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Fudgin [204]
3 years ago
10

A consumer group has determined that the distribution of life spans for gas ranges (stoves) has a mean of 15.0 years and a stand

ard deviation of 4.2 years. The distribution of life spans for electric ranges 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 ranges and a second SRS of 40 electric ranges. Which of the following best describes the sampling distribution of the difference in mean life span of gas ranges and electric ranges?
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.
Mathematics
1 answer:
Art [367]3 years ago
8 0

Answer:

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

Step-by-step explanation:

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.

A consumer group has determined that the distribution of life spans for gas ranges (stoves) has a mean of 15.0 years and a standard deviation of 4.2 years. Sample of 35:

This means that:

\mu_G = 15

s_G = \frac{4.2}{\sqrt{35}} = 0.71

The distribution of life spans for electric ranges has a mean of 13.4 years and a standard deviation of 3.7 years. Sample of 40:

This means that:

\mu_E = 13.4

s_E = \frac{3.7}{\sqrt{40}} = 0.585

Which of the following best describes the sampling distribution of the difference in mean life span of gas ranges and electric ranges?

Shape is approximately normal.

Mean:

\mu = \mu_G - \mu_E = 15 - 13.4 = 1.6

Standard deviation:

s = \sqrt{s_G^2+s_E^2} = \sqrt{0.71^2+0.585^2} = 0.92

So the correct answer is given by option b.

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