Malaika Wauters Math 11 - Lab 7
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University of California, San Diego *
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Course
11
Subject
Mathematics
Date
May 30, 2024
Type
Pages
7
Uploaded by MegaRose1330
Malaika Wauters
Math 11
Denise Rava
Lab 7: SAT Scores and Cloud Seeding
1.
Examine the distributions of the Math and Verbal SAT scores. Do they appear to be
approximately normally distributed? Are there any outliers? What about the distribution
of the GPAs?
a.
The distributions of the Math and
verbal SAT scores appear to be normally
distributed, as seen in the bell shaped
curve on the histograms to the left,
however, there are outliers in both sets of
data on the lower end. The distribution of
the GPAs is also bell shaped and
approximately normal but there are no
evident outliers that can be assumed from
the histogram.
2.
How does the performance of males compare to the performance of females on the Math
SAT test? What about the Verbal SAT test? How do the first-year GPAs of male and
female students compare?
a. The data collected for the Math test section score show that Males have a slightly
higher median in comparison to females but also a wider spread of data. While both
males and females have two outliers, both male outlier scores are on the lower end while
1 outlier score achieved by females is higher than average.
b. In regards to the Verbal test scores, both male and females have approximately the
same median score but the spread of data for females is greater between the median and
upper quartile while the spread is wider for males below the median. The data only shows
1 outlier for females and 3 for males.
c. The boxplot for the GPAs of males and
females shows that the median GPA
among males is lower than that of
females, although females have a greater
interquartile range. The also are no
outlier GPA scores amongst females
while there are 2 outliers for males on the
higher end.
3.
Based on what you observed from examining the data graphically, does it appear that the
assumptions required to conduct t-tests are met?
a.
Yes, independence is assumed and also the histograms in question1 demonstrate a
normal distribution bell shaped curve, therefore the conditions to perform t-tests
should be adequate.
4.
Do macroeconomics students at Vanderbilt score significantly higher on the Math SAT
than the national average (which in the mid-1980s was around 470)?
a.
One sample t-test & one sided
b.
Null hypothesis H
0
: μ = 470
c.
Alternative hypothesis H
A
: μ >470
d.
Degrees of freedom = n-1 = 391-1 = 390
e.
T statistic = 50.80
f.
P-value: P(t > 50.80) < 0.005
g.
Given that the p-value is less than 0.005, which is less than 0.05 (the test
significance level), we can reject the null hypothesis and there is evidence that the
mean Math SAT score at Vanderbilt was higher than the national average.
5.
Do macroeconomics students at Vanderbilt score significantly higher on the Verbal SAT
than the national average (which in the mid-1980s was around 425)?
a.
One sample t-test & one sided
b.
Null hypothesis H
0
: μ = 425
c.
Alternative hypothesis H
A
: μ >425
d.
Degrees of freedom = n-1 = 391-1 = 390
e.
T statistic = 38.45
f.
P-value: P(t > 38.45) < 0.005
g.
Given that the p-value is less than 0.005, which is less than 0.05 (the test
significance level), we can reject the null hypothesis and claim that there is
sufficient evidence that the mean Verbal SAT score at Vanderbilt was higher than
the national average of 425.
6.
Report a 95 percent confidence interval for the true mean Math SAT score. Do 95 percent
of students have Math SAT scores that fall within this interval? Explain your answer.
a.
One sample t-test & two sided
b.
Null hypothesis H
0
: μ = 624.9
c.
Alternative hypothesis H
A
: μ /= 624.9
d.
95% CI = (619.5,630.2)
e.
The 95% confidence interval means that we can claim that the true mean falls in
this interval with 95% confidence, but it does not mean that 95% of the students
scored in this range for the Math section of the SAT.
7.
Is there a statistically significant difference between the Verbal and Math scores of
microeconomics students at Vanderbilt?
a.
Paired t-test & 2 sided alternative
b.
Null hypothesis H
0
: Math = Verbal
c.
Alternative hypothesis H
A
: Math /= Verbal
d.
T statistic = -20.4
e.
P <0.0001, given that the p value is less than 0.05, we reject the null hypothesis.
8.
Is there a statistically significant difference between the performances of males and
females on the Verbal SAT? Construct a 95 percent confidence interval for the difference.
Does it include zero? Relate this to the conclusion of your test.
a.
2 sample t-test
b.
T statistic = 0.60
c.
Degrees of freedom = n-1 = 124-1 = 123
d.
P-value: P(|t| >0.6) = 2 x P(t >0.6) → 2 x P(t >1.289) = 0.20
e.
Given that the p value is actually greater than 0.20, we cannot reject the null
hypothesis and conclude that there is no significant statistical difference between
male and female verbal SAT scores.
f.
