Mathematics, 18.03.2021 01:20, jadeb124

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 - bar XE, 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.
E. Mean = 1.6 years, standard deviation = 0.40 years, shape: moderately right skewed.

Answers: 1

Show answers

Answers

answered: mazzi88

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

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution:

Problems of normal distributions can be solved using 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.

Subtraction of normal variables:

When we subtract normal variables, 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:

$\mu_G = 15, \sigma_G = 4.2, n = 35, s_G = \frac{4.2}{\sqrt{35}} = 0.71$