Question

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.† 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

51.8	44.2	45.0	32.3	45.0
31.6	45.8	38.8	49.5	43.0

Public Colleges

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

(a)

Compute the sample mean (in thousand dollars) and sample standard deviation (in thousand dollars) for private colleges. (Round the standard deviation to two decimal places.)

sample mean$  thousandsample standard deviation$  thousand

Compute the sample mean (in thousand dollars) and sample standard deviation (in thousand dollars) for public colleges. (Round the standard deviation to two decimal places.)

sample mean$  thousandsample standard deviation$  thousand

(b)

What is the point estimate (in thousand dollars) of the difference between the two population means? (Use Private − Public.)

$  thousand

Interpret this value in terms of the annual cost (in dollars) of attending private and public colleges.

We estimate that the mean annual cost to attend private colleges is $  more than the mean annual cost to attend public college

(c)

Develop a 95% confidence interval (in thousand dollars) of the difference between the mean annual cost of attending private and public colleges. (Use Private − Public. Round your answers to one decimal place.)

$  thousand to $  thousand

Answer 1

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.

Private Colleges

51.8	44.2	45.0	32.3	45.0
31.6	45.8	38.8	49.5	43.0

Public Colleges

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

(a)

Compute the sample mean (in thousand dollars) and sample standard deviation (in thousand dollars) for private colleges. (Round the standard deviation to two decimal places.)

sample mean$  thousandsample standard deviation$  thousand

Compute the sample mean (in thousand dollars) and sample standard deviation (in thousand dollars) for public colleges. (Round the standard deviation to two decimal places.)

sample mean$  thousandsample standard deviation$  thousand

Sample Mean =

sample standard deviation =

For Private Colleges

1 = ( 51.8 +44.2+ 45.0 +32.3 +45.0+ 31.6+ 45.8 +38.8+ 49.5 +43.0 ) / 10

   = 427 / 10 = 42.7

Standard deviation =

=

                                = 6.651316

So For Private Colleges

Sample Mean 2 = 42.7                                        (in thousand dollars)

Standard deviation s1 = 6.651316                      (in thousand dollars)

Similarly we calculate sample mean and standard deviation for Public Colleges is

Sample Mean 2 = 22.3                                           (in thousand dollars)

Standard deviation s2 = 4.532308                        (in thousand dollars)

b)

What is the point estimate (in thousand dollars) of the difference between the two population means? (Use Private − Public.)

Private − Public = 1 - 2   = 42.7    - 22.3   = 20.4

Thus point estimate (in thousand dollars) of the difference between the two population means is 20.4

Interpretation

This value in terms of the annual cost (in dollars) of attending private and public colleges.

We estimate that the mean annual cost to attend private colleges is $20400 (in dollars) more than the mean annual cost to attend public college

(c)

Develop a 95% confidence interval (in thousand dollars) of the difference between the mean annual cost of attending private and public colleges. (Use Private − Public. Round your answers to one decimal place.)

For this we fisrt calculate standard Error SE

SE =

where

sd2 =

Here n1 = 10 , n2 = 10 , s1 = 6.651316    , s2 = 4.532308     

sd2 = = 32.39091

Hence SE = = = 2.545227

SE = 2.545227

Now 95% confidence interval of the difference between the mean is given by

CI = { ( 1 - 2 ) - * SE , ( 1 - 2 ) + * SE   }

Here is t-distributed with n1+n2-2 = 18 degree of freedom and =0.05 { for 95% confidence },

It can be computed from statistical book or more accurately from any software like R,Excel

From R

> qt(1-0.05/2,18)
[1] 2.100922

Hence = 2.100922

So 95% confidence interval of the difference between the mean is given by

CI = { ( 1 - 2 ) - * SE , ( 1 - 2 ) + * SE   }

     = { ( 42.7 - 22.3 ) - 2.100922 * 2.545227 , ( 42.7 - 22.3 ) + 2.100922 * 2.545227   }

     = { 15.05268 , 25.74732 }

So 95% confidence interval (in thousand dollars) of the difference between the mean annual cost of attending private and public colleges. is    $15.05 thousand to $ 25.75 thousand