Question

A distribution of values is normal with a mean of 220 and a standard deviation of 13. From this distribution, you are drawing samples of size 35. Find the interval containing the middle-most 48% of sample means:

Answer 1

Answer:

The interval containing the middle-most 48% of sample means is between 218.59 to 221.41.

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 normally distributied random variable X, with mean $\mu$ and standard deviation $\sigma$, the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 220, \sigma = 13, n = 35, s = \frac{13}{\sqrt{35}} = 2.1974$

Find the interval containing the middle-most 48% of sample means:

50 - 48/2 = 26th percentile to 50 + 48/2 = 74th percentile. So

74th percentile

value of X when Z has a pvalue of 0.74. So X when Z = 0.643.

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

By the Central Limit Theorem

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

$0.643 = \frac{X - 220}{2.1974}$

$X - 220 = 0.643*2.1974$

$X = 221.41$

26th percentile

Value of X when Z has a pvalue of 0.26. So X when Z = -0.643

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

$-0.643 = \frac{X - 220}{2.1974}$

$X - 220 = -0.643*2.1974$

$X = 218.59$

The interval containing the middle-most 48% of sample means is between 218.59 to 221.41.