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Sampling Distributions
Standard z
x
z=

REVIEW
Sample Means
x = 
x =

n
Name____________________________
Class period_______
Sample Proportions
 pˆ  p
 pˆ 
p(1  p)
n
Note: Sampling distributions are important. As evidence of their importance, statisticians have special names
for the mean and for the standard deviation of sampling distributions. The mean is called the expected value
and the standard deviation is called the standard error. The term error means deviations or random
variation. The word error is left over from the 19th century, when random variation was referred to as the
“normal law of error.” So the standard error represents the natural variation that occurs when a statistic is
taken from a sample rather than the entire population. Of course, error sometimes means mistake, so you
will have to be alert to the context when this word appears (or you may make an error).
Read and know the Central Limit Theorem. Read all of Chapter 18.
1. The ages of all college students have a mean of 26 years and a standard deviation of 4 years. Find the
probability that the mean age for a random sample of 36 students would be less than 25 years.
2. The time taken to complete a statistics test by all students is normally distributed with a mean of 75
minutes and a standard deviation of 8 minutes. Find the probability that the mean time taken to complete this
test by a random sample of 19 students is more than an hour and half.
3. For a batch of 500 fuses, the breaking points have a mean of 30.25 amperes and a standard deviation of
1.05 amperes. If a sample of 100 of these fuses is obtained without replacement, find the probability that the
mean for this sample is greater than 30.00 amperes.
4. What is the difference between  and  x ?
5. Reclaimed phosphate land in Polk County, Florida, has been found to emit a higher mean radiation level
than other nonmining land in the county. Suppose that the radiation level for the reclaimed land has a
distribution with a mean of 5.0 working levels (WL) and a standard deviation of .5 WL. Forty houses built
on reclaimed land are randomly selected and the radiation level is measured in each. What is the probability
that the sample mean for the 40 houses will exceed 5.2 WL?
6. Psychomotor retardation scores for a large group of manic-depressive patients were found to have mean
930 and standard deviation 130. A group of 42 randomly selected patients were tested and scored. What is
the probability that the mean score of the 42 patients was between 935 and 960?
7. The proportion of students who fail a statistics test is usually 6.5%. After an epidemic of absences for one
reason and another, a random group of 160 students taking a recent test resulted in a failure rate of 11%.
What is the probability the failure rate would be this high or higher by chance alone?
8. Suppose the proportion of bats that live under the Congress Street bridge that have rabies is 15%. If 70
bats are randomly selected and tested for rabies, what is the probability that the proportion of bats in the
sample that test positive for rabies is more than 25%?
9. A community college finds that the mean grade for students taking a basic statistics course is 2.34 with a
standard deviation of .36. The distribution of grades is approximately normal. If a student is randomly
selected, what is the probability that their statistics grade is 3.5 or above?
10. The number of accidents per week at a hazardous intersection varies with mean 2.2 and standard
deviation 1.4. This distribution is discrete and so is certainly not normal.
a) Let x be the mean number of accidents per week at the intersection during a year (52 weeks). What
is the approximate distribution?
b) What is the approximate probability that x is less than 2?
c) What is the approximate probability that there are fewer than 100 accidents at the intersection in a
year? (Hint: Restate this event in terms of x which is in weeks)
11. Every Monday, David’s DVD rental has Spinner Day. A customer may choose to spin the spinner and
rent a second DVD for an amount (in cents) equal to the numbers on the spinner, with the larger number
first. The spinner is divided into quarters and has the numbers 2, 4, 6, and 8 on it. For instance, if the
customer spins a two and a six, a second DVD may be rented for $0.62. If a four and a four are spun, a
second DVD is $0.44. Let X represent the amount paid for a second DVD on Spinner Day. The expected
value of X is $0.66 and the standard deviation of X is $0.04.
a) If a customer spins the spinner and rents a second DVD movie every Monday for 10 consecutive weeks,
what is the total amount that the customer would expect to pay?
b) If a customer spins the spinner and rents a second DVD every Monday for 30 consecutive weeks, what is
the approximate probability that the total amount paid for these second DVDs will exceed $20.40?
12. a) If you flip a coin 10 times, what is the probability that you will get exactly 8 heads?
b) If one trial consists of flipping a coin 10 times and counting the number of heads, and you complete 100
trials, how many trials would you expect to get exactly 8 heads?
13.(for Runi) In a study on blood pressure and diet, a random sample of Seventh Day Adventists were
interviewed at a national meeting. Because many people who belong to this denomination are vegetarians ,
they are a very useful group for studying the effects of a meatless diet.
vegetarian
not vegetarian
totals
a)
b)
c)
d)
Black
32
57
89
White
38
135
173
Asian
21
35
56
Hispanic
8
74
82
Other
1
2
3
Totals
100
303
403
If one person is randomly selected, what is the probability that they are Hispanic?
If one person is randomly selected, what is the probability that they are vegetarian?
If one person is randomly selected, what is the probability that they are white or vegetarian?
If one person is randomly selected, what is the probability that they are Asian given that they are
vegetarian?
e) If two people are selected at random, what is the probability that both are Black?
f) If two people are selected at random, what is the probability that neither is vegetarian?
g) If one person is selected at random, what is the probability that they are Asian and vegetarian?
********ANSWERS***********
NOTE: CONDITIONS MUST BE CHECKED IN ORDER TO USE THE NORMAL DISTRIBUTION AS
A MODEL. ALL WORK MUST BE SHOWN FOR CREDIT….INCLUDING SKETCH.
1. Conditions: Random sample of college students given. Independent since one student’s age does not affect
another’s age. n = 36 ≥ 30. A normal model of the sampling distribution is acceptable. P( x < 25) = 0.0668
2. Conditions: Random sample of statistics students given. Hopefully, student scores are independent. A
normal model of the distribution of x is acceptable since the population is normal….even though n = 19.
P( x > 90) = 0+ or 0.0001
3. Conditions: Since the fuses are taken without replacement, the condition of independence is
compromised. The 10% rule doesn’t help us out here either. So we cannot carry on.
4.  is the standard deviation of a population
 x is the standard deviation of the sampling distribution of sample means
5. Conditions: Random sample of houses given. We’ll assume there’s more than 10(40) or 400 houses on
the reclaimed land so independence is not an issue. n = 40 > 30. A normal model OK. P( x > 5.2) = 0.0057
6. Conditions: Random sample of patients given. One patient’s score shouldn’t affect another’s so
independence is reasonable. n = 42 > 30. A normal model is OK. P(935 < x < 960) = .3345
7. Conditions: Random selection of students from each school given. Again, test scores should be
independent. np = 160(.065) = 10.4 > 10 and n(1 ─ p) = 160(1─.065) = 149.6 > 10 so our sample is large
enough. P( p̂ > .11) = .0104
8. Conditions: Random selection of bats given. We can assume there’s more than 10(70) or 700 bats living
under the Congress Street bridge since we have the world’s largest urban bat population. Large enough
sample since np = 70(.15) = 10.5 > 10 and n(1-p) = 70(.85) = 59.5 > 10. A normal model is appropriate.
P( p̂ > .25) = 0.0096
9. P(x ≥ 3.5) = 0.0006
This is just for a single student.
10. a)  x  2.2
b) P( x < 2) = .1515
 x  .1941
Note: Conditions required for b and c.
b) P( x > .68) = .0031
11. a) $6.60
12. a) .0439 (use the binomial theorem)
b) about 4
13. a) .203
e) .0483
b) .248
c) .583
d) .21
c) P( x < 1.923) = .0764
Conditions required for part b.
f) .565
g) .0521