The average annual cost (including tuition, room, board, books and fees) to attend a public college takes nearly a third of the annual income of a typical family with college-age children (Money, April 2012). At private colleges, the average annual cost is equal to about 60% of the typical family's income. The following random samples show the annual cost of attending private and public colleges. Data are in thousands of dollars.

Private Colleges
52.8 30.6 43.2 45.8 33.3
33.3 50.5 44.0 42.0 37.8

Public Colleges
20.3 22.8 28.2 18.5 15.6 25.6
24.1 14.4 28.5 21.8 22.0 25.8

Required:
a. Compute the sample mean and sample standard deviation for private and public colleges.
b. What is the point estimate of the difference between the two population means?
c. Develop a 95% confidence interval of the difference between the annual cost of attending private and pubic colleges.

Answer:

(a)

[tex]\bar x_1 =41.33[/tex]     [tex]\sigma_1 = 7.48[/tex]

[tex]\bar x_2 =22.3[/tex]       [tex]\sigma_2 = 4.53[/tex]

(b)

[tex]d = 19.03[/tex]

(c)

[tex]CI = 16.33[/tex] to [tex]21.73[/tex]

Step-by-step explanation:

Given

[tex]n_1 = 10\\ n_2 = 12[/tex]

And the accompanying data for private (1) and public (2) colleges

Solving (a): The sample mean and sample standard deviations

Private College

Calculating sample mean

This is calculated as:

[tex]\bar x_i = \frac{\sum x_i}{n_i}[/tex]

[tex]\bar x_1 =\frac{52.8+30.6+43.2 +45.8 +33.3+33.3 +50.5 +44.0 +42.0 +37.8}{10}[/tex]

[tex]\bar x_1 =\frac{413.3}{10}[/tex]

[tex]\bar x_1 =41.33[/tex]

Calculating sample standard deviation

This is calculated as:

[tex]\sigma_i = \sqrt\frac{\sum(x_i - \bar x_i)}{n_i-1}[/tex]

[tex]\sigma_1 = \sqrt\frac{(52.8-41.33)^2+(30.6-41.33)^2+.........+(37.8-41.33)^2}{10-1}[/tex]

[tex]\sigma_1 = \sqrt\frac{503.261}{9}[/tex]

[tex]\sigma_1 = \sqrt{55.9178888889[/tex]

[tex]\sigma_1 = 7.4778264816[/tex]

[tex]\sigma_1 = 7.48[/tex] -- approximated

Public College

Calculating sample mean

[tex]\bar x_2 =\frac{20.3 +22.8 +28.2 +18.5 +15.6 +25.6+ 24.1 +14.4 +28.5 +21.8 +22.0 +25.8}{12}[/tex]

[tex]\bar x_2 =\frac{267.6}{12}[/tex]

[tex]\bar x_2 =22.3[/tex]

Calculating sample standard deviation

[tex]\sigma_2 = \sqrt\frac{(20.3 -22.3)^2+(22.8 -22.3)^2+(28.2 -22.3)^2+(18.5 -22.3)^2+(15.6 -22.3)^2+................+(25.8-22.3)^2}{12-1}[/tex][tex]\sigma_2 = \sqrt\frac{225.96}{11}[/tex]

[tex]\sigma_2 = \sqrt{20.5418181818[/tex]

[tex]\sigma_2 = 4.532308262[/tex]

[tex]\sigma_2 = 4.53[/tex] -- approximated

Solving (b): Point estimate of the difference between the two population means

This is calculated as:

[tex]d = \bar x_1 - \bar x_2[/tex]

[tex]d = 41.33 - 22.3[/tex]

[tex]d = 19.03[/tex]

Solving (c): 95% confidence interval

First, calculate the degrees of freedom:

[tex]df = \frac{(\sigma_1^2/n_1 + \sigma_2^2/n_2)^2}{\frac{(\sigma_1^2/n_1)^2}{n_1 - 1} + \frac{(\sigma_2^2/n_2)^2}{n_2 - 1}}[/tex]

[tex]df = \frac{(7.48^2/10 + 4.53^2/12)^2}{\frac{(7.48^2/10)^2}{10 - 1} + \frac{(4.53^2/12)^2}{12 - 1}}[/tex]

