Mathematics, 13.04.2021 04:40, lisnel

A student researcher compares the ages of cars owned by students and cars owned by faculty at a local state college. A sample of 263 cars owned by students had an average age of 7.25 years. A sample of 291 cars owned by faculty had an average age of 7.12 years. Assume that the population standard deviation for cars owned by students is 3.77 years, while the population standard deviation for cars owned by faculty is 2.99 years. Determine the 90% confidence interval for the difference between the true mean ages for cars owned by students and faculty. Step 1 of 3: Find the point estimate for the true difference between the population means.

Answers: 1

Show answers

Answers

answered: addisynshepherd

The point estimate for the true difference between the population means is 0.13.

The 90% confidence interval for the difference between the true mean ages for cars owned by students and faculty is between -0.35 years and 0.61 years.

Step-by-step explanation:

To solve this question, before building the confidence interval, 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 $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Subtraction between normal variables:

When we subtract two 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 sample of 263 cars owned by students had an average age of 7.25 years. The population standard deviation for cars owned by students is 3.77 years.

This means that:

$\mu_s = 7.25, \sigma_s = 3.77, n = 263, s_s = \frac{3.77}{\sqrt{263}} = 0.2325$

A sample of 291 cars owned by faculty had an average age of 7.12 years. The population standard deviation for cars owned by faculty is 2.99 years.

This means that:

$\mu_f = 7.12, \sigma_f = 2.99, n = 291, s_f = \frac{2.99}{\sqrt{291}} = 0.1753$

Difference between the true mean ages for cars owned by students and faculty.

Distribution s - f. So

$\mu = \mu_s - \mu_f = 7.25 - 7.12 = 0.13$

This is also the point estimate for the true difference between the population means.

$s = \sqrt{s_s^2+s_f^2} = \sqrt{0.2325^2+0.1753^2} = 0.2912$

90% confidence interval for the difference:

We have that to find our $\alpha$ level, that is the subtraction of 1 by the confidence interval divided by 2. So:

$\alpha = \frac{1 - 0.9}{2} = 0.05$

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 = 1.645*0.2912 = 0.48

The lower end of the interval is the sample mean subtracted by M. So it is 0.13 - 0.48 = -0.35 years

The upper end of the interval is the sample mean added to M. So it is 0.13 + 0.48 = 0.61 years.

The 90% confidence interval for the difference between the true mean ages for cars owned by students and faculty is between -0.35 years and 0.61 years.

answered: Guest

Is there a picture of a graph or whatever that goes along with this?

answered: Guest

5.25 ounces

step-by-step explanation:

divided 3/4 and 6

Other questions on the subject: Mathematics

Mathematics, 21.06.2019 12:30, keananashville

Danny draws a transversal, t, on two parallel lines ab and cd, as shown below: he makes the following table to prove that the alternate interior angles are equal: which is the missing justification? (6 points) corresponding angles formed by a transversal on parallel lines are supplementary. corresponding angles formed by a transversal on parallel lines are congruent. co-interior angles formed by a transversal on parallel lines are congruent. co-exterior angles formed by a transversal on parallel lines are congruent.

Answers: 1

continue

Mathematics, 21.06.2019 13:00, danyratilladc

How can we find the measure of an exterior angle if we know the measurement of an interior angle

Answers: 1

continue

Mathematics, 21.06.2019 14:30, ajl05

Christine went shopping and bought each of her six six nephews a? gift, either a video costing? $14.95 or a cd costing? $16.88. she spent ? $ 97.42 97.42 on the gifts. how many videos and how many cds did she? buy?

Answers: 1

continue

Mathematics, 21.06.2019 15:30, rina62

Driving 75 mph, it takes alicia 2 hours to reach the airport to go on a vacation. it then takes her 4 hours to get to her destination with the jet traveling at a speed of 400 mph. how many miles does alicia travel to get to her destination?

Answers: 2

continue

Do you know the correct answer?

A student researcher compares the ages of cars owned by students and cars owned by faculty at a loca...