Chad Taylor
Math 1040: Statistics
Professor Alia Maw
Term Project Part 1
1.) For my Term Project, I have selected Data Set 5: IQ and Lead Exposure. These data come from
the study: “Neuropsychological Dysfunction in Children with Chronic Low-Level Lead
Absorption,” by P. J. Landrigan, R. H. Whitworth, R. W. Baloh, N. W. Staehling, W. F. Barthel, and
B. F. Rosenblum, Lancet, Vol. 1, Issue 7909.
2.) The data which I selected were from a study done to compare Intelligence Quotients (IQ) with
the subjects’ blood lead levels from individuals (children) living near a lead smelter over the
course of two years. Various other variables from each subject were taken from the subjects
including age (in years),and gender (represented as 1 and 2 for males and females respectively).
IQ scores were taken and noted on the standard IQ scale, and blood lead levels were measured
in micrograms per 100 ml of blood once a year for two consecutive years. These measurements
are noted as YEAR1 and YEAR2 for each of the two years. Exactly how all the data was extracted
is unsure, although interviews, IQ tests, and blood extraction were most likely utilized to
procure said data. At first glance, no noticeable correlations exist between IQ and blood lead
levels, but this can be discussed and reproved if necessary post-analysis.
3.) variable
name in
describe what the variable means/how it is
the data
measured (include units)
set
is the variable
what is the level
quantitative or of measurement
qualitative?
for this variable?
Qualitative.
Ordinal
Quantitative.
Ratio
Qualitative.
Nominal
Quantitative.
Ratio
Quantitative.
Ratio
Given on a scale of 1, 2, or 3 representing
low medium and high blood lead levels,
respectively. Low lead represents ˂ 40 μg Pb/
Lead
100 ml Blood for both years. Medium Lead
represents ≥ 40 μg Pb/ 100 ml Blood in one of
two years. High Lead represents ≥ 40 μg Pb/
100 ml blood in both years.
Age
Sex
Year 1
Year 2
Age of subject in years
Gender, represented as 1 or 2 for male and
female respectively.
Blood lead levels for first year of observation
(μg Pb/ 100 ml)
Blood lead levels for second year of
observation (μg Pb/ 100 ml)
IQV
Verbal IQ Score
Quantitative.
Ratio
IQP
Performance IQ Score
Quantitative.
Ratio
IQF
Full IQ Score
Quantitative.
Ratio
Term Project Part 2: Individual Portion
1.)
Figure 1: Randomly Selected Individuals' Genders
Figure 2: 35 Randomly Selected Individuals' Genders
Figure 3: Systematically Selected Individuals' Genders (Every 2nd Subject)
Figure 4: Systematically Selected Individuals' Genders (Every 2nd Subject)
2.) Obtaining the data for question two of the assignment was rather easy. We used a systematic
approach which involves selecting the data from every nth individual from the original set,
where n is an arbitrarily selected value by the person conducting the study (in this case us.)
We decided to select every 2nd person from the list. In order to do this, we first numbered
every individual from 1 to 121 (as there were 121 participants.) We then threw out the data
from every oddly numbered individual to leave us with only the evens. We did this only to the
point at which we had 35 data, per the assignment regulations. These 35 systematically
selected data points were then used in our graphs.
3.) As seen in the graphs above, when utilizing either 35 randomly selected individuals or 35
systematically selected individuals, our results were changed dramatically. In one there were
far more greater males, and in the other, females. This is likely due to the small sample size of
only 35 individuals, and the fact that subjects were merely giving one answer out of two
options, male or female. The difference between the two selection methods would likely be
less noticeable given a larger sample and more complex question.
4.) In this case, the method of systematic selection of data was far more accurate to the
population frequencies than was the random selection. Notice that the systematic selection
gave us 60% males while the whole population was approximately 61.16% males. I believe this
to be merely a coincidence, since the random selection just as easily could have granted the
same ratio given a larger sample size. So all in all, I’ve learned that the larger the sample size,
the more likely your results are to be accurate.
