Question

5. (8 pts) The following data is for a sample of students. College year Senior Weekly study hours 15 GPA 3.2 Junior 20 10 Name Gender County Liam Male Lauderdale Emma Female Choctaw Noah Male Clarke Olivia Female Lauderdale William Male Jasper Female Choctaw James Male Kemper Isabella Female Newton Logan Male Newton اس اس ام سا یہ 15 Sophomore Junior Freshman Senior Sophomore Senior 2.6 3.0 2.8 2.5 30 Ava 40 15 3.5 3.0 10 10 Freshman 3.3 (a) If we want to study how "County" affects "GPA what method should be used? (b) If we want to study whether "Weekly study hours" affects "GPA", what method should be used? (c) If we want to use "Weekly study hours to predict "Gender", what method should be used? (d) If we want to study whether there is a relationship between "County" and "College year", what methodshould be used?

Answer 1

Looking at the list of variables, we may notice that each of them is measured in a particular pattern - some are numeric and others categorical. In statistics, the four common values of measurement are - Nominal, Ordinal, Interval and Ratio scale.

Nominal scale measures the variables plainly in the format of distinct categories - a subject belongs either to one group or the other. Ordinal scale also contains categories, only in that these categories are ordered in a particular sense. In Interval scale, the observations are numeric, with an absolute zero, as in measurement of temperature and equal intervals measure equal real differences. In ratio scale, zero simply means void or null and here, the ratio two observations are meaningful.

From the data, we find that the variables - Gender, County and College are categorical variables measured in nominal scale and the weekly variables - Weekly study hours and GPA are numerical variables. Now, to study relationship between variables measured in different scales, the following statistical tools may be considered approporiate:

Type of Variable		Continuous	Categorical
Categorical	2 Categories	Independent / Paired t - test	Chi-Square test of Independence / Association
	> 2 Categories	Analysis of Variance (ANOVA)	Chi-Square test
Continuous		Correlation / Rehression	t test / ANOVA

(a) We need to study how County affects GPA. Here, if the County had no effect on GPA, the students would score the same GPA, whichever county he belongs to. Since GPA is a continuous measure, comparing the mean GPA across six groups would suffice the comparison of the whole groups. As mentioned in the table, we may apply the Analysis of Variance test to test the hypothesis:

$H_{0}: \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}=\mu_{5}=\mu_{6}$ Vs H₀: No all means are equal

If the data provides sufficient evidence to reject H₀, we may say that County does affect GPA.

(b) Here, we ar studying two continuous variables - No. of hours studied and GPA. As mentioned in the table, the appropriate statistical tool to study the relationship between the two would be Pearson's correlation (which measures the strength and direction of the relationship). But this would only tell us if the two variables are related and not the causal relationship between them. Hence, in order to determine if No. of hours affects the GPA, we may run a Simple linear regression (assuming a linear relationship). On regressing the dependent variable GPA on the independent variable - No. of hours studied per week, we would get a relationship of the form:

GPA = a + b (No. of Hours)

where, the estimated slope coefficient 'b' measures the change in GPA for a unit change in no. of hours studied.

(c) If we wish to use the weekly study hours to predict the GPA, we may simply substitute the no. of hours studies for which the GPA is to be calculated in the fitted regression model above.

(d) Here, we need to study the relation between two categorical variables - County and College year. Here, the data we take into account would be in the form of counts / frequencies.

As mentioned in the table, we need to resort to the Chi square test of association / independence. If the two variables were indeed independent, we would expect equal counts across all the categories in the cross tabulation of data. If the frequencies are not uniformly distributed and differ greatly from the expected counts, we may say that the two variables are dependent/ associated.