Based on records kept at a gas station, the distribution of gallons of gas purchased by customers is skewed to the right with mean 10 gallons and standard deviation 4 gallons. A random sample of 64 customer receipts was selected, and the sample mean number of gallons was recorded. Suppose the process of selecting a random sample of 64 receipts and recording the sample mean number of gallons was repeated for a total of 100 samples. Which of the following is the best description of a dotplot created from the 100 sample means? a. The dotplot is skewed to the right with mean 10 gallons and standard deviation 4 gallons. b. The dotplot is skewed to the right with mean 10 gallons and standard deviation 0.5 gallon. c. The dotplot is skewed to the right with mean 10 gallons and standard deviation 0.4 gallon. d. The dotplot is approximately normal with mean 10 gallons and standard deviation 0.5 gallon. e. The dotplot is approximately normal with mean 10 gallons and standard deviation 0.4 gallon.

The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex].

In this problem, we have that:

[tex]\mu = 10, \sigma = 4, n = 100, s = \frac{4}{\sqrt{100}} = 0.4[/tex]

So the correct answer is:

e. The dotplot is approximately normal with mean 10 gallons and standard deviation 0.4 gallon.

alokdubeyvidyaatech alokdubeyvidyaatech · Answer 2 · 2021-11-16T16:13:40+01:00

The dotplot is approximately normal with a mean of 10 gallons and a standard deviation of 0.4 gallons.

The correct answer is option e

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean μ and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean μ and standard deviation [tex]s=\frac{\sigma}{\sqrt{n} }[/tex]

Mean of 10 gallons and standard deviation of 4 gallons.

This means that, μ = 10, [tex]\sigma=4[/tex]

Random samples of size 100.

This means that [tex]n=100[/tex]

Use the Central Limit Theorem to find the mean and standard error of the mean of the indicated sampling distribution.

The standard deviation is:

[tex]s=\frac{\sigma}{\sqrt{n} }=\frac{4}{\sqrt{100} }=0.4[/tex]

Therefore, The dotplot is approximately normal with a mean of 10 gallons and a standard deviation of 0.4 gallons.

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