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# T Tests for Independent and Related Samples: 6 problems

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t Tests for Independent and Related Samples

Problem Set 2: Chapter 10 & 11: problems:

Chapter 10:

2. Describe what is measured by the estimated standard error in the bottom of the independent -measures t-statistic.
18. In 1974, Lotus and Palmer conducted a classic study demonstrating how the language used to ask a question can influence eyewitness memory. In the study, college students watched a filmed of an automobile accident and then were asked about what they saw. One group was asked, questions what they saw. One group was asked, "About how fast were the cars going when they smashed into each other?" Another group was asked the same question expect the verb was changed to hit" instead of smashed into" group reported significantly higher estimates of speed than the hit" group. Suppose a researcher repeats this study with a sample of today's college students and obtains the following results.

Estimated Speed

Smashed into Hit
n = 15 n = 15
M = 40. 8 M = 34. 0
SS = 510 SS = 414

24. A researcher conducts an independent -measures research study and obtains t =2.070 with df = 28.
a. How many individuals participated in the entire research study?
b. Using a two-tailed test with &#945; = .05, is there a significant difference between the two treatment conditions?
c. Compute r2 to measure the percentage of the variance accounted for by the treatment effect.

Chapter 11:
2. Participants enter a research study with unique characteristics that produce different scores from one person to another. For an independent -measure study, this individual difference can cause problems. Briefly explain how these problems are eliminated or reduced with a repeated -measure study.

4. A researcher conducts an experiment comparing two treatment conditions and obtains data with 10 scores for each treatment condition.
a. If the researcher used an independent-measure design how many subjects participated in the experiment?
b. if the researcher used a repeated-measures design, how many subjects participated in the experiment?
c. If the researcher used a matched -subjects design, how many subjects participated in the experiment?

10. Research has shown that losing even night's sleep can have a significant effect on performance of complex tasks such as problem solving (Linde & Bergstroem, 1992). To demonstrate this phenomenon, a sample of n = 25 college students was given a problem- solving task at noon on one day and again at noon on the following day. The students were not permitted any sleep between the two tests. For each student, the difference between the first and second score was recorded. For this sample, the students averaged MD = 4.7. points better on the first test with a variance of s2 = 64 for the difference scores.
a. Do the data indicate a significant change in problem -solving ability? Use a tailed test with &#945; = .05.
b. Compute an estimated Cohen's d to measure the size of the effect.

22. In a Preview of a chapter, there is a discussion discussed as a report describing the Olympic and how marksmen can be affected by heartbeats. Pelton (1983) reported that Olympic -level marksmen shoot much better if they fire between heartbeats rather than squeezing the trigger during a heartbeat. The small vibration caused by a heartbeat seems to be sufficient to affect the marksmen's aim. The following hypothetical data demonstrate this phenomenon. A sample of n = 8 Olympic marksmen fire a serious of rounds while a researcher records heartbeats. For each marksman, the total score is recorded for shots fired during heartbeats and for shots fired between heartbeats. Do these data indicate a significant difference? Test with &#945; = .05.

Participants During heartbeats Between heartbeats
A 93 98
B 90 94
C 95 96
D 92 91
E 95 97

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## 30 Multiple Choice Problems in Statistics

Problem A
The manager of a grocery store claims that the average time that customers spend in checkout lines is 20 minutes or less. A sample of 36 customers is taken. The average time spent on checkout lines for the sample is 24.6 minutes; and the sample standard deviation is 12 minutes. Conduct a hypothesis test (at 0.05 level of significance) to determine if the mean waiting time for the customer population is significantly more than 20 minutes.

The observed value of the test statistic is:
a. 2.3 b. 0.38 c. -2.3 d. -0.38

The p-value is:
a. 0.5107 b. 0.0214 c. 0.0137 d. 0.4893

Problem E:
A company wants to measure the relationship between its employee productivity (measured in output/employee) and the number of employees. Sample data for the last four months are shown below. Use simple linear regression to estimate this relationship.

Independent Variable Dependent Variable
Number of Employees Employee Productivity
15 5
12 7
10 9
7 11

ANSWER QUESTIONS 16 THROUGH 19 BELOW.

16. The least squares estimate of the slope b1 is:
a. -0.7647 b. -0.13 c. 21.4 d. 16.41

17. The least squares estimate of the intercept b0 is:
a. -7.647 b. -0.13 c. 21.4 d. 16.41

18. The estimated employee productivity when the number of employees is 5 is:
a. 78 b. 12.59 c. 5.8 d. 32.6

19. If the sample covariance is -8.67; estimate the coefficient of correlation between the number of employees and employee productivity:
a. -0.997 b. 0.997 c. 1.23 d. 1.02

Problem F:
Consumer Research is an independent agency that is collecting data on annual income (INCOME) and household size (SIZE), to predict annual credit card charges. It runs a regression analysis on the data and an incomplete MS Excel output is shown below.
ANSWER QUESTIONS 20 THROUGH 30 BELOW.

