Question

(1 point) The cost of unleaded gasoline once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Sixteen gas stations are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations. a) What's the approximate probability that the average price for 16 gas stations is over $4.69? almost zero 0.1587 0.0943 unknown b) Find the probability that the average price for 30 gas stations is less than $4.55.

Answer 1

a) unknown

b) 1.43% probability that the average price for 30 gas stations is less than $4.55.

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 = (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 distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

`\mu = 4.59, \sigma = 0.1`

a) What's the approximate probability that the average price for 16 gas stations is over $4.69?

Sample size less than 30 and unknow distribution. Therefore, the central limit theorem cannot be applied and the answer is unknown.

b) Find the probability that the average price for 30 gas stations is less than $4.55.

Now
`n = 30, s = (0.1)/(√(30)) = 0.0183`

This probability is the pvalue of Z when X = 4.55. So

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

By the Central Limit Theorem

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

`Z = (4.55 - 4.59)/(0.0183)`

`Z = -2.19`

`Z = -2.19` has a pvalue of 0.0143

1.43% probability that the average price for 30 gas stations is less than $4.55.