Question

The amount people who pay for cell phone service varies quite a bit, but the mean

Answer 1

Answer:

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.

Step-by-step explanation:

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:

$Z = \frac{X - \mu}{\sigma}$

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 $s = \frac{\sigma}{\sqrt{n}}$.

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 $\mu = p$ and standard deviation $s = \sqrt{\frac{p(1-p)}{n}}$

In this question:

Mean of the distribution: 55

Standard deviation of the distribution: 22

Sample of 1000.

This means that:

$\mu = 55, \sigma = 22, n = 1000$

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 $s = \frac{22}{\sqrt{1000}} = 0.6957$.

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 $X = 56$. So:

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{56 - 55}{0.6957}$

$Z = 1.44$

$Z = 1.44$ 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.