A consumer group has determined that the distribution of life spans for gas ovens has a mean of 15.0 years and a standard deviation of 4.2 years. The distribution of life spans for electric ovens has a mean of 13.4 years and a standard deviation of 3.7 years. Both distributions are moderately skewed to the right.

Suppose we take a simple random sample of 35 gas ovens and a second simple random sample of 40 electric ovens. Suppose we take a simple random sample of 35 gas ovens and a second SRS of 40 electric ovens. Which of the following best describes the sampling distribution of barXG – barXE, the difference in mean life span of gas and electric ovens?

A. Mean=1.6 years, standard deviation=7.9 years, shape: moderately right skewed.

B. Mean=1.6 years, standard deviation=0.92 years, shape: approximately normal.

C. Mean=1.6 years, standard deviation=0.92 years, shape: moderately right skewed.

D. Mean=1.6 years, standard deviation=0.40 years, shape: approximately normal.

## Answer: B

**Mean=1.6 years, standard deviation=0.92 years, shape: approximately normal.**

## Explanation

We need to understand the normal probability distribution and central limit theorem in order to solve this question.

### Normal Probability Distribution:

We can use z-score formula to solve the problems of normal distributions.

In a set with mean û and standard deviation ó, the z-score of a measure X given by

Z= X-û/ó

The Z-score measures how many standard deviations and the measure is from the mean. After finding 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 P-value, we get the probability that the value of the measure is greater than X.

### Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean û and standard deviation ó, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean û and standard deviation

**S=ó/√n**

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

### Subtraction of normal variables:

When the normal variables are subtracted, the mean is the subtraction of the means, while the standard deviation is the square root of the sum of the variances.

**35 gas ovens**

A consumer group has determined that the distribution of life spans for gas ovens has a mean of 15.0 years and a standard deviation of 4.2 years. This means that:

ûG=15, óG=4.2, n=35, sG=4.2/√35=0.71

**40 electric ovens**

The distribution of life spans for electric ovens has a mean of 13.4 years and a standard deviation of 3.7 years.

ûE=13.4, óE=3.7, n=40, she=3.7/√40=0.585

## Which of the following best describes the sampling distribution of barXG – barXE, the difference in mean life span of gas and electric ovens?

By the central limit theorem, the shape is approximately normal.

**Mean: û=ûG – ûE = 15 – 13.4 = 1.6**

### Standard deviation:

**s= √s²G+s²E = √(0.71)² + (0.585)² = 0.92**

Thus, the correct answer is **option B**.