(-9.38, 17.58)
g.
The 95% confidence interval includes 0, since we failed to reject the null
hypothesis and claimed that there is no difference in verbal SAT performance
between males and females, the confidence interval containing 0 confirms that.
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9.
Is there a statistically significant difference between the freshman GPAs of males and
females? Construct a 95 percent confidence interval for the difference. Does it include
zero? Relate this to the conclusion of your test.
a.
2 sample t-test
b.
T statistic = 2.17
c.
Degrees of freedom = n-1= 124-1 = 123
d.
P-value: P(|t| >2.17) = 2 x P(t >2.17)
e.
2 x P(t>2.358) < 2 x P(t>2.17) < 2 x P(t>1.98) = 0.02 < p < 0.05
f.
Given that the p-value is less than 0.05, we can reject the null hypothesis and
claim that there is sufficient evidence that there is a statistically significant
difference between freshmen male and females
g.
(0.01,0.206)
h.
The confidence interval is lacking 0, which is accurate to the previous claim that
there is a significant difference between male and female freshman GPAs.
10. Is there a statistically significant difference between the performances of males and
females on the Math SAT? Construct a 95 percent confidence interval for the difference.
Does it include zero? Relate this to the conclusion of your test.
a.
2 sample t-test
b.
T statistic = -2.57
c.
Degrees of freedom = n-1 = 124-1 = 123
d.
P-value: P(|t| >-2.57) = 2 x P(t >2.57)
e.
2 x P(t >2.617) < 2 x P(t >2.57) < 2 x P(t >2.358) = 0.01 < p < 0.02
f.
Given that the p-value is less than 0.05, we can reject the null hypothesis and
claim that there is sufficient evidence that there is a statistically significant
difference between female and male performance on the Math section of the SAT.
g.
(-27.62, -3.58)
h.
We should not have 0 as a possible value in the 95% confidence interval because
we reject the null hypothesis with the low p value, and this confidence interval
does not include the 0 which confirms the difference in performance between
females and males on the Math SAT.
11.
First examine and compare the two distributions with histograms and a side-by-side
boxplot. Discuss what you observe in a few sentences.
a.
From the skewed shape of the bars on the histograms above, the distribution is
likely exponential. Outliers are present in both sets of data but the seeded clouds
data exemplifies more extreme outliers
b.
The box plots also show that the
seeded clouds have a much wider
range of data and more extreme
outliers on the upper end of the
data, it also shows that the seeded
clouds have a higher median
rainfall.
12.
Conduct a hypothesis test to test the hypothesis that cloud seeding with silver nitrate
increases rainfall. Use significance level .01. Based on what you observed in answering
the previous question, are you confident that this is a valid test? If not, what assumptions
are violated?
a.
Two sample t-test
b.
T statistic = -2
c.
Degrees of freedom = 26-1 = 25
d.
P-value: P(t>2)
e.
P(t>2.06) < P(t>2) < P(t>1.708) = 0.025 < p < 0.05
f.
Given that the p value is greater than 0.01, we fail to reject the null hypothesis, we
do not have statistically significant evidence that cloud seeding increases rainfall.
13.
Try taking natural logarithms of the rainfall totals. Are the new variables less skewed?
Are the distributions closer to normal? What does a side-by-side boxplot of the variables
show?
a. The distribution shape depicted in the
histograms above resemble a more bell
shape which translates to a normal
distribution and are not really skewed. (C4
being unseeded and C5 being seeded clouds)
The box plot shows a higher median rainfall
value in seeded clouds with a more similar
interquartile range. Both plots have outliers
but the outlier for unseeded is more extreme.
I’m more confident in the normal
distribution in comparison to exponential.
14.
Conduct another formal hypothesis test using the transformed variables, again at
significance level .01. Are you more confident in the results of this test or the results of
the previous test?
a.
T statistic = -2.54
b.
Degrees of freedom = 26-1 = 25
c.
P-value: P(t>2.54)
d.
P(t>2.787) < P(t>2.54) < P(t>1.485) = 0.005 < p < 0.01
e.
Given that the p value is very low and under 0.01, we can reject the null
hypothesis and there is statistically significant evidence to prove that cloud
seeding does increase rainfall
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15.
Write a couple of sentences explaining your overall conclusions about the effect of cloud
seeding with silver nitrate on rainfall.
a.
With the completion of the t-test, and yielding a very low p value, we can reject
the null hypothesis that suggests there is no difference between the rainfall that is
produced by the two types of clouds. Therefore, we can claim that there is
statistically significant evidence for the difference between rainfall in seeded and
unseeded clouds, more specifically the evidence suggests that there is a higher
amount of rainfall in silver nitrate seeded clouds in comparison to unseeded
clouds.