Question

Suppose that a random sample of size 36 is to be selected from a population with mean 43 and standard deviation 6. What is the approximate probability that X will be more than 0.5 away from the population mean?

Answer 1

Answer:

61.70% approximate probability that X will be more than 0.5 away from the population mean

Step-by-step explanation:

To solve this question, we have 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 $\sigma$, a large sample size can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 36, \sigma = 6, n = 36, s = \frac{6}{\sqrt{36}} = 1$

What is the approximate probability that X will be more than 0.5 away from the population mean?

This is the probability that X is lower than 36-0.5 = 35.5 or higher than 36 + 0.5 = 36.5.

Lower than 35.5

Pvalue of Z when X = 35.5. So

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

By the Central Limit Theorem

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

$Z = \frac{35.5 - 36}{1}$

$Z = -0.5$

$Z = -0.5$ has a pvalue of 0.3085.

30.85% probability that X is lower than 35.5.

Higher than 36.5

1 subtracted by the pvalue of Z when X = 36.5. SO

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

$Z = \frac{36.5 - 36}{1}$

$Z = 0.5$

$Z = 0.5$ has a pvalue of 0.6915.

1 - 0.6915 = 0.3085

30.85% probability that X is higher than 36.5

Lower than 35.5 or higher than 36.5

2*30.85 = 61.70

61.70% approximate probability that X will be more than 0.5 away from the population mean