Figure 5: Systematically Selected Individuals' Genders (Every 2nd Subject)
Individual Term Project Part 3
1.) Summary Statistics:
Sample Set
Random
Systematic
2.)
Mean
Std. dev.
Min
Q1
Median
Q3
Max
34.6
13.229201
2
24
36
44
59
32.085714
10.407496
2
27
32
40
53
Randomly Selected
Systematically Selected
3.) The distribution in each of these two samples vary significantly from each other as well as from
the population. Although both the random and systematic selections follow a normal
distribution for the most part, increasing toward the middle, there are stark contrasts to one
another. For instance, their maximum number of frequencies in the middle vary greatly as to
how many people on average had that midrange level of lead exposure, despite the fact that
both samples contained but 35 individuals. In comparison to the population, both selected
samples seemed to have IQR’s significantly lower suggesting that, in these two samples, the lead
levels were generally higher. The differences in median values is astonishingly unique to each of
the sample methods used.
Random Year 1 Row
I.D.
107
118
120
84
114
121
46
76
8
24
98
93
16
95
14
68
88
116
1
32
20
39
70
60
96
73
34
19
79
35
Random Year 1
Lead Value
49
51
44
41
40
42
27
18
24
19
51
40
29
44
36
2
44
48
25
20
21
24
33
10
45
35
36
34
42
24
Systematic Year 1
Row I.D. (3’s)
3
6
9
12
15
18
21
24
27
30
33
36
39
42
45
48
51
54
57
60
63
66
69
72
75
78
81
84
87
90
Year 1
(Systematic)
30
29
24
29
30
28
35
19
22
32
2
38
24
36
33
24
29
28
27
10
34
23
32
35
34
36
40
41
51
43
10
108
11
100
111
31
52
21
50
59
93
96
99
102
105
40
45
42
53
45
Individual Term Project Part 4
1.) The interpretation for each of the confidence intervals is relatively the same, although for
posterity’s sake, I will proceed to type them all out.
a. Our first confidence interval states that: given the sample proportion of men to women
taken from our “population” we are 95% confident that the population parameter of
proportion will fall into the confidence interval which we calculated (theoretically,
omitting any calculating errors.)
b. Our second confidence interval states that: given the sample lead-blood levels taken
from our “population” we are 95% confident that the population parameter of mean (μ)
will fall into the confidence interval (30.06 μg Pb/100 ml, 39.14 μg Pb/100 ml)
c. Our third confidence interval states that: given the sample lead-blood levels taken from
our “population” we are 95% confident that the population parameter of standard
deviation (σ) will fall into the confidence interval (11.26, 18.83).
2.)
a. Unfortunately, there was a slight decimal error which ensued in the calculation of our
first interval for proportion of men to women. Before I could mention and correct this
slight error, the group paper had already been submitted. Nevertheless, I calculated the
interval to be (.33, .64) which just barely encompasses the true proportion of men to
women. This value, taken from the total population was 74/121 or .612.
b. Our second confidence interval was calculated accurately, since the mean of the
population was calculated from the entire data set to be 34.6 which wonderfully falls
into our interval of (30.06 μg Pb/100 ml, 39.14 μg Pb/100 ml).
c. Finally, our third confidence interval designed for standard deviation, which was (11.26,
18.83), completely enshrouded and included standard deviation taken from the entire
population which was found to be 13.36 using StatCrunch.
Term Project Part 5: Individual Portion
1.) For this portion of the term project, I have selected a value of α= 5% or .05. This corresponds to
a 95% confidence interval, and seems to be a very common confidence level to select.
