Respuesta :
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 [tex]\mu[/tex] 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 [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
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:
[tex]\mu_G = 15[/tex]
[tex]s_G = \frac{4.2}{\sqrt{35}} = 0.71[/tex]
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:
[tex]\mu_E = 13.4[/tex]
[tex]s_E = \frac{3.7}{\sqrt{40}} = 0.585[/tex]
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:
[tex]\mu = \mu_G - \mu_E = 15 - 13.4 = 1.6[/tex]
Standard deviation:
[tex]s = \sqrt{s_G^2+s_E^2} = \sqrt{0.71^2+0.585^2} = 0.92[/tex]
So the correct answer is given by option b.
