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MAT 201 Midterm (Online Class)
Please work out all of the given problems. Credit with be based on the work
that is shown that contributes towards the final answer. If you are using the
calculator to solve a problem, be sure to write down what you are putting in
your calculator. Also show all normal and Student T diagrams.
Problem 1 (20 Points) (For each part be sure to write down the sampling distribution for each,
put in your normal curve diagram, answer the question and determine whether the normality
assumption was needed.) Suppose that the amount of time that first time moms spend in labor is
normally distributed with mean 10.3 hours and standard deviation 2.4.
A. If fifty first time moms are surveyed, what is the probability that the total of all of
their times in labor is less than 500 hours?
X ~ N(10.3x50,2.4xroot(50)) = N(515,16.97)
Picture of a normal curve with mean 515 and shading from the farthest left to 500.
normalcdf(-999999,500,515,16.97) = 0.188
The normality assumption was not needed since the sample size was 50 > 30.
B. Find the first quartile for the time in labor for first time moms.
X ~ N(10.3,2.4)
Picture of a normal curve with mean at 10.4, shading in the left 25% of the area
with this area labeled as 0.25 and Q1 marked on the X axis.
invNorm(0.25,10.3,2.4) = 8.68
The normality assumption was needed since only a single individual is being
looked at.
C. If 6 first time moms are randomly selected, find the probability that their average
time in labor is between 10 and 11 hours.
XBar ~ N(10.3,2.4/root6) = N(10.3,0.9798)
Picture of a normal curve with mean at 10.3 and the part between 10 and 11
shaded.
normacdf(10,11,10.3,0.9798) = 0.383
The Normality assumption was needed since the sample size was 6 < 31.
Problem 2 (20 Points)
A study was done to see if giving alcoholics a very painful electric shock for a week each
time they touch any glass or bottle of alcohol will help cure them of their alcoholism.
Suppose that with no such treatment 10% of alcoholics are cured of their alcoholism in a
year.
A. Discuss the implications of a Type I Error.
You end up deciding that the shock treatment is a cure. You go ahead and zap
countless alcoholics and they remain alcoholics at the same rate at had they not
been shocked. Countless people experience the very painful electric shock for no
reason.
B. Discuss the implications of a Type II Error.
You are not able to determine that the shock therapy is effective. Many alcoholics
could have been cured, but they don’t receive the treatment. People die as a result
of DUI accidents.
C. Suppose that 300 alcoholics were given the electric shock treatment and 35 of
them were cured a year later. Test the hypothesis and state your conclusion in the
context of the study. Be sure to state the null and alternative hypothesis, the pvalue, and the test you are using in your calculator. Then state your conclusion
using a complete sentence in the context of the study.
Ho: p = 0.10
H1: p > 0.10
Since np = 35 > 5 and nq = 265 > 5 we can use the 1-PropZTest. This gives a pvalue of 0.17. There is insignificant evidence to make a conclusion about whether
more than 10% of alcoholics who are treated with the shock therapy will be cured.
Problem 3 (3 Points Each) Please circle the following True or False.
A. The expected value for the number of babies South Korean women have is 1.2.
Therefore more South Korean women have one baby than 2 babies.
.
True
False
B. Three hundred randomly selected college students will be asked if they have more
than one year left of college to complete for their intended degree. Then this
procedure results in a binomial distribution. True
False
C. If x is a random variable that follows a normal distribution with mean 20, then
P(15 < x < 17) = P(23 < x < 25).
True
False
D. If the distribution of exam scores for all Statistics final exams throughout
California has mean 76 and standard deviation 9, then the distribution of all
possible means of random samples 100 students’ exam scores also has mean 76
and standard deviation 9.
True
False
E. Since 30% of all California Community College (CCC) students are Hispanic, if
you want to survey 500 CCC students stratified by ethnicity, then you must survey
exactly 150 Hispanics.
True
False
F. A researcher surveyed 45 randomly selected Americans to construct a confidence
interval for the proportion of all Americans who have been to Yellowstone. Since
the sample size is larger than 30, the Central Limit Theorem can be used to
conclude that the sampling distribution is normal.
True
False
G. If A and B are events such that P(A) = 0.3 and P(B) = 0.8, then
P(A and B) = 0.24.
True
False
H. The number of times that a person has been in a car accident is a discrete variable
and the amount of time that a person spends in a car each day is a continuous
variable.
True
False
I. A manager will be giving a presentation on the amount of money that customers
spend in the store. The audience knows statistics well. A box plot should be used
to best display the middle 50% of the amount of money that customers spend and
a histogram should be used to best display the fact that the amount of money
customers spend follows a bimodal distribution.
True
False
Problem 4 (10 Points) Suppose that the weight in pounds of ripe red delicious apples
on an apple tree follows a uniform distribution on [0.2,0.5].
A. What is the probability that a randomly selected red delicious apple weighs
between 0.33 and 0.41 pounds? Be sure to include your diagram.
Sketch of a rectangle between 0.2 and 0.5. The height of the rectangle is h.
(0.5 – 0.2)h = 1, 0.3h = 1 h = 1/0.3 = 3.3333
Then shade in the inner rectangle between 0.33 and 0.44. The probability is
(0.41 – 0.33)(3.3333) = 0.267
B. If 20 red delicious apples are randomly selected what is the probability that at
least 14 of them will weigh less than 0.4 pounds?
