Description
The following 10 items are based on the following summarized research:
(*you will need your z-tables from Howell to complete many of these questions)
A psychologist studied self-esteem scores and found the sample data set to be normally distributed with a mean of 50 and a standard deviation of 5.
Part A**What raw score cuts off the bottom 10.03% of this distribution? Or to phrase another way, what raw score is associated with the 10.03 percentile?
Steps: (IT HELPS TO DRAW A GRAPH AND SHADE THE AREA(S) IN QUESTION)
To avoid mis-scoring by Blackboard, enter only numbers and, if required, negative signs (-) and decimal points (.).
What is the z-score that cuts off the bottom 10.03% of this distribution? Or to phrase another way, what z-score is associated with the 10.03 percentile? (HINT: Don’t forget that z scores BELOW the mean are negative!)
What is the raw score that cuts off the bottom 10.03% of this distribution? Or to phrase another way, what raw score is associated with the 10.03 percentile?
Part B**What percentage of the scores is between 57.5 and 65?
Steps:
What is the z-score that corresponds to the raw score of 57.5?
What is the z-score that corresponds to the raw score of 65?
What percentage of the scores is between 57.5 and 65?
Part C:**A raw score of 35 is associated with what percentile?
Steps:
What is the z-score associated with a raw score of 35?
A raw score of 35 is associated with what percentile?
Part D:**What raw scores mark the middle 34% of this distribution? (HINT: DRAW AND SHADE IN THE MIDDLE 34% TO VISUALIZE! Remember…..the middle 34% means that 17% is BELOW the mean and 17% is ABOVE the mean……)
Steps:
What are the z-scores that mark the middle 34% of this distribution? The z-score below the mean is The z-score above the mean is
What is the raw score below the mean?
A nurse studied PSA readings and found the sample data set to be normally distributed with a mean of 87 and a standard deviation of 7.25. What is the median of this distribution?
(Hint: this question has to do with knowing the characteristics of a normal distribution and where the mean, mode, and median lie in relation to one another in such a distribution. Check out the “Skewed and Normal Distributions” document under the Module 4 Instruction Materials.)
The following 9 questions (Q3 to Q11) are primarily conceptual questions based on Modules 3 and 4
In a negatively skewed distribution, Alisha scored the mean, Ken scored the median, and Catherine scored the mode. Who had the highest score?
Alisha
Ken
Catherine
All three scored approximately the same
In a normal distribution, Alisha scored the mean, Ken scored the median, and Catherine scored the mode. Who had the lowest score?
Alisha
Ken
Catherine
All three scored approximately the same
The standard normal distribution has a mean of ___ and a variance of ____
0; 0
1; 1
1; 0
0; 1
On a comprehensive final exam, the mean of the class exam scores was 517, and the standard deviation of the scores was 6. One particular student’s z-score was 3. What was this student’s raw score?
The standard deviation for the sample consisting of 26, 48, and 70 is
A medical association randomly selected 10 trauma surgeons and 10 medical researchers and asked them how flexible their schedule is on a 1 (not at all flexible) to 9 (very flexible) scale. The data are summarized below:
Mean
Variance
Trauma surgeons
2
1
Medical researchers
7
3
Select the statement that accurately reflects these results:
(Hint: For this problem, the key is in the variance and in knowing what the variance represents! No calculations are necessary.)
All medical professionals have similar levels of flexibility in their schedules.
Trauma surgeons have more flexibility than medical researchers.
Medical researchers agree more with each other about their level of flexibility than trauma surgeons.
Trauma surgeons agree more with each other about their level of flexibility than medical researchers.
none of the above
A sample of data has a standard deviation of 35. If you were to multiply all of the scores in the data set by factor (or constant) of 1, what would the new standard deviation be?
(Hint: The “Working with Constants in Equations” document under Instruction Materials will be helpful to you in answering this question.)
Select the statement that is false.
Given a raw score, its z-score is the number of standard deviations above or below the mean.
The variance for a set of data can be a negative value.
A normal curve with mean 0 and standard deviation 10 has a greater spread of scores than a normal curve with mean 0 and standard deviation 1.
The standard deviation for a set of data can be 0
Along with the mean, which of the following completely characterizes a standard normal distribution?
(Hint: Refer to page 1 of the Module 4 course notes as well as section 6.2 of Howell, 7th edition, and section 8.3 of Utts, 3rd edition.)
