Download Warm-up 7.2 Generating Sampling Distributions

Warm-up
7.2 Generating Sampling Distributions
Free Response Question
Simple random samples of young and older adults in a large city were surveyed as to credit card
debt, and the data are summarized:
n
Mean Median Min
Max
Q1
Q3
SD
Young: 100 1500 1500
1300
1700 1400
1600
120
Older:
1000
2100 1400
1655
200
100 1520 1500
(a) Is there evidence that either of these samples were drawn from populations with normal
distributions? Explain.
(b)Assume that both samples are drawn from normally distributed populations. Would a greater
percentage of younger or older adults more likely be able to pay off their credit debt with
$1,550? Explain.
E #1 - 4
E1. a.
b. The larger the sample size, the more trials listed under
frequency for the true mean of the data.
c. The larger the sample size the smaller the spread of the
sampling distribution
E 3. a. b. c. The midrange for the entire page is 1 + 18 = 19/2 =
9.5 As shown in the table the true range is 7.5. The midrange
of the samples will be higher than the average rectangle area.
Answers to H.W. E #1 - 4
7.2 Generating Sampling Distributions
Sampling Distribution of a Mean
If you toss two fair dice 10,000, how would you expect the
histogram to look like if you were graphing the results of the
average of two die?
Central Limit Theorem
The sampling distribution of any mean becomes more
nearly Normal as the sample size grows. Most
importantly the observations need to be independent
and collected with randomization.
FYI “central” in the theorem name means “fundamental”
CLT and Equations
• The CLT requires essentially the same assumptions and
conditions from modeling proportions: Independence,
Sample size, Randomization, 10% and large enough sample
The standard deviation of the sampling distribution is
sometimes called the standard error of the mean.
x 

n
standard dev. of pop. / sample size
Properties of Sampling Distribution of Sample Mean
pg 430
If a random sample of size n is selected from a population
with mean  and standard deviation , then
• The mean  x of the sampling distribution of x equation
equals the mean of the population,  :
• The standard deviation  x , the sampling distribution of
x , sometimes called the standard error of the mean:
x 

standard dev. of pop. / sample size
n
• Using the formula above you can find standard error of
the sample mean without simulation
Pg 429
Number of Children Problem
Physical Education Department and BMI study
A college physical education department asked a random sample of 200
female students to self-report their heights and weights, but the
percentage of students with body mass indexes over 25 seemed
suspiciously low. One possible explanation may be that the respondents
“shaded“ their weights down a bit. The CDC reports that the mean
weight of 18-year-old women is 143.74 lb, with a standard deviation of
51.54 lb, but these 200 randomly selected women reported a mean
weight of only 140 lb.
Question: Based on the Central Limit Theorem and the 68-95-99.7 Rule,
does the mean weight in this sample seem exceptionally low or might
this just be random sample-to-sample variation?
7.1 to 7.2 Review
Example 1: The Centers for Disease Control and Prevention reports
that the average weight of a man is 190 lb with a standard deviation
of 59 lb. An elevator in our building has a weight limit of 10 persons
or 2500 lb. What’s the probability that if 10 men get on the elevator,
they will overload its weight limit?
Sampling Distribution Method
Combining Data Method
Example 2: Harold fails to study for his statistics final.
The final has 100 multiple choice questions, each with
5 choices. Harold has no choice but to guess randomly
at all 100 questions. What is the probability that
Harold will get at least 30% on the test?
Sampling Distribution Method
Binomial Distribution Method
H.W. 7.2 P #10 E#15, #23, and 24
Common Test Mistakes
2. Suppose you buy a raffle ticket in each of 25 consecutive
weeks in support of your favorite charity. One of the 1200
raffle tickets sold each week pays $2000. What do you expect
to win for those 25 weeks, and with what standard deviation?
A.win about $0, give or take about $58
B. win about $25, give or take about $58
C. win about $42, give or take about $58
D.win about $42, give or take about $289
E. win about $210, give or take about $1443
3. Suppose Alex rolls a fair die until either a one or three
appears on top. What is the probability that it will take Alex
more than three rolls to get either a one or three the first
time?
2
4
3
3
A.  6   1   2  B.
C.
 2
 1
1  
1  
    
3
 2 3   3 
3
2
D. 2  2   2 
   
3 3 3
3
E.
2
 1   2  1   2   1 
1            
 3   3  3   3   3 
FORM B 10. d. was not capable of being solved using Ch. 6
concepts.
Common Mistakes on Test
what is the mean and standard deviation for the number of defects for pairs of shoes produced by this company.
Form A
1. This table gives the percentage of women who ultimately have a
given number of children. For example, 19% of women ultimately
have 3 children. What is the probability that two randomly selected
women will have a combined total of exactly 2 children?
0 and 2,
1 and 1,
2 and 0
0.18* 0.35 + 0.17*0.17+ 0.35* 0.18 = 0.159
Form B 2. For the sake of efficiency, a shoe company decides to produce
the left shoe of each pair at one site and the right shoe at a
different site. If the two sites produce shoes with a number of
defects reflected by 1  0.002, 1  0.15 and 2  0.005,  2  0.18,
 X Y   X  Y
0.002  0.005  0.007
 X2  Y   X2   Y2 0.152  0.182  0.0549  .234
Expected Number of Success and Expected Number of Trials
Binomial Distribution: Flipping a coin 6 times, about how
many times flip head on average.   n p  6 (0.5)  3
x
In a simple random sample of 15 students, how many are expected to be
younger than 20 . 60% of students are under 20.
Geometric Distribution: Flipping a coin, when do you expect
1
(on average) to have your first success.
x 
p
What is the expected number of interview before the second person
without health insurance is found? (16% no health insurance)
Expected Number of Success and Expected Number of Trials
Binomial Distribution: Flipping a coin 6 times, about how
many times flip head on average.   n p  6 (0.5)  3
x
What is the average number of students without laptops you would
expect to find after sampling 5 random students? (60% w/ laptops)
Geometric Distribution: Flipping a coin, when do you expect
(on average) to have your first success.
1
x 
p
On average how many otters would biologists have to check before
finding an infected otter? (20% are infected)