# T Test for Independent Groups and Dependent Groups (Two-Group Designs)

COUN 503

**Web Calculator Exercise 3**

**T test for independent groups and dependent groups (Two-group designs)**

- An LU professor is interested in whether there is a difference between undergraduate students and graduate students in the amount of time spent praying each day. The professor gathers information from random samples of undergraduate and graduate students on the LU campus. The amount of time praying is normally distributed and is measured on an interval/ratio scale.

Graduate / Undergraduate

15 / 9

17 / 11

10 / 9

13 / 6

11 / 5

17 / 6

- What statistical test should be used to analyze the data?

- Is this a one- or two tailed test?

- Identify H0 and Ha for this study.

- Conduct the appropriate analysis. Should H0 be rejected?

- A teacher wants to investigate whether there is a difference between male and female students in the amount of time they spend studying for statistics. The table below shows the amount of time students spend studying statistics each week. The amounts of time spent studying are normally distributed.

Male / Female

27 / 25

25 / 29

19 / 18

10 / 23

16 / 20

17 / 15

15 / 19

- What statistical test should be used to analyze the data?

- Is this a one- or two tailed test?

- Identify H0 and Ha for this study.

- Conduct the appropriate analysis. Should H0 be rejected?

- A researcher is interested in whether listening to music helps or hinders test-performance. To control for differences in cognitive level, this researcher decides to use a within-participants design. He selects a random sample of participants and has them study different material of equal difficulty in both the music and no music conditions. Participants take a 20-item quiz on the material. The table below shows the scores on the quiz. The study is completely counterbalanced to control for order effects. The scores obtained are measured on an interval-ratio scale and are normally distributed.

Music / No Music

17 / 17

16 / 18

15 / 17

16 / 17

18 / 19

18 / 18

- What statistical test should be used to analyze the data?

- Is this a one- or two tailed test?

- Identify H0 and Ha for this study.

- Conduct the appropriate analysis. Should H0 be rejected? What should the researcher conclude?

- Calculate the 95 confidence interval.

**Web Calculator Exercise 4**

**Non-parametric tests (Chi-square tests, Wilcoxon-rank-sum Wilcoxon signed-ranks tests)**

**Note: When searching for a web calculator for the Wilcoxon-rank-sum test, please be aware that this test is also called the “Mann-Whitney U test”. This test is used for Ordinal data and for an independent groups design. **

1.A high school counselor wants to examine whether the teenage pregnancy rate at her school is different from the rate nationwide. She knows that the rate nationwide is 15 %. She randomly selects 80 female students from her high school. She finds that 9 out of 80 students are either pregnant now or have been pregnant previously.

- What statistical test should be used to analyze the data?
- Identify H0 and Ha for this study
- Conduct the appropriate analysis
- Should H0 be rejected? What should the researcher conclude?

2.You notice at the gym that it appears more women tend to work out together, whereas more men tend to work out alone. To determine whether this difference is significant, you collect data on the workout preferences for a sample of men and women at your gym. The data follows.

Males / FemalesTogether / 5 / 8

Alone / 7 / 6

a.What statistical test should be used to analyze the data?

b.Identify H0 and Ha for this study

c.Conduct the appropriate analysis.

d.ShouldH0 be rejected? What should the researcher conclude?

3.A researcher is interested in comparing the Christian maturity level of students who volunteer for community service versus those who do not. The researcher assumes that those who perform community service will have higher Christian maturity scores. The maturity scores tend to be skewed (not normally distributed). Higher scores indicate higher Christian maturity.

No Community Service / Community Service32 / 47

40 / 48

54 / 59

13 / 72

20 / 80

26 / 55

- What statistical test should be used to analyze these data?
- Is this a one- or two- tailed test?
- Identify H0 and Ha for this study
- Conduct the appropriate analysis
- Should H0 be rejected? What should the researcher conclude?

4.A researcher is interested in comparing tastiness of different kinds of cheese in grilled cheese sandwiches. The researcher expects that Cheddar cheese grilled sandwiches are rated tastier than Swiss cheese grilled sandwiches. This researcher wants to use a within-subjects design. Each participant tastes both Cheddar and Swiss cheese grilled sandwiches. Except for the types of cheese used in the grilled sandwiches, there is no difference between the two kinds of sandwiches. Participants indicate the degree of tastiness on a 10-point scale and the ratings appear in the following table. Tastiness scores tend to be skewed.

