Download sat verbal scores

SAT VERBAL SCORES
Math 116 – Reviewing for the Final Exam
This is all about Means
Chapter 6 – Normal Distributions
1) According to the College Board, the mean SAT verbal score of students whose first language is
English is 515. Assume SAT verbal scores are normally distributed with a population standard deviation
of 112.
a) Describe in words the population and the variable under consideration.
Students whose first language is English
b) Identify usual and unusual scores according to the range rule of thumb.
SAT verbal score
c) What is the probability that a student selected at random has a mean SAT verbal score of
(i)
At most 458? Is this an unusually low score?
P( x  458)  P( z 
458  515
)  P( z  .51)  0.3050
112
It’s not unusually low; the probability is more than 0.05. We may observe such a result or lower in
about 305 samples out of 1000
With calculator: normalcdf(-10^9, 458, 515, 112) = 0.3050
(ii)
At least 745? Is this an unusually high score?
P( x  745)  P( z 
745  515
)  P( z  2.05)  1  0.9798  .0202
112
It’s an unusually high score; the probability is less than 0.05. We may observe such a result or
higher in about 202 samples out of 10000
With calculator: normalcdf(745, 10^9, 515, 112) = 0.0200
(iii)
Between 500 and 600?
P(500  x  600)  P(
500  515
600  515
z
)  P(.13  z  .76)  0.7764  0.4483  .3281
112
112
With calculator: normalcdf(500, 600, 515, 112) = 0.3293
d) Lisa scored on the 60th percentile. What was her score?
The area to the left of the 60th percentile score is 0.600. Use the table, from inside out to find the z
score. The z-score is 0.25. Then, use the formula: x    z  515  0.25*112  543
With calculator: invNorm(0.6, 515, 112) = 543.37
e) A college in the Northeast accepts applicants who scored in the top 5%. What is the lowest SAT
verbal score accepted?
This score separates the top 5% from the bottom 95%. The area to the left of the score is 0.9500.
Use the table, from inside out to find the z score. The z-score is 1.645. Then, use the formula:
x    z  515  1.645*112  699.24
With calculator: invNorm(0.95, 515, 112) = 699.22
f) The cut-off for the unusually low scores is__330.78____. (This is the score that separates the bottom
5% of the scores)
With calculator: invNorm(0.05, 515, 112) = 330.78
(You do it showing all steps as done in part (e). The z-score will be -1.645)
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SAT VERBAL SCORES
Chapter 7 – Section 7.2 – Distribution of Sample Means
2) The mean SAT verbal score of students whose first language is English is 515, on the basis of data
obtained from the College Board. SAT verbal scores are normally distributed with a population standard
deviation of 112.
a) Give the shape, mean and standard deviation of the distribution of sample means for samples of size
20.
X is Normal with   515 and   112
According to the Central Limit Theorem, the distribution of sample means, for samples of size 20,
is also normal with
 x    515
x 

n

112
 25.04
20
b) What is the probability that a sample of size 20 has a mean of at most 458?
P( x  458)  P( z 
458  515
)  P( z  2.28)  0.0113
112
20
With calculator: normalcdf(-109 ,458,515,
112
) = 0.0114~0.01
20
c) The probability of observing a sample of size 20 with a mean of 458 or less when the population
mean is 515 is 0.01_______.
So, for samples of size 20, about __1____ sample(s) in 100 will result in a sample mean of 458 or less
when the population mean is 515.
Because this event only happens __1___ out of __100___ times, we consider it to be usual/unusual.
d) What may this result suggest?
If the population mean is 515, it’s very unusual to observe a sample of size 20 with a mean of 458
or less. This unusually low result may suggest that “probably” the sample was selected from a
population with mean lower than 515.
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SAT VERBAL SCORES
Chapter 8 – Confidence Intervals about a Population Mean μ
3) We want to estimate the SAT verbal score of all students whose first language is not English.
Suppose a simple random sample of 20 students from that group have a mean SAT verbal score of 458.
Assume SAT verbal scores are normally distributed with a population standard deviation of 112.
a) What is the point estimate?
b) Verify that the requirements for constructing a confidence interval about x-bar are satisfied.
c) Construct a 90% confidence interval estimate for the SAT verbal score of all students whose first
language is not English. (Are you using z or t? Why?)
d) The statement “90% confident” means that, if 100 samples of size _____ were taken, about
_____ intervals will contain the parameter μ and about ____ will not.
e) We are _____% confident that the mean SAT verbal score of all students whose first language is
not English is between _____ and ______
f) With 90% confidence we can say that the mean SAT verbal score of all students whose first
language is not English is ______ with a margin of error of _______
g) For 90% of such intervals, the sample mean would not differ from the actual population mean by
more than ________.
h) How can you produce a more precise confidence interval?
i) According to the College Board, the mean SAT verbal score of students whose first language is
English is 515. Does the interval constructed in part (c) suggest that for students whose first
language is not English, the mean score is lower? Explain.
j) How large of a sample should be selected in order to be 90% confident that the point estimate xbar will be within 30 units of the true population mean?
k) Circle the correct choice:
 Increasing the confidence level produces a longer/shorter
 Increasing the confidence level
increases/decreases
 Increasing the sample size
increases/decreases
interval.
the precision.
the precision.
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SAT VERBAL SCORES
Chapter 9 – Testing a Mean μ
4) A school administrator claims that students whose first language is not English score worse on the
verbal portion of the SAT exam than students whose first language is English. The mean SAT verbal
score of students whose first language is English is 515, on the basis of data obtained from the College
Board. Suppose a simple random sample of 20 students whose first language was not English results in a
sample mean SAT verbal score of 458. SAT verbal scores are normally distributed with a population
standard deviation of 112.
a) Why is it necessary for SAT verbal scores to be normally distributed to test the claim using the
methods from chapter 9?
b) Test the researcher’s claim at the 0.05 level of significance. (Are you using z or t? Why?)
 Set both hypothesis

