}{3.2}$

$X - 60 = -1.175*3.2$

$X = 56.2$

88th percentile

value of X when Z has a pvalue of 0.88. So X when Z = 1.175

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

$1.175 = \frac{X - 60}{3.2}$

$X - 60 = 1.175*3.2$

$X = 63.8$

So the answer is:

$\left[56.2, 63.8]\right$