Population Analysis

Part 1

1. Make an argument for why it would make sense to give students a z-score instead of a grade for their classes, especially when there are multiple sections of the same class taught by different instructors.
2. Researchers comparing the IQ scores for younger adults and older adults typically find greater differences (diversity) in the older adults. Following are typical data showing IQ scores (that have been standardized) for two groups (samples) of participants.
   Older Adults Younger Adults
   89 111 100 130 98 110 111 102
   87 99 121 91 96 106 114 95
   a. Compute the mean, the range, the variance, and the standard deviation for each group.
   b. Is one group of scores more variable than the other?
3. In a population distribution, a score of X = 36 corresponds to a z = -5.00 and a score of 81 corresponds to a z = 2.50. Find the mean and the standard deviation for the population.
4. A distribution with a mean of 93 and a standard deviation of 12 is transformed into a standardized distribution with a mean of 154 and a standard deviation of 9. Find the new standardized score for each of the values from the original population.
   a. X = 80
   b. X = 53
   c. X = 100
   d. X = 105
5. Find the z-score boundaries that separate a normal distribution as described in each of the following.
   a. The middle 36% from the 64% in the tails.
   b. The middle 60% from the 40% in the tails.
6. S.M.A.R.T. test scores are standardized to produce a normal distribution with a mean of 140 and a standard deviation of 35. Find the proportion of the population in each of the following S.M.A.R.T. categories.
   a. Genius: Score of greater than 180
   b. Superior intelligence: Score between 133 and 161.
   c. Average intelligence: Score between 110 and 120.
7. A population of scores forms a normal distribution with a mean of 180 and a standard deviation of 13.
   a. What proportion of the scores have values between 175 and 185?
   b. If samples of n = 4 are selected from the population, what proportion of the samples have means between 175 and 185?
   c. If samples of n = 16 are selected from the population, what proportion of the samples will have means between 175 and 185?
8. For a population with a standard deviation of 40, how large a sample is necessary to have a standard error that is:
   a. less than or equal to 10 points?
   b. less than or equal to 4 points?
   c. less than or equal to 2 points?
9. Why is it possible to define the probability of committing a Type I error but impossible to know the probability of committing a Type II error? Which one is a worse error to commit? Explain why you think that (hint: there is no “right” answer, you get marks for arguing your side convincingly).
10. Montarello and Martin (2005) found that 5th grade students completed more math problems correctly when simple problems were mixed with their regular math assignments. To further explore this phenomenon, suppose that a researcher selects a standardized math test that produces a normal distribution of μ = 90 and a standard deviation of 8. The researcher modifies the test by inserting a set of very easy problems among the standardized questions, and gives the modified test to a sample of 25 students. If the average test score for the sample is M = 98, is this result sufficient to conclude inserting the easy questions improves student performance? Use a one-tailed test with α = 0.05. Also name the independent and dependent variables for this experiment.
11. Find an example in a media article from 2020 (newspaper or magazine; cite your resource) where correlational data was incorrectly presented as if it were causal data. Explain why, for this data, there could be another reason for the data to correlate besides there being a causal relation between the two variables.
12. Compute the Pearson correlation for the following data. X Y__
    23 9
    25 13
    11 15
    13 17
    14 15
    27 14
    10 9
    12 16

Part 2
Question 1 (1 point)

A set of scores ranges from a high of X = 34 to a low of X = 24 and has a mean of 29. Which of the following is the most likely value for the standard deviation for these scores?
Question 1 options:
2 points
5 points
10 points
34 points

Question 2 (1 point)

What is the value of SS for the following set of scores? Scores: 1, 1, 4, 0
Question 2 options:
18
10
9
6

Question 3 (1 point)

What is the variance for the following population of scores? Scores: 5, 2, 5, 4
Question 3 options:
6
2
1.5
1.22

Question 4 (1 point)