Sketch graph, shade rejection region, label, and indicate possible locations of the point
estimate in the graph.
****You should be wondering: Is x-bar = 458, lower than 515 by chance, or is
it significantly lower? The p-value found below will help you in answering this.

Use a feature of the calculator to test the hypothesis. Indicate the feature used and the
results:
Test statistic =
p-value =
***How likely is it observing an x-bar = 458 or less when you select a sample
of size 20 from a population that has a mean µ of 515?
very likely,
likely,
unlikely,
very unlikely
*** Is x-bar = 458, lower than 515 by chance, or is it significantly lower?

What is the initial conclusion with respect to Ho and H1?

Write the conclusion using words from the problem
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SAT VERBAL SCORES
Chapters 8 and 9 – Hypothesis Testing and Confidence Intervals for 1  2
(Independent Samples)
5) Do students whose first language is not English score lower on the verbal portion of the SAT than
students whose first language is English? A researcher selects a simple random sample from each group
and obtains the following results.
First Language not
English
First language English
Sample size
20
Mean
458
Standard deviation
120
100
515
112
Assume the SAT verbal scores are normally distributed.
a) Test the claim at the 5% level of significance. (Are you using z or t? Why?)
 Set both hypothesis

Sketch graph, shade rejection region, label, and indicate possible locations of the point
estimate in the graph.
****You should be wondering: Is the difference between the x-bars lower than
zero by chance, or is it significantly lower? The p-value found below will help
you in answering this.

Use a feature of the calculator to test the hypothesis. Indicate the feature used and the
results:
Test statistic =
p-value =
***How likely is it observing such a difference between the x-bars (or a more
extreme one) when the means of the two populations are equal?
very likely,
likely,
unlikely,
very unlikely
*** Is the difference between the x-bars lower than zero by chance, or is it
significantly lower?

What is the initial conclusion with respect to Ho and H1?

Write the conclusion using words from the problem
b) Construct a 90% confidence interval estimate for the mean difference between the SAT verbal
scores for the two groups. What does the interval suggest? (Are you using z or t? Why?)
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SAT VERBAL SCORES
Chapter 9 – Testing 1  2 (Dependent Samples)
6) In a certain school, during the spring semester, 20 students whose first language was not English took
the SAT. During the summer they attended a three-month SAT preparation course, and in the Fall they
took the SAT again. The before and after verbal SAT scores were compared. The mean of the
differences was –76.9 and the standard deviation 153.7.
a) At the 0.05 significance level can we conclude that the intensive course helped the students improve
their score?
 Set both hypothesis

Sketch graph, shade rejection region, label, and indicate possible locations of the point
estimate in the graph.
****You should be wondering: Is the sample mean difference d-bar = -76.9
lower than zero by chance, or is it significantly lower? The p-value found below
will help you in answering this.

Use a feature of the calculator to test the hypothesis. Indicate the feature used and the
results:
Test statistic =
p-value =
***How likely is it observing such a value of d-bar (or a more extreme one)
when the population mean difference is zero?
very likely,
likely,
unlikely,
very unlikely
*** Is the mean difference d-bar = -76.9 lower than zero by chance, or is it
significantly lower?

What is the initial conclusion with respect to Ho and H1?

Write the conclusion using words from the problem
b) Construct a 90% confidence interval about the population mean difference. What does the interval
suggest?
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