Question

A simple random sample of size equals 49 is obtained from a population with mu equals 84 and sigma equals 14. ​(a) Describe the sampling distribution of x overbar. ​(b) What is Upper P (x overbar greater than 87.8 )​

Answer 1

(a) The sampling distribution of
`\bar X` is given by;

`\bar X` ~ Normal (
`\mu =84, s = (\sigma)/(√(n) ) =(14)/(√(49) )` )

(b) P(
`\bar X` > 87.8) = 0.0287

Explanation:

We are given that a simple random sample of size equals 49 is obtained from a population with mu equals 84 and sigma equals 14.

Let
`\bar X` = sample mean

The z-score probability distribution for sample mean is given by;

Z =
`(\bar X-\mu)/((\sigma)/(√(n) ) )` ~ N(0,1)

where,
`\mu` = population mean = 84

`\sigma` = standard deviation = 14

n = sample size = 49

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) 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.

(a) The sampling distribution of
`\bar X` is given by;

`\bar X` ~ Normal (
`\mu =84, s = (\sigma)/(√(n) ) =(14)/(√(49) )` )

(b) Probability of
`\bar X` greater than 87.8 is given by = P(
`\bar X` > 87.8)

P(
`\bar X` > 87.8) = P(
`(\bar X-\mu)/((\sigma)/(√(n) ) )` >
`(87.8-84)/((14)/(√(49) ) )` ) = P(Z > 1.90) = 1 - P(Z
`\leq` 1.90)

= 1 - 0.9713 = 0.0287

The above probability is calculated by looking at the value of x = 1.90 in the z table which has an area of 0.9713.