Question

Parking lots; A survey of autos parked in student and staff lots at a large university classified the brands by country of origin, as seen in the table

                               Driver

                       Student      Staff

American         104            105

European         33               11

Asian                55               48

a) What percentage of all cars surveyed wcre American?

b) What percentage of the American cars were owned by students?

c) What percen of the students owned American

d) What is the marginal distribution of origin ?

e) What is the condition of drivers American cars?

f) Do you think that the origin of the car is independent of the type of driver? Explain

Answer 1

Based on the given data,

Percentage of all cars surveyed were American = No. of American cars surveyed / Total No. of cars surveyed

=

Hence, about 59% of all cars surveyed wcre American.

b)  Percentage of the American cars were owned by students:

Percentage of the American cars were owned by students

= No. of cars owned by American students / Total no. of American cars

=

About 50% of the American cars were owned by students.

c) Percentage of the students owned American cars:

Percentage of the students who owned American cars

= No. of American cars owned by students / Total No. of students

=

About 54% of the students owned American cars.

d) Marginal distribution of origin:

Let the variable X denote the origin.

Based on origin, the marginal distribution is given by the figures marked in red:

In percentages,

Cars owned = 59% by Americans

= 12% by Europeans

= 29% by Asians

e) About equal proportion of American cars are owned by students and staff i.e. half of the American cars are owned by students and the other half by the staff.

f) To test the independence, chi square test may be used:

First, expected frequencies are obtained by multiplying the row and column totals of a cell, and dividing it by total (=356).

To test: H₀ : The origin of the car is independent of the type of driver Vs

H_a: The origin of the car and the type of driver are dependent.

Test statistic used is chi square, given by,

From the table,

= 9.02

Comparing the statistic with the tabled value for (No. of rows-1)(No. of columns - 1) degrees of freedom i.e. for (3-1)(2-1) = (2)(1) = 2 df, at alpha = 5%,

Chi squared critical value = 5.99

Since, the test statistic value 9.02>5.99, H₀ may be rejected at 5% level of significance, since there is not sufficient evidence to support the null hypothesis.

Hence, H_a is accepted.

We may conclude that origin of the car is not independent of the type of driver.