Question

A gaming company would like to test the hypothesis that the average age of someone who uses gaming console A is different from the average age of someone who uses gaming console B. A random sample of 36 gaming console A users had an average age of 34.2 years while a random sample of 30 gaming console B users had an average age of 32.7 years. Assume that the population standard deviation for the age of gaming console A and gaming console B users is 3.9 and 4.0​ years, respectively. This company would like to set α = 0.10.

a) What is the critical value for this hypothesis​ test?
a) 2.33
b) 1.28
c) 1.645
d) 1.96
b) What is the 90% confidence interval for the difference in population means?
a) (-0.11, 3.11)
b) (-1.70, 4.70)
c) (0.65, 2.35)
d) (1.18, 1.82)

Answer 1

Question a:

c) 1.645

Question b:

a) (-0.11, 3.11)

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
`\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 = (\sigma)/(√(n))`.

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
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

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
`\mu_A = 34.2, s_A = (3.9)/(√(36)) = 0.65`

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
`\mu_B = 32.7, s_B = (4)/(√(30)) = 0.73`

Distribution of the difference in population means:

`\mu = \mu_A - \mu_B = 34.2 - 32.7 = 1.5`

`s = √(s_A^2+s_B^2) = √(0.65^2+0.73^2) = 0.9774`

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
`\alpha` level, that is the subtraction of 1 by the confidence interval divided by 2. So:

`\alpha = (1 - 0.9)/(2) = 0.05`

Now, we have to find z in the Ztable as such z has a pvalue of
`1 - \alpha`.

That is z with a pvalue of
`1 - 0.05 = 0.95`, so Z = 1.645.

Now, find the margin of error M as such

`M = zs`

`M = 1.645*0.9774 = 1.61`

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.