The same initial rectangle is drawn and the region between 0.2 and 0.4 is shaded.
That area is (0.4 – 0.2)(3.3333) = 0.667.
Next use binomcdf(20, 0.667, 14) = 0.702
Problem 5 (10 Points) Suppose that the mean height of professional male basketball
players is 79 inches and the standard deviation of 3 inches and that the average height of
a professional female soccer player is 71 inches with a standard deviation of 2 inches.
You observe a 72-inch-tall professional male basketball player walking with a 68-inchtall professional female soccer player. Use z-scores to determine who is more unusually
short compared to the person’s possible teammates.
For the basketball player, z = (72 – 79)/3 = -2.33
For the soccer player, z = (68 – 71)/2 = -1.5
The basketball player is more unusually short since the z-score is farther from 0.
Problem 6 (20 Points) What is the average wait time in emergency rooms? Ten people
who had to wait in an emergency room were observed.
Time in 34
1
46
27
41
38
27
32
30
16
minutes
A. Determine the appropriate 95% confidence interval.
Use TInterval with Data to get [19.935,38.465]
B. Write a sentence that explains your findings.
We are 95% confident that the population mean wait time in the emergency room
is between 20 and 38 minutes.
C. Use a sentence or two to explain what it means in the context of this study to be
95% confident.
If many samples of 10 emergency wait times are taken then each sample would
produce its own confidence interval. 95% of these confidence intervals will
contain the population mean wait time in the emergency room.
D. Was it necessary to make any assumptions about the underlying distribution of the
population? Explain.
Yes, since the sample size is 10 < 31, we must assume that the population of wait
times is normally distributed.
Problem 7 (20 Points) A researcher wanted to estimate the proportion of people who
purchase at least one item online each month.
A. If the researcher wants to construct a 86% confidence interval with a margin of
error of plus or minus 3%, at least how many people must be surveyed?
Use n = 0.25(z/E)^2. Find z by sketching a normal curve with 0.86 in the middle,
so 0.14 on the outside or 0.07 on the left. Then use z = invNorm(0.07) = -1.4758.
E = 0.03, so we get 0.25(-1.4758/0.03)^2 = 604.996. Survey 605 people.
B. Suppose that the researcher instead surveyed 400 people and 82 of them purchase
at least one item online each month. Determine the appropriate 95% confidence
interval.
Use 1-PropZInt to get (0.16544,0.24456)
C. Write a sentence that explains your findings from part B.
We are 95% confident that between 17% and 24% of all people purchase at least
one item online each month.
Problem 8 (20 Points) The average California resident disposes of 4.7 pounds of garbage
per day. Your area has a new recycle program and wants to see if the daily disposal is
less in this area. The 55 area residents that were surveyed disposed an average of 3.9
pounds per day and the standard deviation for these residents was 2.6 pounds.
A.
Write down the appropriate null and alternative hypotheses and find the test
statistic and the P-Value for this study.
Ho:  = 4.7, H1:  < 4.7
Test statistic: t = -2.28, P-Value = 0.013
B.
Use a complete sentence to state the results in the context of the survey using
a level of significance of 0.05.
There is statistically significant evidence to conclude that the population mean
amount of garbage residents in the area where the new recycle program has
occurred is less than 4.7 pounds per resident.
C.
The P-Value represents a probability. Write a few sentences that interpret this
probability in the context of the study.
If the population mean amount of garbage residents in the area where the new
recycle program has occurred is equal to 4.7 and a new sample of 55 area
residents were surveyed then there is a 1.3% chance that the sample mean
garbage amount for these 55 residents would be less than 3.9 pounds per day.
D.
The level of significance represents a probability. Write a few sentences that
interpret this probability in the context of the study.
If the population mean amount of garbage residents in the area where the new
recycle program has occurred is equal to 4.7 and a new sample of 55 area
residents were surveyed then there is a 5% chance that the new study would
result in the false conclusion that the population mean amount of garbage area
residents dispose of per day is less than 4.7.
Problem 9 (20 Points) The relative frequency histogram below shows the age
distribution of customers at a local ski resort.
A. What is the probability that a customer selected at random will be younger than
30?
P(x < 30) = 0.4 + 0.3 + 0.1 = 0.8
B. What is the probability that a customer selected at random will be under 21 given
that that customer is over 18 years old?
P(x<21|x>18) = P(x < 21 and x > 18) / P(x > 18) = 0.3/(0.3 + 0.4 + 0.2) =
0.3333
C. If a one year old baby skier was added to the population of customers, would the
standard deviation go up, down, or stay the same? Explain.
The one year old would be an extreme outlier which is far from the mean. Thus
the standard deviation would go up.
Problem 9 (Extra Credit) Credit will be given for any honest answers.
A. Write down one thing that your instructor can do or could have done for you to
help you better succeed in this online course.
B. Write down one thing that your instructor did for you that helped you succeed in
this online course.
C. Write down one thing that you can do for yourself to help you better succeed in
this online course.
D. Write down one thing that you have done for yourself to help you better succeed
in this online course.