Practice Problem: t-test

A research study was conducted to examine the differences between older and younger adults on perceived life satisfaction. A pilot study was conducted to examine this hypothesis. Ten older adults (over the age of 70) and ten younger adults (between 20 and 30) were give a life satisfaction test (known to have high reliability and validity). Scores on the measure range from 0 to 60 with high scores indicative of high life satisfaction; low scores indicative of low life satisfaction. The data are presented below. Compute the appropriate t-test.

Independent t-test

1. What is your computed answer? tobs = 4.317

2. What would be the null hypothesis in this study? The null hypothesis would

be that there are no significant differences between younger and older adults

on life satisfaction.

3. What would be the alternate hypothesis? The alternate hypothesis would be

that life satisfaction scores of older and younger adults are different.

4. What probability level did you choose? .05

5. What is your tcrit? tcrit = 2.101

6. Is there a significant difference between the two groups? Yes, the tobs is in

the tail. In fact, even if one uses a probability level the t is still in the tail.

Thus, we conclude that we are 99.9 percent sure that there is a significant

difference between the two groups.

7. Interpret your answer. Older adults in this sample have significantly higher

life satisfaction than younger adults.

Practice Problem: ANOVA

A research study was conducted to examine the clinical efficacy of a new antidepressant. Depressed patients were randomly assigned to one of three groups: a placebo group, a group that received a low dose of the drug, and a group that received a moderate dose of the drug. After four weeks of treatment, the patients completed the Beck Depression Inventory. The higher the score, the more depressed the patient. The data are presented below. Compute the appropriate test.

1. What is your computed answer? F = 10.74 (2,12) p < .01

2. What would be the null hypothesis in this study? There will be no difference

in depression levels between the three groups. The groups taking the drug

will not be different than the groups taking the placebo.

3. What would be the alternate hypothesis? There will be a difference

somewhere in depression levels between the three levels of drug groups.

4. What probability level did you choose? p = .01.

5. What is your Fcrit? Fcrit = 6.93

6. Is there a significant difference between the groups? Yes - a significant

difference exists somewhere between the three groups.

7. If there is a significant difference, where specifically are the differences?

There is a significant difference between the placebo group and the low dose

group (tcomp = 3.22 and tcrit = 2.31, p = .05). There is a significant difference

between the placebo group and the moderate dose group (tcomp = 4.63 and tcrit

= 2.31, p = .05). There is no significant difference between the low dose and

the moderate dose groups (tcomp = 1.23 and tcrit = 2.31, p = .05).

8. Interpret your answer. The drug appears to help alleviate depression.

However, as there is no significant difference between taking a low or

moderate dose, a low dose would be recommended.

Practice Problem: CHI SQUARE

In these times of educational reform, attention has been focused on pre-school for all children. Since many districts are facing budget cuts, funding pre-

school programs may impact other offerings. Before making their recommendations, administrators in a large urban district take a random sample of 50 seventh graders and compare the pre-algebra achievement levels of those who attended pre-school and those who did not. If achievement is independent of attending pre-school then the proportions at each level should be equal. Use the counts in the frequency table to determine if there is an association between attending pre-school and pre-algebra achievement.

We use a Chi-squared test since we have one random sample,  the sample size is less than 10% of the population of all 7th graders in the district, and each expected cell count is large enough (greater than 5).

1. What is your computed chi-squared? Chi-squared= 2.85

2. What would be the null hypothesis in this study? There is no relationship

between Pre-algebra achievement and pre-school attendance. The Pre-algebra

achievement is not dependent on pre-school attendance.

3. What would be the alternate hypothesis? There is a relationship between Pre-

algebra achievement and pre-school attendance. The Pre-algebra achievement is

dependent on pre-school attendance.

4. What significance level did you choose? a= .05

5. What is your critical value? X2crit = 5.99

6. Is there relationship between between pre-algebra achievement and pre-school

attendance.? No - there is no relationship between Pre-algebra achievement and

pre-school attendance.

7. Interpret your answer. Since our chi-squared is less than the critical value, our

sample does not provide significant evidence that pre-algebra achievement is

related to pre-school attendance. This study alone would not support funding pre-

school education for all students in the district.

Practice Problem: Correlation and Linear Regression

It is assumed that achievement test scores should be correlated with student's classroom performance. One would expect that students who consistently perform well in the classroom (tests, quizes, etc.) would also perform well on a standardized achievement test (0 - 100 with 100 indicating high achievement). A teacher decides to examine this hypothesis. At the end of the academic year, she computes a correlation between the students achievement test scores (she purposefully did not look at this data until after she submitted students grades) and the overall g.p.a. for each student computed over the entire year. The data for her class are provided below.

1. Compute the correlation coefficient. r = .524127623 or .52

2. What does this statistic mean concerning the relationship between

achievement test performance and g.p.a.? There is a moderate correlation

between achievement test performance and g.p.a. As the achievement test

scores go up, the g.p.a.s tend to increase as well and vice versa.

3. What percent of the variability is accounted for by the relationship between

the two variables and what does this statistic mean? r2 = .27 The percent a

variability is relatively low. Only 27 percent of the achievement test

performance is related to the g.p.a (and vice versa). Seventy-three percent of

the variability is left unexplained.

4. What would be the slope and y-intercept for a regression line based on this

data? The slope would be .028430629 and the y-intercept would

be .62711903.