Positive Skew
Variance/Standard Deviation
Asymmetry
Negative Skew
Unformatted Attachment Preview
MODULE 4
Lesson 4: The Normal Distribution
TEXTBOOK REFERENCES:
1. Howell—Chapter 6 The Normal Distribution
2. Howell—Appendix D: Table D10–Z-tables (For the newer 7th edition of Howell Appendix E10 instead)
3. Utts—Chapter 8 Bell-Shaped Curves and Other Shapes
COURSE NOTES:
Bell-Shaped Curves—The Normal Distribution—Z-Scores
The Normal Distribution:
-a specific distribution having a characteristic bell-shape form
Uses: 1) Convert raw scores into percentiles
2) Determine probability of obtaining certain scores
3) Convert scores in one distribution to scores in another distribution
Convert histogram to frequency polygon (line graph) to a normal distribution.
-where a line graph is a graph in which straight lines connect a series of points
(Please see Howell)
Abscissa—horizontal axis
Ordinate—vertical axis
Density—-height of a curve for a given value of X; And is closely related to the probability of an
observation in an interval around X
Characteristics of the Normal Curve
-there are many possible normal curves, each differing in its value for the mean and standard deviation.
-For a given mean, the larger the standard deviation, the greater the spread of scores. That is,
although each normal curve is bell-shaped, as the standard deviation increases, the curve
becomes flatter.
-Shared characteristics
-each normal curve is symmetric
-that is, if you draw a line down the middle and fold the right half over the left half, the
two halves will coincide.
-each normal curve is unimodal
-the tails never actually touch the X-axis (they stretch to +/- infinity)
The Standard Normal (Curve) Distribution
-a normal distribution with a mean equal to ―
0‖ and a variance equal to ―
1‖; denoted as N(0,1)
-where N=normal
-0 refers to mu (mean)μ
-1 refers to sigma squared (variance) σ2
Linear Transformation
-a transformation involving addition, subtraction, multiplication, or division of or by a constant.
**-a procedure to convert a mean to “0” and the standard deviation to “1”**
PSY120: Module 4 Course Notes
Page 1
Z-score (standard score):
-number of standard deviation units above or below the mean
-is the deviation of a raw score from the mean in standard deviation units.
-that is, a z-score tells us how far from a mean a score is in standard deviation units and in which
direction
-positive z-scores correspond to scores above the mean
-negative z-scores correspond to scores below the mean
Formulas:
1)
z score from a raw score
sample:
population:
Example: Assume IQ scores are normally distributed with μ =100, σ =15
Suppose someone had scored 120 on an IQ Test. What does this equal in terms of zscores?
Therefore, this is above the mean, and 1.33 refers to 1.33 standard deviation units above
the mean
Suppose another person scored 75. What is z?
Therefore, this is below the mean, and 1.67 refers to 1.67 standard deviation units below the
mean
2)
Raw score from a z score
Sample
Population
Example:
PSY120: Module 4 Course Notes
z = -1.00 (IQ scale)
= 100 + (15)(-1.00)
= 100-15
= 85
Page 2
Interpretation of z-scores for Mound-shaped distributions of data (Empirical Rule):
1)
Approximately 68% of the measures will have a z-score between –1 and +1
2)
Approximately 95% of the measures will have a z-score between –2 and +2
3)
All or almost (99.7%) all of the measures will have a z-score between –3 and +3
Steps For Calculation:
You will need a combination of skills to answer the following questions in this section. Keep in mind,
most questions will go through a 6-step process.
1)
Graph
2)
Shade the area(s) in question
3)
Calculate the z-score(s)
4)
Look up the z-score(s) in the Howell tables
5)
Answer the question
6)
Interpret your response
Make sure you can read the Howell tables and understand what a ‗larger portion‘, a ‗smaller portion‘, and a
‗mean to z‘ area is under the curve.
Larger Portion—any area under the curve greater than 50% of the distribution and includes 1 of the 2 tails
Smaller Portion—any area under the curve less than 50% of the distribution and includes 1 of the 2 tails
Mean to Z—any area under the curve that goes from a z-score up/back to the mean
Examples:
1) What z-score is associated with the 33rd percentile?
-look under ‗smaller portion‘ until you find 0.33
-go back to the z-score (remember the 33rd percentile is below the mean)
-0.44
2) What z-score is associated with the 67th percentile?
-look under ‗larger portion‘ until you find 0.67
-go back to the z-score (remember the 67th percentile is above the mean)
+0.44
3) Assume that women‘s shoe sizes are normally distributed with a μ= 8.25 and σ = 1.17.
What is the z-score corresponding to a size 6 shoe size?
Z = (6 – 8.25)/1.17
-1.92
What shoe size falls 2.5 standard deviation units above the mean?
X = 8.25 + (1.17)(2.5)
11.18
4) Suppose Verbal SAT scores are normally distributed with a μ= 500 and σ = 100.
A person scores 637 on the verbal SAT. What is her percentile rank?
