Question

A simple random sample of size nequals64 is obtained from a population with mu equals 75 and sigma equals 8.

Answer 1

Answer:

a) Approximately normal with mean 75 and standard deviation s = 1.

b) 0.0606

c) 0.0179

d) 0.9110

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, 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:

$\mu = 75, \sigma = 8, n = 64, s = \frac{8}{\sqrt{64}} = 1$

​(a) Describe the sampling distribution of x overbar. ​

Approximately normal with mean 75 and standard deviation s = 1.

(b) What is Upper P (x overbar greater than 76.55 )​? ​

This is 1 subtracted by the pvalue of Z when X = 76.55.

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

By the Central Limit Theorem

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

$Z = \frac{76.55 - 75}{1}$

$Z = 1.55$

$Z = 1.55$ has a pvalue of 0.9394.

1 - 0.9394 = 0.0606

(c) What is Upper P (x overbar less than or equals 72.9 )​?

This is the pvalue of Z when X = 72.9. So

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

$Z = \frac{72.9 - 75}{1}$

$Z = -2.1$

$Z = -2.1$ has a pvalue of 0.0179.

(d) What is Upper P (73.5 less than x overbar less than 77.05 )​?

Pvalue of Z when X = 77.05 subtracted by the pvalue of Z when X = 73.5 So

X = 77.05

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

$Z = \frac{77.05 - 75}{1}$

$Z = 2.05$

$Z = 2.05$ has a pvalue of 0.9778

X = 73.5

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

$Z = \frac{73.5 - 75}{1}$

$Z = -1.5$

$Z = -1.5$ has a pvalue of 0.0668

0.9778 - 0.0668 = 0.9110