[tex]df = \frac{53.3647}{31.3045/9 + 2.9244/11}[/tex]

[tex]df = \frac{53.3647}{3.7441}[/tex]

[tex]df = 14.2530[/tex]

[tex]df = 14[/tex]

Calculate the critical value (t)

At 95% confidence interval and df = 14;

[tex]df = 14[/tex]

The confidence interval is then calculated as:

[tex]CI = (\bar x_1 - \bar x_2) \± \sqrt{\sigma_1^2/n_1 + \sigma_2^2/n_2}[/tex]

[tex]CI = (41.33 - 22.3) \± \sqrt{7.48^2/10 + 4.53^2/12}[/tex]

[tex]CI = 19.03 \± \sqrt{7.305115[/tex]

[tex]CI = 19.03 \± 2.70[/tex]

Split

[tex]CI = 19.03 - 2.70[/tex] to [tex]19.03 + 2.70[/tex]

[tex]CI = 16.33[/tex] to [tex]21.73[/tex]

This implies that:

Private colleges have population mean annual cost of $16.33 to $21.73 more expensive than public colleges

Do u know rulers <3???

I really want to know what 2+2 is.

Sixty percent of the flowers in Karen's garden are roses. Which of the following shows 60% roses?

A rose and a tulip
A. 2 tulips and 3 roses
B. 3 tulips and a rose
C. A tulip and 3 roses
D. A tulip and 4 roses

What is the total surface area of the triangular prism shown?

Problem 5 Statement: 1. Andre received and spent money in the following ways last month. For each example write a positive or negative number to represent the change in money from his perspective. a. His uncle gave him $25 as a present. b. He earned $18 dollars tutoring. c. He spent $10 on a book. 2. What does 0 mean in this situation?

The dot plots show the heights of the players on two basketball teams.



Which statement is best supported by these data?

Select all expressions equivalent to 6x + 12 – 4x – 3 + 3x.

1. 2x+3x+15
2. 2x+3x+9
3. 5x+15
4. 5x+9

Select all expressions equivalent to 7x + (x + 2) + 4x.


1. 12x+2

2. 8x+2+4x

3. 11x+2

4. 7x+2+4x

2 friends equally share 5 cookies. How many cookies
does each friend get?​

Find the distance between the pair of points (-1, 4) and (-3, 7)

no links

what is the area of the trapozoid? pls help :)

One leg of a right triangle has a slope of -4. What is the slope of the other leg?

Please help me solve this! Thank you so much!

I need help with this

for any three dimensional figure the sum of the areas of all its faces is its

Write an equation of the line in slope-intercept form that
passes through each pair of points.
a) (-2, -1) and (5, 13)
b) (0, -3) and (5,-5)

16, 48,144,...
Find the 6th term.

NEED HELP ASAP PLEASE! BRAINLIEST TO RIGHT ANSWER! For two similar triangles ABC and DEF, the scale factor of ∆ABC to ∆DEF is 2 : 3. If AB = 2(BC) and DE = 3(BC), what is EF?
A. 6BC
B. 2BC
C. 3BC
D. 1.5BC

Use the distributive property to remove the parentheses
5(x-3)

Plz guys help me plz

I need major help thank you

One kilogram is approximately 2.2 pounds. Write a direct variation equation that relates x kilograms to y pounds.
I need help please.

Classify the angle pair using all names that apply

Please help I’ll mark you as brainliest if correct!

What are all the values for u that will make this equation true?
3u - 9 = 3

PLEASE ANSWER THIS QUESTION FAST Please qucickllyyy

draw the image of ABC under a translation by five units to the left

The function g(x) = 20,000(1.20)* represents the population of a rapidly growing
city x years after the year 2000.
What does the value g(10) - g(9) represent in terms of the given context?
A. The population was 103,196 people in 2009
B. The population was 123,835 people in 2010
C. There was an increase of 20,639 people from 2009 to 2010.
D. There was an increase of 227,031 people from 2009 to 2010.

find the area enclosed by the figure. Use 3.14 for π. (The figure is not to scale).

​

Rectangle MNOP with vertices M(-7, 1),
N(4, 1), 0(-4, 4), and P(-7, 4) in the line
y = -x

please help me i really need help please help me please