Participant / Cheddar / Swiss1 / 7 / 4

2 / 6 / 5

3 / 9 / 6

4 / 10 / 8

5 / 6 / 7

6 / 7 / 6

7 / 8 / 9

- What statistical test should be used to analyze these data?
- Identify H0 and Ha for this study.
- Conduct the appropriate analysis.
- Should H0 be rejected? What should the researcher conclude?

**Web Calculator Exercise 5**

ANOVAs

- A researcher is interested in comparing the effectiveness of three different kinds of therapy for anger problems. Eight participants are randomly assigned to three treatment conditions: Cognitive Therapy (CT), Cognitive-Behavioral Therapy (CBT), and Interpersonal Therapy (IT). All participants complete the weekly treatment for 8 weeks. Participants take an anger test before and after the 8-week treatments. Each participant’s difference score (= anger score after the treatment minus (–) anger score before the treatment) is shown in the following table. The three therapists in this experiment have equal educational levels and clinical experiences.

CT / CBT / IT

3 / 5 / 7

4 / 4 / 8

4 / 6 / 9

4 / 6 / 10

3 / 5 / 6

8 / 4 / 7

6 / 9 / 11

2 / 7 / 10

- What statistical test should be used to analyze the data?
- Identify H0 and Ha for this study.
- Conduct the appropriate analysis.
- Should H0 be rejected? What should the researcher conclude?

- A teacher is interested in comparing the effects of three different types of rehearsal methods (Rote rehearsal, Imagery, and Story) on number of words recalled. Participants are randomly assigned into three different groups. In each group, participants are asked to study a list of words using the assigned method. The following table shows the result.

Rote rehearsal / Imagery / Story

5 / 5 / 7

5 / 4 / 6

4 / 6 / 8

6 / 7 / 9

5 / 7 / 6

4 / 6 / 9

3 / 8 / 8

5 / 5 / 10

6 / 7 / 9

- What statistical test should be used to analyze the data?
- Identify H0 and Ha for this study.
- Conduct the appropriate analysis.
- Should H0 be rejected? What should the researcher conclude?
- If there is a significant difference, conduct the post hoc test.

- A professor is interested in whether there is a difference in counseling students’ statistics competency scores among 1) those who have never taken any statistics course, 2) those who have only taken an undergraduate statistics course, 3) those who have taken COUN 503, and 4) those who have taken an advanced statistic course while working on a Ph. D. The professor gathers information from a random sample of students and obtains the statistics competency scores. The scores are normally distributed.

1) / 2) / 3) / 4)

7 / 11 / 13 / 15

6 / 11 / 14 / 14

8 / 10 / 15 / 15

5 / 12 / 12 / 15

4 / 13 / 13 / 10

4 / 15 / 13 / 15

- What statistical test should be used to analyze these data?

- Identify H0 and Ha for this study.

- Conduct the appropriate analysis.

- Should H0 be rejected? What should the researcher conclude?

**Web Calculator Exercise 2**

**Z test & One sample t-test**

- A researcher is interested in whether students who attend private high schools have higher average SAT Scores than students in the general population. A random sample of 90 students at a private high school is tested and and a mean SAT score of 1030 is obtained. The average score for public high school student is 1000 (σ= 200).

- Is this a one- or two tailed test?

- What are H0 and Ha for this study?

- Compute Z obt

- What is the Z critical value (Z cv ) using a 0.05 alpha level?

- Should H0 be rejected? What should the researcher conclude?

- Determine the 95 % confidence interval for the population mean, based on the sample mean.

- A researcher hypothesized that the pulse rates of long-distance athletes differ from those of other athletes. He believed that the runners’ pulses would be slower. He obtained a random sample of 10 long-distance runners. He measured their resting pulses. Their pulses were 45, 45, 64, 50, 58, 49, 47, 55, 50, 52 beats per minute. The average resting pulse of athletes in the general population is normally distributed with a pulse rate of 60 beats per minute.

- What statistical test should be used to analyze the data?

- Is this a one- or two- tailed test?

- What are H0 and Ha for this study?

- Find tcv from appendix A in Jackson’s text.

- Compute t obt

- Should H0 be rejected? What should the researcher conclude?

- A researcher hypothesizes that people who listen to music via headphones have greater hearing loss and will thus score lower on a hearing test than those in the general population. On a standard hearing test, the overall mean for the general population is 22.5. The researcher gives this same test to a random sample of 12 individuals who regularly use headphones. Their scores on the test are 15, 14, 20, 20, 25, 22, 21, 19, 16, 17, 21, 22.

