Appendix B
Basic Review Problems

The problems in this document were designed to review fundamental concepts. They are not placed in any descriptive context,but are meant to check whether one is able to carry out basic calculations. The problems are listed in the same order as the corresponding material in the text. Several problems include multiple samples. The answers may be found in a companion document at this website.

1. Assume an entire population is surveyed and the following data reported.Find the population mean, the uncorrected sum of squares, the correction term, the corrected sum of squares, the population variance, and the population standard deviation.

82 38 78 31 38
23 55 78 65 66
80 48 67 53 83

2. From a population with mean µ = 50.0 and standard deviation σ = 12.0 eight random samples of 10 individuals are drawn. For each, find the sample mean, the uncorrected sum of squares, the correction term, the corrected sum of squares, the sample variance, and the standard error.

Sample
1 2 3 4 5 6 7 8
52 59 50 39 44 50 67 55
43 59 41 42 47 45 78 58
55 43 46 60 60 42 43 50
44 52 35 54 41 52 38 61
45 56 40 41 49 52 73 41
59 58 36 53 38 58 45 50
38 50 48 46 58 62 54 25
48 36 57 44 68 47 41 41
41 49 58 43 51 30 51 59
61 50 48 63 51 47 57 55

3. A population has a normal distribution with mean µ = 50 and standard deviation σ = 12.
   (a) Calculate the probability that X is greater than 62.
   (b) Calculate the probability that X is less than 25.
   (c) Calculate the probability that X is greater than 50 but less than 79.
   (d) Calculate the probability that X is less than 50 but greater than 43.
   (e) Calculate the probability that X is less than 70.
   (f)  Calculate the probability that X is greater than 27.
   (g) Calculate the probability that X is not between 23 and 77.
   (h) Calculate the probability that X is between 23 and 77.
   (i)  Calculate the probability that X is between 34 and 57.
   (j)  Calculate the probability that X is not between 35 and 60.

4. From a population with mean µ = 50.0 and standard deviation σ = 12.0 eight random samples of 10 individuals are drawn. For each find the sum, the uncorrected sum of squares, the sample mean, the sample variance, and the standard error. Then find 90%, 95%, 98%, and 99% confidence intervals for the population mean.

Sample
1 2 3 4 5 6 7 8
42 61 45 28 50 51 46 54
52 55 60 51 58 55 35 52
38 44 62 36 50 54 62 78
46 50 28 20 41 27 70 46
47 47 45 62 53 51 45 53
58 51 38 40 52 57 56 76
25 70 67 55 61 24 44 38
48 48 53 68 22 51 48 43
68 45 42 56 50 68 50 36
70 60 48 62 44 51 39 47

5. Eight random samples of 10 individuals are drawn from a population. For each, find the sum, the uncorrected sum of squares, the sample mean, the sample variance, and the standard error. Then find 90%, 95%, 98%, and 99% confidence intervals for the population variance.

Sample
1 2 3 4 5 6 7 8
48 42 53 52 33 44 45 65
58 44 41 68 61 44 49 47
47 61 42 56 59 63 58 70
35 47 57 54 48 44 55 25
67 42 32 51 50 45 59 27
62 58 61 49 35 67 45 48
53 75 47 53 43 42 49 55
48 60 57 35 48 57 57 62
50 20 35 48 51 41 50 41
70 49 51 53 42 44 60 38

6. Eight random samples of 10 individuals are drawn from a population. For each, find the sum, the uncorrected sum of squares, the sample mean, the sample variance, and the standard error. For each sample test (i) H0: µ ≤ 50 versus Ha: µ ≥ 50 (ii) H0: µ ≥ 50 versus Ha: µ ≤ 50 (iii) H0: µ = 50 versus Ha: µ ≠ 50 all at the ∝ = 0.05 level.

Sample
1 2 3 4 5 6 7 8
66 37 44 47 45 47 63 51
53 45 75 55 34 18 41 33
37 52 55 47 51 20 34 56
31 40 52 52 23 61 43 31
75 21 44 60 33 68 57 47
54 40 72 14 45 30 13 37
37 82 50 72 57 29 49 47
75 43 61 54 72 55 49 44
48 72 46 66 29 51 40 68
49 30 24 54 53 62 47 57

7. Population 1 has a mean of 50.0 and a standard deviation of 12.0. Population 2 has a mean of 50.0 and a standard deviation of 6.0. Each population is sampled four times. For each of the eight samples, find the sum, the uncorrected sum of squares, and the sample variance. Then for each pair of samples (one from each population), test at the ∝ = 0.05 level whether they have significantly different variances.
Example 1 Example 2 Example 3 Example 4
Sample 1 Sample 2 Sample 1 Sample 2 Sample 1 Sample 2 Sample 1 Sample 2
42.0 50.0 42.0 49.0 51.0 53.5 56.0 54.5
56.0 53.5 40.0 51.0 52.0 50.5 52.0 52.5
40.0 45.0 42.0 61.0 44.0 51.0 42.0 48.5
72.0 50.5 57.0 43.5 46.0 50.0 37.0 52.0
51.0 48.5 48.0 47.5 53.0 39.0 33.0 37.0
53.0 56.5 44.0 48.5 65.0 39.0 38.0 52.5
70.0 51.5 22.0 50.5 35.0 45.5 44.0 45.5
61.0 50.0 47.0 52.5 50.0 47.0 47.0 60.0
54.0 56.0 64.0 54.0 42.0 42.5 39.0 50.5
40.0 53.0 58.0 55.0 35.0 46.0 28.0 48.0