Under what circumstances is the computational formula preferred over the definitional formula when computing SS, the sum of the squared deviations, for a sample?
Question 4 options:
When the sample mean is a whole number
When the sample mean is not a whole number
When the sample variance is a whole number
When the sample variance is not a whole number

Question 5 (1 point)

A sample of n = 25 scores has M = 20 and s2 = 9. What is the sample standard deviation?
Question 5 options:
3
4.5
9
81

Question 6 (1 point)

All the possible samples of n = 3 scores are selected from a population with µ = 30 and σ = 5 and the mean is computed for each of the samples. If the average is calculated for all of the sample means, what value will be obtained?
Question 6 options:
30
Greater than 30
Less than 30
Impossible to predict

Question 7 (1 point)

What z-score corresponds to a score that is above the mean by 2 standard deviations?
Question 7 options:
+2
+10
+20
Impossible to determine without knowing the value of the standard deviation

Question 8 (1 point)

For a population with µ = 80 and σ = 6, what is the z-score corresponding to X = 68?
Question 8 options:
–0.50
–2.00
+2.00
–12.00

Question 9 (1 point)

In a population with a standard deviation of σ = 5, a score of X = 44 corresponds to a z-score of z = 2.00. What is the population mean?
Question 9 options:
49
54
39
34

Question 10 (1 point)

Last week Tim got a score of X = 54 on a math exam with µ = 60 and σ = 8. He also got X = 49 on an English exam with µ = 55 and σ = 3, and he got X = 31 on a psychology exam with µ = 37 and σ = 4. For which class should Tim expect the best grade?
Question 10 options:
Math
English
Psychology
All 3 grades should be the same.

Question 11 (1 point)

A distribution with µ = 55 and σ = 6 is being standardized so that the new mean and standard deviation will be µ = 50 and σ = 10. When the distribution is standardized, what value will be obtained for a score of X = 58 from the original distribution?
Question 11 options:
X = 53
X = 55
X = 58
X = 61

Question 12 (1 point)

For a sample with s = 12, a score of X = 73 corresponds to z = –1.00. What is the sample mean?
Question 12 options:
M = 61
M = 67
M = 79
M = 85

Question 13 (1 point)

A class consists of 10 males and 30 females. A random sample of n = 3 students is selected. If the first two students are both females, what is the probability that the third student is a male?
Question 13 options:
10/37
10/38
10/40
8/38

Question 14 (1 point)

What proportion of a normal distribution is located in the tail beyond z = –1.00?
Question 14 options:
0.8413
0.1587
–0.3413
–0.1587

Question 15 (1 point)

What proportion of a normal distribution is located between z = –1.25 and z = +1.25?
Question 15 options:
0.8944
0.2112
0.3944
0.7888

Question 16 (1 point)

A normal distribution has a mean of µ = 40 with σ = 10. If a vertical line is drawn through the distribution at X = 55, what proportion of the scores are on the right side of the line?
Question 16 options:
0.3085
0.6915
0.0668
0.9332

Question 17 (1 point)

A normal distribution has a mean of µ = 100 with σ = 20. If one score is randomly selected from this distribution, what is the probability that the score will have a value between X = 90 and X = 120?
Question 17 options:
0.1498
0.4672
0.5328
0.2996

Question 18 (1 point)

IQ scores form a normal distribution with µ = 100 and σ = 15. Individuals with IQs above 140 are classified in the genius category. What proportion of the population consists of geniuses?
Question 18 options:
0.9962
0.5038
0.4962
0.0038

Question 19 (1 point)

Samples of n = 16 scores are selected from a population. If the distribution of sample means has an expected value of 40 and a standard error of 2, what are the mean and the standard deviation for the population?
Question 19 options:
µ = 40 and σ = 2
µ = 40 and σ = 8
µ = 40 and σ = 32
µ = 160 and σ = 2
Question 20 (1 point)

What symbol is used to identify the standard error of M?