Z = (637-500)/100
1.37
th
Look up 1.37 and go to ‗larger portion‘
91.47 percentile
What is the percentile rank of somebody who got 412 on the verbal SAT?
Z = (412-500)/100
-0.88
Look up 0.88 and go to ‗smaller portion‘
18.94th percentile
PSY120: Module 4 Course Notes
Page 3
5) A second grade teacher had to administer the state IQ test to their students.
Assume the IQ scores for the population are normal with a μ= 100 and σ = 17.
What proportion of students will likely score 120 or above?
Z = (120-100)/17
1.18
Look up 1.18 and go to ‗smaller portion‘
0.119
*remember areas, probabilities, and proportions are decimal values between 0 and 1 including 0 and
1…therefore, you just leave 0.119 in decimal format.
What percentage of students will score below 85 or above 105?
First thing we need to do is convert both raw scores to z-scores
Z = (85-100)/17
Z = (105- 100)/17
Second thing we need to do is convert both z-scores to areas
-0.88 smaller portion
Area = .1894
+0.29 smaller portion
Area = .3859
Finally we need to add these areas together
-0.88
+0.29
57.53%
What is the probability of obtaining at IQ score between 110 and 125?
Again, first thing we need to do is convert both raw scores to z-scores
Z = (125-100)/17
Z = (110-100)/17
Second thing we need to do is convert both z-scores to areas
1.47 mean to z
Area = .4292**
Area = .2224**
0.59 mean to z
Finally we need to subtract these areas
1.47
0.59
0.2068**
**When working with proportions (that is, values less than 1) it is important to round to four decimal places
to ensure reasonable precision.**
What IQ score corresponds to the 10th percentile?
The z-score that corresponds to the 10th percentile is
Therefore, X = 100 + (17)(-1.28)
PSY120: Module 4 Course Notes
-1.28
78.24
Page 4
Review Questions:
1.
Suppose the mean number of hours of television watched by Americans is 25.4 hours/week, with a
standard deviation of 6.1. let us also suppose that the distribution is nearly normal in shape so that we can use
the normal distribution as a model.
What proportion of people watch more than 25.4 hours/week?
What percentage of people watch 26.925 hours/week or more?
What proportion of people watch less than 26.925 hours/week?
What percentage of people watch less than 28.45 hours/week?
What proportion of people watch more than 28.45 hours/week?
What percentage of people watch between 19.3 to 31.5 hours/week?
What percentage of people watch between 25.4 to 26.925 hours/week?
What proportion of people watch between 13.2 to 37.6 hours/week?
What proportion of people watch between 7.1 to 43.7 hours/week?
Compute the 90th percentile. Interpret.
How many hours of television do you have to watch to be in the top 10%?
How much television do you have to watch to be in the bottom 10%?
Compute the 70th percentile. Interpret.
How many hours of television do you have to watch to be in the top 30%?
How many hours of television do you have to watch to be in the top 20%?
2. Recall, that the GRE (graduate entrance exam) has a mean of 497 with a standard
deviation of 115.
What percentage of the population will score between 450 and 465 on the GRE?
What proportion of the population will score between 500 and 612 on the GRE?
What percentage of the population will score between 382 and 439 on the GRE?
3. The following refer to a distribution with mean = 60 and standard deviation = 4.3.
The raw score corresponding to a Z-score of 0.00 is __________.
The raw score corresponding to a Z-score of –1.51 is __________.
The Z-score corresponding to a raw score of 68.7 is __________.
The standard deviation of a Z-score distribution is ___________.
What is the area between a Z-score of .43 and a Z-score of 1.33?
What is the area between a Z-score of –1.25 and a Z-score of .36?
4. In a normal distribution of test scores with a mean equal to 57 and a standard deviation equal to 6.5, what
percentage of the scores on the test will be greater than 65?
5. The scores on a personality test are normally distributed with
and
What proportion of people taking the test can be expected to score between 229 and 325?
6. In a distribution of scores with a mean of 1500 and a standard deviation of 250, what raw score corresponds
with the 67th percentile?
PSY120: Module 4 Course Notes
Page 5
Answers to Review Questions:
1.
.50
40.13% .5987
69.15% .3085
68.26% 9.87% .9544
33.21; 90% of the population watches television for 33.21 hours or less
33.21 hours or more
17.59 hours or less
28.57; 70% of the population watches television for 28.57 hours or less
28.57 hours or more
30.52 hours or more
2.
3.
4.
5.
6.
4.88%
.33
15%
60
53.51 2.02 1
10.93%
.7518
1610
PSY120: Module 4 Course Notes
.9974
.2418 .5350
Page 6
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