# 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

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.

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.