Practice Problems for the Cumulative Final Exam

1. A researcher wonders whether memory will be better for studying items when under hypnosis(group 1) or not under hypnosis (group 2). The data for group 1 was, X-bar₁ = 21; n1=11; s1=5.745 and the data for group 2 was, X-bar2 =27; n2=11; s2=6.000.
a) go through the formal hypothesis testing steps to determine if the people under hypnosis have worse memory than those people not under hypnosis. Assume alpha = .05.
b) State if your p-value is greater or less than alpha. What error could you have made and state it in terms of the problem.
c) Construct a 95% confidence interval for the difference in memory for those under hypnosis and those not under hypnosis, and draw a conclusion.

2. A researcher is interested in whether there is a relationship between personality type (Type A or Type B) and whether the person has had a heart attack (heart attack or no heart attack). Assume alpha=.01. The data are as follows:

a)Conduct a formal hypothesis test to determine if having a heart attack is dependent on personality type.
b)State if your p-value is greater or less than alpha. What type of error could you have made in terms of the problem?

3. A researcher was interested in how perceived difficulty of math problems effects the number of problems that people solve. Every participant was given the same set of math problems but was told that they were either easy, medium, or difficult. The number of problems solved was then recorded for each group. The following data were obtained: X-bare=8, ne=5, se=2.92; X-barm=6, nm=5, sm=3.16; X-bard=3, nd=5, sd=1.58.
a)conduct a formal hypothesis test to determine if the number of problems solved differs with perceived difficulty. (assume alpha = .05)
b) State if your p-value is greater or less than alpha. What type of error could you have made in terms of the problem?

4.A graduate admissions committee wants to know if their graduate applicant's GPAs are significantly higher than 3.0. The 13 applicants had a mean of 3.45 with a standard deviation of .32.
a) Test the hypothesis that the sample mean is greater than 3.0. Alpha = .05.
b) State if your p-value is greater or less than alpha. What type of error could you have made in terms of the problem?
c) Find the 99% CI of the population mean of all graduate applicants GPAs. Draw a conclusion regarding if the sample mean's GPA is significantly different from 3.0.

5.A therapist wants to examine if a treatment for phobias is effective. 5 people are given a phobia test before and after treatment. Higher scores indicate more phobia. The following data were obtained: X-bar before=9.2; X-bar after = 5.8; sD=2.8.
a) Construct a 99% CI for the difference in before and after phobia scores, and interpret it.
b) Conduct a formal hypothesis test to determine if the phobia treatment lowers the after treatment phobia scores. Assume alpha = .05.
c) State if your p-value is greater or less than alpha. What error could you have made and state it in terms of the problem.
d) Compare your conclusions for the CI and Hypothesis testing. Why were the conclusions different?

6. A researcher investigates whether a period of time feels longer or shorter when people are bored compared to when people are not bored. The researcher obtains the following estimates of the time period (in minutes). X-barb=14.5; sb=3.20; nb=31; X-barnb=12.0; snb=3.82; nnb=34.
a) Conduct a formal hypothesis test to determine if people have different time estimates when bored and not bored. Assume alpha=.05.
b) Calculate your p-value for this test and compare it to alpha. State what error you could have made in terms of the problem.
c) Construct a 90% CI for the difference in time estimates when bored and when not bored, and interpret it.

7. A researcher wants to know if there is a relationship between the number of glasses of juice consumed per day and the number of doctor visits per year. The data are as follows: n = 10; X-bar = 1.7; Y-bar = 4.7; SSxy = -27.9; SSxx = 25.1; SSyy = 54.1; Sy.x = 1.7.
a) Find the slope of the regression line and interpret it.
b) Find the intercept of the regression line and interpret it.
c) Find the 90% CI for the slope, and interpret it.
d) Test the hypothesis that amount of glasses of juice consumed per day and number of doctor visits are negatively related. Assume alpha = .025. State if your p-value is greater or less than alpha. What error could you have made and state it in terms of the problem.
e) Test the hypothesis that number of glasses of juice consumed per day can predict number of doctor's visits. Assume alpha = .01. State if your p-value is greater or less than alpha. What error could you have made and state it in terms of the problem.
f) Compute r2, and make an interpretation of the stat.

8. Janet is a lumberjack and can chop 22 logs in a minute. She wants to know how fast she is relative to all the other lumberjacks. The other lumberjacks have a mean of 20 logs chopped in a minute with a standard deviation of 1.8.
a) Compute Janet's relative standing using z-scores.
b) Compute Janet's relative standing using percentile rank.
c) What proportion of scores fall between 20.2 and 20.9?
d) What raw score corresponds to a z-score of -1.2?

9. A researcher wonders if students scores on the GRE verbal test are related to their scores on the GRE quantitative test. The following data were collected, but the researchers noted that the assumption of normality had been violated.

a) conduct a formal hypothesis test to determine if scores on the verbal test are positively associated with scores on the quantitative test. Assume alpha = .05.
b) State if your p-value is greater or less than alpha. What type of error could you have made in terms of the problem?

