Question

The amount people pay for cable service varies quite a bit but the mean monthly fee is $142 and the standard deviation is $29. the distribution is not normal many people pay about $76 for basic cable and about $160 for premium service but some pay much more. a sample survey is designed to ask a simple random sample of 1,500 cable service customers how much they pay. let x hat be the mean amount paid.

Part a: what are the mean an standard deviation of the sample distribution of x hat show your work and justify your reasoning.
Part b: what is the shape of the sampling distribution of x hat justify your answer.
Part C: what is the probability that the average cable service paid by the sample of cable service customers will exceed $143? show your work.

Answer 1

a) By the Central Limit Theorem, the mean is $142 and the standard deviation is $0.7488.

b) By the Central Limit Theorem, approximately normal.

c) 0.0901 = 9.01% probability that the average cable service paid by the sample of cable service customers will exceed $143

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

When the distribution is normal, we use 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.

The mean monthly fee is $142 and the standard deviation is $29.

This means that
`\mu = 142, \sigma = 29`

Part a: what are the mean an standard deviation of the sample distribution of x hat show your work and justify your reasoning.

Sample of 1500(larger than 30).

By the Central Limit Theorem

The mean is $142

The standard deviation is
`s = (29)/(√(1500)) = 0.7488`

Part b: what is the shape of the sampling distribution of x hat justify your answer.

By the Central Limit Theorem, approximately normal.

Part C: what is the probability that the average cable service paid by the sample of cable service customers will exceed $143?

This is 1 subtracted by the pvalue of Z when X = 143. So

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

By the Central Limit Theorem

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

`Z = (143 - 142)/(0.7488)`

`Z = 1.34`

`Z = 1.34` has a pvalue of 0.9099

1 - 0.9099 = 0.0901

0.0901 = 9.01% probability that the average cable service paid by the sample of cable service customers will exceed $143