Question 20 options:
σM
µ
σ
MM
Question 21 (1 point)

Samples of size n = 9 are selected from a population with μ = 80 with σ = 18. What is the standard error for the distribution of sample means?
Question 21 options:
80
18
6
2
Question 22 (1 point)

For a particular population, a sample of n = 4 scores has an expected value of 10. For the same population, a sample of n = 25 scores would have an expected value of _.
Question 22 options:
4
8
10
20
Question 23 (1 point)

If a sample of n = 4 scores is obtained from a population with μ = 70 and σ = 12, what is the z-score corresponding to a sample mean of M = 73?
Question 23 options:
z = 0.25
z = 0.50
z = 1.00
z = 2.00

Question 24 (1 point)

A sample is selected from a normal population with μ = 50 and σ = 12. Which of the following samples would be considered extreme and unrepresentative for this population?
Question 24 options:
M = 53 and n = 16
M = 53 and n = 4
M = 56 and n = 16
M = 56 and n = 4
Question 25 (1 point)

What is measured by the denominator of the z-score test statistic?
Question 25 options:
The average distance between M and µ that would be expected if H0 was true
The actual distance between M and µ
The position of the sample mean relative to the critical region
Whether or not there is a significant difference between M and µ
Question 26 (1 point)

If a hypothesis test produces a z-score in the critical region, what decision should be made?
Question 26 options:
Reject the alternative hypothesis
Fail to reject the alternative hypothesis
Reject the null hypothesis
Fail to reject the null hypothesis

Question 27 (1 point)

When is there a risk of a Type I error?
Question 27 options:
Whenever H0 is rejected
Whenever H1 is rejected
Whenever the decision is "fail to reject H0"
The risk of a Type I error is independent of the decision from a hypothesis test.

Question 28 (1 point)

You complete a hypothesis test using  = .05, and based on the evidence from the sample, your decision is to fail to reject the null hypothesis. If the treatment actually does have an effect, which of the following is true?
Question 28 options:
You have made a Type I error.
You have made a Type II error.
You might have made a Type I error, but the probability is only 5% at most.
You have made the correct decision.

Question 29 (1 point)

A researcher administers a treatment to a sample of participants selected from a population with µ = 80. If a hypothesis test is used to evaluate the effect of the treatment, which combination of factors is most likely to result in rejecting the null hypothesis?
Question 29 options:
A sample mean near 80 with  = .05
A sample mean near 80 with  = .01
A sample mean much different than 80 with  = .05
A sample mean much different than 80 with  = .01

Question 30 (1 point)

If a hypothesis test is found to have power = 0.70, what is the probability that the test will result in a Type II error?
Question 30 options:
0.30
0.70
p > 0.70
Cannot determine without more information

Question 31 (1 point)

For which of the following Pearson correlations would the data points be clustered most closely around a straight line?

Question 31 options:
r = –0.10
r = +0.40
r = –0.70
There is no relationship between the correlation and how close the points are to a straight line.

Question 32 (1 point)

The results from a research study indicate that adolescents who watch more violent content on television also tend to engage in more violent behaviour than their peers. The correlation between amount of TV violent content and amount of violent behaviour is an example of _.
Question 32 options:
a positive correlation
a negative correlation
a correlation near zero
a correlation near one

Question 33 (1 point)

A set of n = 10 pairs of scores has ΣX = 20, ΣY = 30, and ΣXY = 74. What is the value of SP for these data?
Question 33 options:
74
24
14
–14

Question 34 (1 point)

Suppose the correlation between height and weight for adults is +0.40. What proportion (or percent) of the variability in weight can be explained by the relationship with height?
Question 34 options:
40%
60%
16%
84%
Question 35 (1 point)

For the linear equation Y = 2X + 4, if X increases by 1 point, how much will Y increase?
Question 35 options:
1 points
2 points
3 points
4 points

Question 36 (1 point)

The following X and Y scores produce SSX = 2 and SP = 8. What is regression equation for predicting Y?

Question 36 options:
Y = 0.25X + 4.5
Y = 0.25X – 4.5
Y = 4X + 3
Y = 4X – 3