Question

Furnace repair bills are normally distributed with a mean of 267 dollars and a standard deviation of 20 dollars. If 64 of these repair bills are randomly selected, find the probability that they have a mean cost between 267 dollars and 269 dollars.

Answer 1

Answer: 0.7881446

Explanation:

Given : Mean :
`\mu = 267\text{ dollars}`

Standard deviation :
`\sigma =20 \text{ dollars}`

Sample size :
`n=64`

The formula to calculate the z-score :-

`z=(x-\mu)/((\sigma)/(√(n)))`

For x= 267 dollars

`z=(267-267)/((20)/(√(64)))=0`

For x= 269 dollars.

`z=(269-267)/((20)/(√(64)))=0.80`

The P-value :
`P(0<z<0.8)=P(z<0.8)-P(z<0)`

`= 0.7881446-0.5= 0.2881446\approx 0.7881446`

Hence, the probability that they have a mean cost between 267 dollars and 269 dollars.= 0.7881446

Answer 2

There is a 28.81% probability that they have a mean cost between 267 dollars and 269 dollars.

Explanation:

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
`(\sigma)/(√(n))`

Problems of normally distributed samples can be 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 = (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.

In this problem, we have that:

`\mu = 267, \sigma = 20, n = 64, s = (20)/(√(64)) = 2.5`.

Find the probability that they have a mean cost between 267 dollars and 269 dollars.

This probability is the pvalue of Z when X = 269 subtracted by the pvalue of Z when X = 267. So:

X = 269

`Z = (X - \mu)/(s)`

`Z = (269 - 267)/(2.5)`

`Z = 0.8`

`Z = 0.8` has a pvalue of 0.7881.

X = 267

`Z = (X - \mu)/(s)`

`Z = (267 - 267)/(2.5)`

`Z = 0`

`Z = 0` has a pvalue of 0.50.

So there is a 0.7881 - 0.50 = 0.2881 = 28.81% probability that they have a mean cost between 267 dollars and 269 dollars.