Regression Statistics
Multiple R 0.88038239
R Square
Adjusted R Square
Standard Error 510.495493
Observations

ANOVA
df SS MS F Significance F
Regression 2 17960368.3 3.31446E-07
Residual 20 260605.648
Total 22

Coefficients Standard Error t Stat P-value Lower 95%
Intercept 352.694714 4.15578994 0.00048872 730.0172039
INCOME 25.062956 8.47147285 2.95851223 0.00776734 7.391781505
SIZE 408.400776 71.808401 1.447E-05 258.6111461

20. The sample size is:
a. 23 b. 22 c. 20 d. 21

21. The coefficient of determination is:
a. 0.88 b. 0.775 c. 0.92 d. -0.38

22. The Sum of Squares for Error (i.e., Residual) is:
a. 17960368.3 b. 5212112.97 c. 23172481.3 d. 260605.648

23. The Sum of Squares for Total (SST) is:
a. 17960368.3 b. 5212112.97 c. 23172481.3 d. 260605.648

24. The Mean Square for Regression is
a. 17960368.3 b. 5212112.97 c. 260605.648 d. 8980184.17

25. The observed or computed F-value is:
a. 34.459 b. 0.029 c. 3.445 d. 0.29

26. The hypothesis to be tested is:
H0: B1 = B2 = 0
Ha: At least one of the B is not equal to 0.
The hypothesis is to be tested at the 5% level of significance. The null hypothesis is:
a. not rejected
b. rejected
c. the test is inconclusive
d. none of the above answers are correct

27. The hypothesis to be tested is:
H0: B1 = 0
Ha: B1 &#8800; 0

The hypothesis is to be tested at the 1% level of significance. The null hypothesis is:
a. not rejected
b. rejected
c. the test is inconclusive
d. none of the above answers are correct

28. The estimate of the intercept b0 is:
a. 10010.2 b. 2810.3 c. 1465.5 d. 2641.5

29. The observed or computed t-stat (i.e., t-value) for the independent variable SIZE is:
a. 2.96 b. 3.445 c. 4.16 d. 5.687

30. What is the estimated annual credit charges if INCOME = 20, and SIZE = 3?
a. 9700 b. 12600 c. 3189 d. 5300

Problem G:
Last year, the student body of a local university consisted of 30% freshmen, 24% sophomores, 26% juniors, and 20% seniors. A sample of 300 students taken from this year's student body showed the number of students in each classification.
Freshmen 83
Sophomores 68
Juniors 85
Seniors 64

We want to know if there has been a significant change in the proportions of student classifications between the two years.
ANSWER QUESTIONS 31 THROUGH 34 BELOW.

31. The expected number of freshmen in this year is:
a. 83 b. 90 c. 30 d. 10

32. The number of degrees of freedom is:
a. 4 b. 2 c. 3 d. 1

33. The hypothesis is to be tested at the 5% level of significance. The critical chi-square value from the table equals:
a. 1.645 b. 1.96 c. 2.75 d. 7.815

34. If the chi-square value that is calculated equals 1.6615, then the null hypothesis is:
a. not rejected
b. rejected
c. the test is inconclusive
d. none of the above answers are correct

Problem H: Use the following Excel Output to answer questions 35-39:

Source Sum of Squares d.f.
Between Groups 213.88125 3
Within Groups 11.208333 20
Total 225.0895 23

35. Consider the above one-way ANOVA table. What is the treatment mean square?

A) 71.297 B) 0.5604 C) 1.297 D) 213.881 E) 9.7

36. Consider the above one-way ANOVA table. What is the mean square error?

A) 71.297 B) 0.5604 C) 1.297 D) 213.8810 E) 9.7

37. Consider the above one-way ANOVA table. How many groups (treatment levels) are included in the study?
A) 3 B) 4 C) 6 D) 20 E) 24

38. Consider the above one-way ANOVA table. If there are equal number of observations in each group, then each group (treatment level) consists of ______ observations.

A) 3 B) 4 C) 6 D) 20 E) 24

39. What is the critical F-value at an alpha of 0.05?

A) 3.1 B) 3.86 C) 14.17 D) 4.94 E) 8.66

Problem I:
Use the following to answer questions 40-42:

The following results were obtained from a simple regression analysis:

= 37.2895 - (1.2024) * X
r = - 0.6774

40. For each unit change in X (independent variable), the estimated change in Y (dependent variable) is equal to:
A) -1.2024 B) 0.6774 C) 37.2895 D) 0.2934

41. When X (independent variable) is equal to zero, the estimated value of Y (dependent variable) is equal to:
A) -1.2024 B) 0.6774 C) 37.2895 D) 0.2934

42. __________ is the proportion of the variation explained by the simple linear regression model:
A) 0.8230 B) 0.6774 C) 0.4589 D) 0.2934 E) 37.2895

43. Given the following information about a hypothesis test of the difference between two means based on independent random samples, which one of the following is the correct rejection region at a significance level of .05?

HA: &#956;A >&#956;B , &#956;1 = 12, &#956;2 = 9, s1 = 4, s2 = 2, n1 = 13, n2 = 10.

A) Reject H0 if Z > 1.96
B) Reject H0 if Z > 1.645
C) Reject H0 if t > 1.721
D) Reject H0 if t > 2.08
E) Reject H0 if t > 1.734

Problem K: Business travelers were asked to rate Miami Airport (on a scale of 1-10). Similarly business travelers were asked to rate Los Angeles airport. A hypothesis test (at alpha = 0.05) is conducted for any difference in the population means in the ratings. The Excel output is shown below. Use the following to answer questions 47- 48:

t-Test: Two-Sample Assuming Unequal Variances
Miami Los Angeles
Mean 6.34 6.72
Variance 4.677959184 5.63428571
Observations 50 50
Hypothesized Mean Difference 0
df 97
t Stat -0.836742811
P(T<=t) one-tail 0.202396923
t Critical one-tail 1.660714588
P(T<=t) two-tail 0.404793846
t Critical two-tail 1.984722076

48. A 95% confidence interval of the difference between the mean ratings is:
a. - 0.52 to 1.25
b. 1.67 to 2.43
c. -0.51 to 1.27
d. -1.28 to 0.52
e. -2.43 to 1.67

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