Respuesta :
Answer:
Question a:
c) 1.645
Question b:
a) (-0.11, 3.11)
Step-by-step explanation:
Before answering the question, we need to understand the central limit theorem and subtraction between normal variables:
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.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
Subtraction between normal variables:
When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.
A random sample of 36 gaming console A users had an average age of 34.2 years, with a standard deviation of 3.9 years.
This means that [tex]\mu_A = 34.2, s_A = \frac{3.9}{\sqrt{36}} = 0.65[/tex]
A random sample of 30 gaming console B users had an average age of 32.7 years, with a standard deviation of 4 years.
This means that [tex]\mu_B = 32.7, s_B = \frac{4}{\sqrt{30}} = 0.73[/tex]
Distribution of the difference in population means:
[tex]\mu = \mu_A - \mu_B = 34.2 - 32.7 = 1.5[/tex]
[tex]s = \sqrt{s_A^2+s_B^2} = \sqrt{0.65^2+0.73^2} = 0.9774[/tex]
a) What is the critical value for this hypothesis test?
We test if the means are different, which means that we have a two-tailed test.
We have the standard deviations for the population, which means that we have a Z test.
Since it is a two-tailed Z-test, the critical value is Z with a p-value of 1 - (0.1/2) = 1 - 0.05 = 0.95, so, looking at the z-table, Z = 1.645, which is option C.
b) What is the 90% confidence interval for the difference in population means?
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1 - 0.9}{2} = 0.05[/tex]
Now, we have to find z in the Ztable as such z has a pvalue of [tex]1 - \alpha[/tex].
That is z with a pvalue of [tex]1 - 0.05 = 0.95[/tex], so Z = 1.645.
Now, find the margin of error M as such
[tex]M = zs[/tex]
[tex]M = 1.645*0.9774 = 1.61[/tex]
The lower end of the interval is the sample mean subtracted by M. So it is 1.5 - 1.61 = -0.11
The upper end of the interval is the sample mean added to M. So it is 1.5 + 1.61 = 3.11
The correct answer is given by option A.