10. A professor thinks that learning statistics will increase a student's IQ. In general, people have a mean IQ of 100. The IQs of 40 students were measured after they look a statistics course. The mean was 106.6 with a standard deviation of 15.
a) Calculate a 94% CI to determine if the IQ of statistics students is different than the norm of 100. (Interpret the CI & draw a conclusion)
b) Conduct a formal hypothesis test to determine if the statistics students have significantly higher IQ scores than the established norm of other people. Alpha = .01
c) Calculate the p-value for the test and compare it to alpha. State what type of error could have been made in terms of the problem.

11. A researcher wants to examine if the amount of relaxation experienced from a bath is lower when the bath is cold than when hot. In the study, 40 people took baths in both cold and hot water. Relaxation scores were then obtained. The following data on relaxation were produced: X-barc=40.4; nc = 40; X-barh = 41.5; nh=40; sD=4.38.
a) Construct a 92% CI for the difference in relaxation scores for cold and hot baths, and draw a conclusion.
b) Conduct a formal hypothesis test to determine if relaxation scores are lower for cold baths than hot baths. Assume alpha=.01.
c) Calculate the p-value for the test and compare it to alpha. State what type of error could have been made in terms of the problem.

12. It is known that cats on average sleep 16.8 hours per day with a standard deviation of 1.4. We sample 45 newborn cats and find the mean hours spent sleeing is 15.2. What is the probablility of getting a sample mean lower than this one?

13. The following are the number of words remembered off of a list in a memory study: 23, 8, 30, 21, 14, 18, 14, 24
a) Compute the mode, median, and mean
b) Compute the range, variance, and standard deviation

Answers:

1a) Hyp-testing/small n/ indep samp/ 2 groups -- upper tailed test.
Ha: mu1-mu2 < 0; Ho: mu1-mu2 >=0.
t-crit = -1.725
Reject Ho if t-calc < -1.725; Do not reject Ho if t-calc >= -1.725.
t-calc = -2.394; Reject the null. Memory is lower when under hypnosis than when not under hypnosis.
b)p<.05; Type I error: Concluding that memory is lower when under hypnosis than when not under hypnosis when actually memory is not worse under hypnosis.
c) CI/small n/ indep samp/ 2 groups--- 95% CI = (-11.23, -.77); 0 is NOT in the CI, so we conclude that there is a significant difference between the hypnosis and no hypnosis groups. Since all the values are negative, we know that the hypnosis group (group 1) had a worse memory than the no hypnosis group (group 2).

2a) Askng a question of association and the data are nominal. Therefore use a Chi-square test.
Ha: having a heart attack is dependent on personality type. Ho: having a heart attack is independent of personality type.
Chi-squarecrit = 6.63
Reject Ho if Chi-squarecalc > 6.63; Do not reject Ho if Chi-squarecalc <=6.63.
Chi-squarecalc=30.56
Reject Ho; conclude that having a heart attack is dependent on personality type.
b) p<.01; Type I error: Concluding that having a heart attack is dependent on personality type, when actually it is not.

3a) Hyp-testing/ 3 groups -- do an ANOVA
Ha: at least 2 of the group means differ in number of problems solved; Ho: mue = mum = mud
F-crit (2,12) = 3.89
Reject Ho if F-calc > 3.89. Do not reject Ho if F-calc <=3.89.
F-calc = 4.52 (SSBG=63.32, SSWG=84.04) Reject Ho and conclude that at least 2 groups differ on the number of problems solved.
b) p < .05. Type I error: concluding at least 2 groups differ on the number of problems solved, when actually there is no difference between all 3 groups.

4a) Hyp-testing/one sample/small n - directional test
Ha: mu > 3.0; Ho: mu <= 3.0
tcrit = 1.782
Reject Ho if tcalc > 1.782; Do not reject Ho if tcalc <= 1.782
tcrit = 5.0
Reject Ho; conclude that the average GPA of the grad applicants is significantly higher than 3.0.
b) p < .05; Type I -conclude that the average GPA of the grad applicants is significantly higher than 3.0, when it actually is not.
c) 99% CI = (3.175, 3.725) -- We are 99% confident that the true population mean for GPA is between 3.175 and 3.725. 3.0 (a) is not in the CI. Since it is below that range, we conclude that the true population GPA of graduate applicants is significantly higher than 3.0.

5a) CI/small n/dep samp/2 groups -- 99% CI = (-2.355,9.155). 0 is in the CI, so we conclude that with 99% confidence there is not a significant difference between the before and after treatment phobia scores.
b) Ha: mub-mua > 0; Ho: mub-mua <=0.
t-crit = 2.132
Reject Ho if t-calc > 2.132; Do no reject Ho if t-calc <= 2.132.
t-calc = 2.72; Reject the null. The treatment did significantly lower the phobia scores.
c) p<.05; Type I error: Concluding that phobia scores were significantly lower after treatment than before treatment, when actually the phobia scores were not lower.
d) The CI was using a much more conservative alpha level (.01) than the hypothesis test (.05). The rejection region is larger for the hypothesis test, and so it found a significant difference whereas the more conservative CI did not. (If they both used the same alpha, the conclusions would always be the same).