- What statistical test should be used to analyze the data?

- Is this a one- or two- tailed test?

- What are H0 and Ha for this study?

- Compute t obt

- Should H0 be rejected? What should the researcher conclude?

**Web Calculator Exercise 1**

**Descriptive Statistics **

- The table below presents data for a sample of people who completed a religious survey.

Age / Gender / Denomination / Church Attendance

56 / 1 / 7 / 4

46 / 2 / 6 / 5

49 / 2 / 6 / 5

49 / 1 / 1 / 5

27 / 2 / 9 / 5

51 / 1 / 4 / 2

47 / 2 / 2 / 3

67 / 1 / 5 / 4

49 / 2 / 2 / 6

33 / 1 / 12 / 6

55 / 2 / 9 / 5

40 / 1 / 7 / 5

62 / 1 / 8 / 6

47 / 2 / 6 / 3

56 / 2 / 9 / 5

22 / 1 / 10 / 2

50 / 2 / 4 / 5

51 / 1 / 10 / 6

50 / 1 / 7 / 6

43 / 1 / 10 / 3

In this table, the numbers in the gender, denomination, and church attendance columns represent the following.

Gender

- Male
- Female

Denomination

- Episcopal
- Lutheran
- Methodist
- Presbyterian
- Other mainline Protestant
- Baptist
- Other Evangelical Protestant
- Pentecostal
- Charismatic
- Non-denominational
- Catholic
- Other

Church Attendance

- less than once a month
- once a month
- a few times a month
- once a week
- twice a week
- three or more times a week

- What is the mean age of this sample? What is the standard deviation?

- Create a frequency distribution table for denomination.

- What is the percentage of people who identify themselves as Baptist in this sample?

- What is the mode of church attendance?

- The results of a recent survey indicate that the average new car costs $23,000, with a standard deviation of $3,500. The price of cars is normally distributed.

- What is a Z score for a car with a price of $ 33,000?

- What is a Z score for a car with a price of $30,000?

- At what percentile rank is a car that sold for $30,000?

- In one elementary school, 200 students are tested on the subject of Math and English. The table below shows the mean and standard deviation for each subject.

Mean / SD

Math / 67 / 9.58

English / 78 / 12.45

One student’s Math score was 70 and the same individual’s English score was 84. On which exam did the student do better?

- Suppose you administered an anxiety test to a large sample of people and obtained normally distributed scores with a mean of 45 and standard deviation of 4. Do not use web-calculator to answer the following questions. Instead, you need to use the Z distribution table in Appendix A in Jackson’s book.

- If Andrew scored 45 on this test. What is his Z score?

- If Anna scored 30 on this test. What is her Z score?

- If Bill’s Z score was 1.5, what is his real score on this test?

- There are 200 students in a sample. How many of these students will have scores that fall under the score of 41?

- The table below shows Psychology exam scores, Statistics Exam scores, and IQ scores for a random sample of students. What can you observe in the relationship between IQ and psychology, psychology and statistics, and IQ and statistics? Using a web-calculator, obtain the Pearson’s r and coefficient of determination for the following relationships.
- Between the IQ and psychology scores
- Between the IQ and statistics scores
- Between the psychology scores and statistics scores.

Student number / IQ / Psychology / Statistics

101 / 142 / 49 / 49

102 / 100 / 30 / 32

103 / 103 / 36 / 38

104 / 121 / 44 / 41

105 / 120 / 35 / 42

106 / 115 / 47 / 43

107 / 101 / 37 / 35

108 / 109 / 45 / 47

109 / 111 / 30 / 43

110 / 115 / 49 / 46

- In a study on caffeine and stress, college students indicated how many cups of coffee they drink per day and their current stress level on a scale of 1 to 10. The table shows the survey results. Using a web-calculator, obtain the appropriate correlation coefficients.

Number of cups of coffee / Stress level

3 / 5

2 / 3

4 / 3

6 / 9

5 / 4

1 / 2

7 / 10

3 / 5

- Discussed how they arrived at the answers (process).

- Answers were easy to find.
- Explanations of how students arrived at the answers (process) were clear.

- Identified appropriate statistical procedures.

- Attempted to find the web-calculator link.
- If a link was broken, students documented the problem.

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