Icef 2004/2005

STATISTICS

FIRST YEAR

EXAM

Solutions

April 5, 2005

SECTION II

Part A

Problems 1  5

Problem 1

The given scatterplot shows the advertised prices (in thousands of dollars) plotted against ages (in years) for random sample of Plymouth Voyagers on several dealers’ lots. A computer printout showing the results of fitting a line to the data by the method of lest squares gives

Dependent Variable: PRICE
Method: Least Squares
Included observations: 13
Variable	Coefficient		Std. Error of Coeff.
C	12.33		1.005
AGE	-1.17		0.224
R-squared	0.7140

In other words,

Price = 12.33  1.17 Age,

R-squared = 71.40%

(a) Find the correlation coefficient for relationship between price and age of Voyagers based on these data.

(b) Do these results give the evidence of connection between Age and Price? Test the corresponding hypothesis.

(c) How will the size of the correlation coefficient change if the 10-year-old Voyager is removed from the data set? Explain your answer.

(d) How will the size of the slope of the least squares regression line change if the 10-year-old Voyager is removed from the data set?

Solution.

(a) It is known that where r is a correlation coefficient and . Obviously , so .

(b) We should test the null hypothesis H₀: versus H_a: . Test-statistics is . Under H₀ has tdistribution with 11 (= number of observations  2) degrees of freedom. Pvalue is less than 0.001 (in fact Pvalue = 0.0003). Hence, null hypothesis should be rejected at any reasonable significance level. In other words, these results show strong connection between Age and Price.

(с) If the 10-year-old Voyager is removed from the data set the remaining points are much more close to straight line, the linear relationship between Price and Age looks more strong, and, of course, it is negative. So, the absolute value of correlation coefficient will increase. (In fact, the new correlation coefficient is equal to 0.966.)

(d) If the 10-year-old Voyager is removed from the data set the least square line will be steeper, so the slope will remain negative and the absolute value of the slope will increase (In fact, the new slope is equal to 1.97.)

Problem 2

All college students were asked in order to assess the usage of the college computer classes for the self-study purposes. The numbers of users of several categories, classified by the year of study, are shown in the following table,

Almost every day Sometimes Never

First year 22 35 15

Second year 41 27 7

Third year 36 15 3

Fourth year 33 13 1

Use these data to investigate the association, if any, between usage of computers and year of study, interpret any association that exists. Name another factor that may have an influence on these data.

Solution. The null hypothesis: there is no association between usage of computers and year of study. In order to test this hypothesis we may use contingency tables. By direct calculations we get the table of observed values:

	Almost every day	Sometimes	Never	Total	Margin
First year	22	35	15	72	0.290
Second year	41	27	7	75	0.302
Third year	36	15	3	54	0.218
Fourth year	33	13	1	47	0.190
Total	132	90	26	248
Margin	0.532	0.363	0.105

and the table of expected values:

	Almost every day	Sometimes	Never
First year	38.32	26.13	7.55
Second year	39.92	27.22	7.86
Third year	28.74	19.60	5.66
Fourth year	25.02	17.06	4.93

(50 points).

The corresponding statistics is equal to 28.25 and P-value is closed to 0. The null hypothesis is rejected at any reasonable significance level.

Looking through the table we may conclude that there are two tendencies: the higher the year the higher the proportion of students using computers almost every day, and the higher the year the lower the proportion of students never using computers.

There are another factors such that the different subjects for different years, the possible increasing number of practical works, the increasing of computer skill, that may have influence on these data.