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

Answer from: 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.

Answer from: Quest

The first option would be correct if it were one half and not one

Answer from: Quest

x + y = 90 5x – 10y = 90 90 – 10y = 5x 90 + 10y = 5x

step-by-step explanation:

last month maria hiked the 5-mile mountain trail a number of times and she hiked the 10-mile canal trail several times. let x represent the number of times she hiked the 5-mile trail, and let y represent the number of times she hiked the 10-mile trail. if she hiked a total of 90 miles, which equation can be used to find the number of times maria hiked each trail? x + y = 90 5x – 10y = 90 90 – 10y = 5x 90 + 10y = 5x

Another question on Mathematics

Mathematics, 21.06.2019 13:00

Myron put $5000 in a 2-year cd playing 3% interest, compounded monthly. after 2 years, he withrew all his money. what was the amount of the withdrawal?

Answers: 1

Answer

Mathematics, 21.06.2019 19:00

Graph g(x)=2cosx . use 3.14 for π . use the sine tool to graph the function. graph the function by plotting two points. the first point must be on the midline and closest to the origin. the second point must be a maximum or minimum value on the graph closest to the first point.

Answers: 1

Answer

Mathematics, 21.06.2019 21:30

Is triangle jkl congruent to triangle rst? if so, what is the scale factor?

Answers: 1

Answer

Mathematics, 22.06.2019 00:10

How do i take any fraction and make it into a decimal or percentage?

Answers: 1

Answer

You know the right answer?

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