Question

The amount people who pay for cell phone service varies quite a bit, but the mean monthly fee is $55 and the standard deviation is $22. The distribution is

not Normal. Many people pay about $30 for plans with 2GB data access and about $60 for 5GB of data access, but some pay much more for unlimited
data access. A sample survey is designed to ask a simple random sample of 1,000 cell phone users how much they pay. Let X be the mean amount paid.
Part A: What are the mean and standard deviation of the sample distribution of X? Show your work and justify your reasoning. (4 points)
Part B: What is the shape of the sampling distribution of X? Justify your answer. (2 points)
Part C: What is the probability that the average cell phone service paid by the sample of cell phone users will exceed $562 Show your work, 4 points)

Answer 1

a) The mean is $55 and the standard deviation is $0.6957

b) The shape of the sampling distribution of x is approximately normal.

c) 0.0749 = 7.49% probability that the average cell phone service paid by the sample of cell phone users will exceed $56.

Explanation:

Normal probability distribution

When the distribution is normal, we use the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

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 and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

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

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean and standard deviation

In this question:

Mean of the distribution: 55

Standard deviation of the distribution: 22

Sample of 1000.

This means that:

Part A: What are the mean and standard deviation of the sample distribution of X?

By the Central Limit Theorem, the mean is 55 and the standard deviation is .

Part B: What is the shape of the sampling distribution of x?

By the Central Limit Theorem, the shape of the sampling distribution of x is approximately normal.

Part C: What is the probability that the average cell phone service paid by the sample of cell phone users will exceed $56?

This is 1 subtracted by the pvalue of Z when . So:

By the Central Limit Theorem

has a pvalue of 0.9251

1 - 0.9251 = 0.0749

0.0749 = 7.49% probability that the average cell phone service paid by the sample of cell phone users will exceed $56.