8. In each of the four examples below, (unpaired) samples were drawn from two populations. Use an appropriate t test to test the hypotheses: H0: µ1 = µ2 versus Ha: µ1 ≠ µ2 at the ∝ = 0.05 level.
Example 1 Example 2 Example 3 Example 4
Sample 1 Sample 2 Sample 1 Sample 2 Sample 1 Sample 2 Sample 1 Sample 2
44.0 47.0 48.0 51.0 63.0 37.0 59.0 48.0
52.0 54.0 58.0 47.0 53.0 52.5 43.0 48.0
55.0 38.5 55.0 49.5 56.0 44.0 50.0 44.5
54.0 46.5 55.0 49.0 39.0 55.0 53.0 49.5
49.0 51.0 39.0 47.5 47.0 36.0 39.0 46.5
52.0 56.0 57.0 49.0 54.0 62.5 41.0 63.0
43.0 37.5 56.0 42.5 47.0 47.5 49.0 48.5
54.0 48.0 62.0 51.5 42.0 55.0 45.0 45.0
46.0 48.0 54.0 53.0 28.0 44.5 43.0 57.5
48.0 42.5 53.0 50.0 52.0 50.0 58.0 54.5

9. In each of the four examples below, paired samples were drawn from two populations. Use an appropriate t test to test the hypotheses: H0: µ1 = µ2 versus Ha: µ1 ≠ µ2 at the ∝ = 0.05 level.
Pair Example 1 Example 2 Example 3 Example 4
Sample 1 2 1 2 1 2 1 2
1 44 57 45 75 42 37 44 73
2 57 57 42 50 48 49 22 44
3 22 52 78 54 57 44 60 53
4 48 57 55 73 46 68 52 43
5 53 37 60 44 22 50 32 33
6 33 59 40 49 43 65 50 50
7 48 54 50 58 56 24 30 42
8 47 55 49 47 57 56 42 48
9 53 45 56 51 40 58 42 32
10 53 59 52 55 61 46 47 51

10. Carry out a one-way analysis of variance at the ∝ = 0.05 level using the following data.
Sample 1 Sample 2 Sample 3 Sample 4 Sample 5
45 63 38 83 52
49 65 36 98 41
40 61 56 58 62
48 39 64 57
75
38
59
42

11. Carry out a randomized complete block design ANOVA using the data below to test the
Treatment (ith)
Block (jth) 1 2 3 4
1 52 54 67 41
2 55 51 48 36
3 63 53 56 54
4 52 56 31 40
5 21 30 60 33

12. Carry out a regression analysis for each of these four sets of paired data.
Observation Example 1 Example 2 Example 3 Example 4
X Y X Y X Y X Y
1 19 26 17 30 14 28 21 36
2 16 23 18 33 20 35 24 35
3 22 31 22 28 17 22 16 24
4 14 30 20 31 19 28 21 31
5 16 27 25 33 21 33 13 27
6 22 30 17 29 21 26 15 29
7 21 37 18 24 17 28 20 27
8 17 31 23 34 18 32 21 37
9 20 31 19 30 17 33 17 28
10 19 25 30 34 19 27 17 29

13. Carry out a Pearson correlation analysis for each of these four sets of paired data testing H0: ρ = 0 versus Ha: ρ ≠ 0.
Observation Example 1 Example 2 Example 3 Example 4
X Y X Y X Y X Y
1 26 30 15 26 17 33 17 30
2 18 33 17 31 20 34 17 33
3 24 38 12 24 15 22 21 26
4 25 28 19 33 15 23 16 29
5 17 34 20 30 21 31 17 24
6 20 31 16 26 26 35 16 33
7 20 32 25 32 23 32 22 29
8 25 30 25 34 25 30 25 34
9 19 25 21 32 20 29 27 39
10 19 26 22 29 21 29 18 35

14. For the data in the table below,test the null hypothesis H0: The ratio of Class 1 to Class 2 to Class 3 is 2 to 1 to 1, respectively.
Class Observed
1 49
2 30
3 21
Total 100

15. Test the null hypothesis H0: There is no association between X and Y classifications (they are independent) for the data in the following 2 × 2 contingency table.
X(1) X(2) Total
Y(1) 14 13 27
Y(2) 12 19 31
Total 26 32 58

16. Test the null hypothesis H0: There is no association between X and Y classifications (they are independent) for the data in the following 4 × 3 contingency table.
X(1) X(2) X(3) Total
Y(1) 15 23 25 63
Y(2) 17 15 21 53
Y(3) 27 29 13 69
Y(4) 22 28 10 60
Total 81 95 69 245

17. Clearly indicate true (T) or false (F) for each.

  1.  The normal distribution is really an infinite family of curves utilizing different
    valuesof µ and σ.
  2. E(Xi) = E() = E(µ) and E[(X - µ)2] = σ 2.
  3.      All analysis of variance F tests are right-tailed regardless of the alternative
    hypotheses.
  4. If two events are independent, they are also mutually exclusive.
  5.  Conditional probabilities are useful only when the two events in question are
    independent.
  6.  In a test of hypothesis it is generally easier to know the probability of a Type I
    error than the probability of a Type II error.
  7. The binomial test can be used when count data falls into two mutually exclusive categories.
  8.  The F test is used to test the equality of two sample variances while the chi-square test can be used to test whether a sample variance deviates significantly from a claimed value.
  9. According to the Central Limit Theorem,the distribution of sample means is approximately normally shaped for large samples regardless of the shape of the distribution of the X i's
  10. To completely describe a binomial distribution one must know n and p, but to
    completely describe a Poisson distribution one only needs to know µ.
  11. If a Type I error is denoted by α, then a Type II error can be denoted as 1 - α.
  12. The sign test can be used to test H0: M = c when the distribution of X is unknown and thought to be nonnormal while the Wilcoxon signed-rank test should be used for symmetrical but nonnormal distributions.

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