Respuesta :
Answer:
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 [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.
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:
[tex]\mu_G = 15, \sigma_G = 4.2, n = 35, s_G = \frac{4.2}{\sqrt{35}} = 0.71[/tex]
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.
[tex]\mu_E = 13.4, \sigma_E = 3.7, n = 40, s_E = \frac{3.7}{\sqrt{40}} = 0.585[/tex]
Which of the following best describes the sampling distribution of barXG - bar XE, the difference in mean life span of gas and electric ovens?
By the Central Limit Theorem, the 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.