6a) Hyp-testing/large n/indep samp/2 groups -- nondirectional test
Ha: mub-mun not= 0; Ho: mub-mun = 0.
z-crit = -1.96 and 1.96
Reject Ho if z-calc > 1.96 or z-calc < -1.96; Do not reject Ho if -1.96 <= z-calc <= 1.96.
z-calc = 2.87; Reject the null. There was a significant difference in time estimations when bored and not bored.
b) p=.0042 (.0042<.05) We could have made a Type I error. Concluding there is a difference in time estimations when bored and not bored, when actually there is no difference.
c)CI/large n/ indep samp/ 2 groups; 90% CI = (1.06, 3.94). 0 is not in the interval, so we conclude there is a significant difference. Since all the numbers are positive we conclude that time estimations are longer for the first group (bored) than the second group (not bored).

7a) b1 = -1.11; For every one extra glass of juice consumed per day, the number of doctor's visits decreases by 1.11.
b) b0 = 6.59; When number of glasses of juice consumer per day is zero, the number of doctor's visits per year is 6.59.
c) 90% CI = (-1.75, -.48). Zero is not in the interval, so we conclude that the slope is significantly different than zero. We are 90% confident that the true population slope is somewhere between -1.75 and -.48. Because all the numbers in the interval are negative, we conclude the slope is negative.
d) Pearson correlation
Ha: rho < 0; Ho: rho >= 0
rcrit = -.6319
Reject Ho if rcalc < -.6319; Do not reject Ho if rcalc >= -.6319
rcalc = -.76
Reject Ho; conclude that number of glasses of juice consumed per day is strongly negatively correlated with number of doctor's visits per year. p<.025; Type I error: concluding that the number of juice consumed per day is strongly negatively correlated with number of doctor's visits per year, when actually it is not.
e) Regression analysis
Ha: beta1 not= 0; Ho: beta1 = 0
tcrit = -2.896 and 2.896
Reject Ho if tcalc < -2.896 or tcalc > 2.896; Do not reject Ho if -2.896 <= tcalc <= 2.896
tcalc = -3.26
Reject Ho; conclude that number of glasses of juice consumed per day can predict the number of doctor's visits a year.
p < .01; Type I: concluding that number of glasses of juice consumed per day can predict the number of doctor's visits a year, when actually it cannot.
f)r2 = .58. 58% of the variability in number of doctor's visits per year can be explained by the number of glasses of juice consumed per day.

8a) z=1.11
b) percentile rank = 86.65% (86.65% of the lumberjacks chop less wood than Janet)
c) .1477
d) raw score = 17.84 logs chopped

9a) data are not normally distributed, so we have to use the Spearman-rank-order test (can handle numerical data).
Ha: rhos > 0; Ho: rhos <= 0
rscrit = .504;
Reject Ho if rscalc > .504; Do not reject Ho if rscalc <= .504.
rscalc = .26
Do not reject Ho; Concude that scores on the GRE verbal test are not positively associated with scores on the GRE quantitative test.
b) p > .05; Type II error: Conclude that scores on the GRE verbal and quantitiative tests are not associated, when actually they really are.

10a) CI/large n/single sample -- 94% CI = (102.9, 110.3) We are 94% confident that the true IQ of the statistics students is between 104.9 and 112.3. Since 100 (a) is not in the interval (it is below it), we conclude that the IQ of statistics students is higher than the value of 100.
b) Hyp-test/large n/single sample
Ha: mu > 100; Ho: mu <= 100
zcrit = 2.33
Reject Ho if zcalc > 2.33; Do not reject Ho if zcalc <= 2.33
zcalc = 2.78; Reject Ho and conclude that students who take a statistics course have significantly higher IQ scores than the norm of 100.
c) p = .0027; (p < .01) Type I error: Concluding that students who take a statistics course have significantly higher IQ scores than the norm of 100, when actually they don't.

11a) CI/large n/dep samp/ 2 groups -- 92% CI = (-2.31, .11). 0 is in the CI, so we conclude that with 92% confidence there is no difference in relaxation scores for cold and hot baths.
b) Hyp test/large n/ dep samp/ 2 groups
Ha: muc-muh < 0; Ho: muc-muh >= 0.
z-crit = -2.33
Reject Ho if z-calc < -2.33; Do not reject Ho if z-calc >= -2.33.
z-calc = -1.59; Do not reject the null. Conclude that relaxation scores are not significantly lower for cold baths than hot baths.
c) p=.056 (.056>.01) We could have made a Type II error. Conclude that relaxation scores are not significantly lower for cold baths than hot baths when they actually are lower. --Note: we had a large sample here so low power probably wasn't a factor.

12. probability = 0 (found by getting z=-7.62; remember the question is asking for the probability of getting a particular mean in a distribution.)

13a) mode = 14; median = 19.5; mean = 19
b) range = 22; variance = 48.29; standard deviation = 6.95