2.) In a randomly selected sample of 35 individuals from part 2 of this assignment, we found that
the proportion of men was 45.71%. This seems a little low to me, as I would expect the
proportion of men to women to be 61.16% (from the population). My claim is that: the
proportion p of the population will be equal to p=0.6116. Therefore:
Ho: p=0.6116
H1: p≠0.6116
I used my TI-83 calculator to compute both the z and p values, although for posterity’s sake,
here is the calculation for the test statistic z: z= (p^ - p)/(√((pq)/n) = (.4571-.6116)/
√((0.6116*0.3884)/35)= -2.56
Z= -2.56 and p=.0106. We therefore reject Ho. Since p is less than a, we can conclude that
there is not sufficient evidence to support the claim that the proportion of men in the
population is 0.6116.
3.) From part 3 of this assignment we took random individual lead/blood levels from 35 randomly
selected individuals. We calculated the mean of lead in the blood of the individuals was 34.6 μg
Pb/100 ml blood. The standard deviation in these 35 individuals was 13.229 μg Pb/100 ml blood.
The mean calculated from the population of all participating individuals was 34.603 μg Pb/100
ml blood. My claim is that the population mean based off of the sample mean will equal
μ=34.603 μg Pb/100 ml blood. Therefore:
Ho: μ=34.603
H1: μ≠34.603
I used my TI-83 calculator to compute both the T and p values, although for posterity’s sake,
here is the calculation for the test statistic T: T= (xbar – μx)/(s/√n) = (34.6-34.603)/(13.229/√35)
= -0.0013.
T= -0.0013 and p=.9989. We therefore fail to reject Ho. Since p is much greater that a, we can
conclude that there is not sufficient evidence to warrant rejection of the claim that the
population mean: μ=34.603.
a. Although hypothesis testing can often be accurate and helpful, we have read and been
taught that they are not always 100% accurate. We see this phenomenon in the
proportion hypothesis test in question 2. In this case we actually knew the population
proportion, but the simple random sample which we took simply did not accurately
represent the population. In fact, it was wildly off by about 20%. This was a fantastic
example of when-hypothesis-testing-goes-wrong.
b. On the other hand, the hypothesis test for the population mean went much better
according to plan and design. Our random sample almost exactly matched that of the
population mean, making the calculations very extreme (low T value, and very high p
value.) I felt that this example, in the shadow of the last, was a great example of how
hypothesis testing can be very reassuring when determining if the population parameter
can be estimated using a sample statistic. All in all, I learned a great deal in this
assignment, and it really helped me piece together module 8 in one flowing idea.
Term Project Part 6: Reflection
I am very glad that I have the opportunity at the end of this project to express my
feelings and thoughts concerning the quality and helpfulness of this assignment. I think I
learned many things from this project, especially how to work with other people. Oddly
enough I think, besides the content of the assignment itself, the most important thing I
learned is that working with peers can be difficult. Some people make promises to
accomplish something, others promise nothing. Some can meet your standards of quality
and hard work, and others can’t. I found a delicate balance between delegation and selfmotivation that held our group together and allowed us to finish assignments on time.
This was a great learning experience for me that will help me, not only in my academic
career, but my future career as a surgeon as well.
Unfortunately, I made the decision of taking Genetics last semester, before I had a
firm grasp on Statistic concepts which I learned this summer. Since genetics mostly
revolves around statistics and chances of events happening, it would have been very
helpful to understand the actual meaning of critical values, p-values, regression
coefficient values, and how they all can be analyzed together to create a larger picture.
Lastly, I believe this project greatly changed the way in which I view statistics,
and its importance in real world situations. I used to think that statistics simply meant
“what is the likelihood that _____ will happen?” whereas now I see deeper applications.
For instance, my group did IQ vs. blood lead values. Not only could you test to see if
there is a correlation between them, but also calculate theoretical values, and potentially
change health and safety codes to harbor a more intelligent populace.
All in all I found this project very helpful to learn and apply the statistics concepts
which we strove to learn all summer. Thank you for your organization and dedication,
and sorry for going over one page; I just had so much to say!