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Math 2311
Final review
1) The probability that a randomly selected person has high blood pressure (the
event H) is P(H) = 0.4 and the probability that a randomly selected person is a
runner (the event R) is P(R) = 0.4. The probability that a randomly selected person
has high blood pressure and is a runner is 0.1. Find the probability that a
randomly selected person
a)Either has high blood pressure or is a runner or both.
b)Has high blood pressure given he is a runner
2) Given the following sampling distribution:
X
-1
P(X)
1
What is mean and the variance?
4
2
1
5
⁄25 3⁄50 1⁄20 1⁄100 ___
3) Suppose you have a distribution, X, with mean = 10 and standard deviation = 3.
Define a new random variable Y = 5X - 5. Find the mean and standard deviation of
Y.
Ans: E[Y] = 45; σY = 15
4) In testing a new drug, researchers found that 5% of all patients using it will
have a mild side effect. A random sample of 7 patients using the drug is selected.
Find the probability that exactly two will have this mild side effect. What is the
mean and variance of this distribution
Ans: 0.04062
5) A quarter back completes 23% of his passes. We want to observe this
quarterback during one game to see how many pass attempts he makes before
completing one pass. What is the probability that the quarterback throws 6
incomplete passes before he has a completion?
Ans: 0.0479
6) The length of time needed to complete a certain test is normally distributed
with mean 52 minutes and standard deviation 11 minutes. Find the probability
that it will take less than 38 minutes to complete the test.
Ans:0.1016
7) Find a value of c so that P(Z ≤ c) = 0.64.
Ans: 0.36
8) A random sample of 625 12-ounce cans of fruit nectar is drawn from among all
cans produced in a run. Prior experience has shown that the distribution of the
contents has a mean of 12 ounces and a standard deviation of 0.12 ounce. What
is the probability that the mean contents of the 625 sample cans is less than
11.994 ounces?
Ans: 0.48
9) In a large population, 91% of the households have cable tv. A simple random
sample of 196 households is to be contacted and the sample proportion
computed. What is the probability that the sampling distribution of sample
proportions is less than 87%?
Ans: 0.0252
10)Find LSRL for the given data and determine the correlation coefficient for the
data shown in this table: Find the residual value for x = 2.
x
y
2
22
Ans: y= 20.479+1.0625x
Res = -0.604
Correlation coefficient=0.8946
8
29
8
28
13
40
16
33
19
41
11) Lloyd's Cereal company packages cereal in 1 pound boxes (16 ounces). A
sample of 25 boxes is selected at random from the production line every hour,
and if the average weight is less than 15 ounces, the machine is adjusted to
increase the amount of cereal dispensed. If the mean for 1 hour is 1 pound and
the standard deviation is 0.1 pound, what is the probability that the amount
dispensed per box will have to be increased?
Ans: 0.0008
12)In a large population, 67% of the households have cable tv. A simple random
sample of 256 households is to be contacted and the sample proportion
computed. What is the mean and standard deviation of the sampling distribution
of the sample proportions?
Ans: [0.67, 0.0294]
13) The following two-way table describes the preferences in movies and fast
food restaurants for a random sample of 100 people.
McDonald's
Iron Man
20
Despicable Me 10
Harry Potter
9
Taco Bell
13
11
11
Wendy's
7
7
12
What percent of the Iron Man lovers also like Taco Bell?
Ans: 33%
14) A simple random sample of 100 8th graders at a large suburban middle school
indicated that 82% of them are involved with some type of after school activity.
Find the 95% confidence interval that estimates the proportion of them that are
involved in an after school activity.
Ans: [0.745, 0.895]
15) It has been observed that some persons who suffer renal failure, again suffer
renal failure within one year of the first episode. This is due, in part, to damage
from the first episode. The performance of a new drug designed to prevent a
second episode is to be tested for its effectiveness in preventing a second
episode. In order to do this two groups of people suffering a first episode are
selected. There are 103 people in the first group and this group will be
administered the new drug. There are 123 people in the second group and this
group will be administered a placebo. After one year, 15% of the first group has a
second episode and 17% of the second group has a second episode. Select a 95%
confidence interval for the difference in true proportion of the two groups.
Ans: [-0.116, 0.076]
16) The gas mileage for a certain model of car is known to have a standard
deviation of 4 mi/gallon. A simple random sample of 49 cars of this model is
chosen and found to have a mean gas mileage of 27.5 mi/gallon. Construct a
96.5% confidence interval for the mean gas mileage for this car model.
Ans: [26.295, 28.705]
17) An oil company is interested in estimating the true proportion of female truck
drivers based in five southern states. A statistician hired by the oil company must
determine the sample size needed in order to make the estimate accurate to
within 2% of the true proportion with 99% confidence. What is the minimum
number of truck drivers that the statistician should sample in these southern
states in order to achieve the desired accuracy?
Ans: 4147(4146.71)
18) Mars Inc. claims that they produce M&Ms with the following distributions:
Brown
20%
Red
25%
Yellow
25%
Orange
5%
Green
15%
Blue
10%
A bag of M&Ms was randomly selected from the grocery store shelf, and the color
counts were:
Brown
Orange
23
14
Red
Green
21
15
Yellow
Blue
19
13
Using the χ2 goodness of fit test (α = 0.05) to determine if the proportion of M&Ms
is what is claimed. Select the [test statistic, p-value, Decision to Reject (RH0) or
Failure to Reject (FRH0)].
Ans: [χ2 = 18.457, p-value = 0.002, RH0]
19)
a) Identify the most appropriate test to use for the following situation:
A national computer retailer believes that the average sales are greater for
salespersons with a college degree. A random sample of 14 salespersons with a
degree had an average weekly sale of $3542 last year, while 17 salespersons
without a college degree averaged $3301 in weekly sales. The standard deviations
were $468 and $642 respectively. Is there evidence to support the retailer's
belief?
Ans: Two sample t test
b) Identify the most appropriate test to use for the following situation:
Quart cartons of milk should contain at least 32 ounces. A sample of 22 cartons
was taken and amount of milk in ounces was recorded. We would like to
determine if there is sufficient evidence exist to conclude the mean amount of
milk in cartons is less than 32 ounces.
Ans: One sample t test
c) Identify the most appropriate test to use for the following situation:
In an experiment on relaxation techniques, subject's brain signals were measured
before and after the relaxation exercises. We wish to determine if the relaxation
exercise slowed the brain waves.
Ans: Matched pairs
20) It has been observed that some persons who suffer renal failure, again suffer
renal failure within one year of the first episode. This is due, in part, to damage
from the first episode. The performance of a new drug designed to prevent a
second episode is to be tested for its effectiveness in preventing a second
episode. In order to do this two groups of people suffering a first episode are
selected. There are 31 people in the first group and this group will be
administered the new drug. There are 45 people in the second group and this
group will be administered a placebo. After one year, 11% of the first group has a
second episode and 9% of the second group has a second episode. Conduct a
hypothesis test to determine, at the significance level 0.1, whether there is reason
to believe that the true percentage of those in the first group who suffer a second
episode is more than the true percentage of those in the second group who suffer
a second episode? Select the [Alternative Hypothesis, Value of the Test Statistic].
Ans: [p1 > p2 , 0.2880