Download CHAPTER 11 NOTES: INFERENCE FOR A DISTRIBUTION

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AP STATISTICS CHAPTER 11: INFERENCE FOR A DISTRIBUTION
Chapter 11.1
Inference for the Mean of a Population
Ex: One concern employers have about the use of
technology is the amount of time that employees
spend each day making personal use of company
technology, such as phone, e-mail, internet, and
games. The Associated Press reports that, on average,
workers spend 72 minutes a day on such personal
technology uses. A CEO of a large company wants
to know if the employees of her company are
comparable to this survey. In a random sample of 10
employees, with the guarantee of anonymity, each
reported their daily personal computer use. The
times are recorded below.
Employee Time
1
66
2
70
3
75
4
88
5
69
6
71
7
71
8
63
9
89
10
86
Does this data provide evidence that the mean for this
company is greater than 72 minutes?
What is different about this problem, compared to
what we were given in chapter 10?
When the
of a statistic is
estimated from the data, the result is called the
of the statistic, and is
given by
When we use this estimator, the statistic that results
does not have a normal distribution, instead it has a
new distribution, called the
.
One-sample z-statistic
One-sample t-statistic
SUMMARY/QUESTIONS TO ASK IN CLASS
.
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AP STATISTICS CHAPTER 11: INFERENCE FOR A DISTRIBUTION
The variability of the t-statistic is controlled by the
.
The number of
is equal to
.
ASSUMING NORMALITY?
USINGTHE t PROCEDURES
1.
2.
3.
4.
Example 2: The Degree of Reading Power (DRP) is a
test of the reading ability of children. Here are DRP
scores for a random sample of 44 third-grade students
in a suburban district:
40 26 39 14 42 18 25 54 48
43 46 27 19 47 19 26 45 22
35 34 15 44 40 38 31 14 33
46 52 25 35 35 33 29 27 41
34 41 49 28 52 47 35 51
At the α = .1, is there sufficient evidence to suggest
that this district’s third graders reading ability is
different than the national mean of 34?
Back to Example 1, Does this data provide evidence
that the mean for this company is greater than 72
minutes?
Employee
1
2
3
4
5
6
7
8
9
10
Time
66
70
75
88
69
71
71
63
89
86
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 11: INFERENCE FOR A DISTRIBUTION
Example 3: The Wall Street Journal (January 27,
1994) reported that based on sales in a chain of
Midwestern grocery stores, President’s Choice
Chocolate Chip Cookies were selling at a mean rate
of $1323 per week. Suppose a random sample of 30
weeks in 1995 in the same stores showed that the
cookies were selling at the average rate of $1208 with
standard deviation of $275. Does this indicate that
the sales of the cookies is different from the earlier
figure?
Example 4: The times of first sprinkler activation
(seconds) for a series of fire-prevention sprinklers
were as follows:
27
30
24
41
33
22
24
27
27
23
28
35
22
Construct a 95% confidence interval for the mean
activation time for the sprinklers.
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AP STATISTICS CHAPTER 11: INFERENCE FOR A DISTRIBUTION
MATCHED PAIRS PROCEDURES
Form 1:
Form 2:
Is this an example of matched pairs?
1)A college wants to see if there’s a difference in
time it took last year’s class to find a job after
graduation and the time it took the class from five
years ago to find work after graduation. Researchers
take a random sample from both classes and measure
the number of days between graduation and first day
of employment
2) In a taste test, a researcher asks people in a random
sample to taste a certain brand of spring water and
rate it. Another random sample of people is asked to
taste a different brand of water and rate it. The
researcher wants to compare these samples
3) A pharmaceutical company wants to test its new
weight-loss drug. Before giving the drug to a random
sample, company researchers take a weight
measurement on each person. After a month of using
the drug, each person’s weight is measured again.
Example: A whale-watching company noticed that
many customers wanted to know whether it was
better to book an excursion in the morning or the
afternoon. To test this question, the company
collected the following data on 15 randomly selected
days over the past month. (Note: days were not
consecutive.)
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AP STATISTICS CHAPTER 11: INFERENCE FOR A DISTRIBUTION
Ex: The effect of exercise on the amount of lactic
acid in the blood was examined in journal Research
Quarterly for Exercise and Sport. Eight males were
selected at random from those attending a week-long
training camp. Blood lactate levels were measured
before and after playing 3 games of racquetball, as
shown in the table.
What is the parameter of interest in this problem?
Construct a 95% confidence interval for the mean
change in blood lactate level.
Based on the data, would you conclude that there is a
significant difference, at the 5% level, that the mean
difference in blood lactate level was over 10 points?
SUMMARY/QUESTIONS TO ASK IN CLASS
Player
1
2
3
4
5
6
7
8
Before
13
20
17
13
13
16
15
16
After
18
37
40
35
30
20
33
19
Name _______________________________
AP STATISTICS CHAPTER 11: INFERENCE FOR A DISTRIBUTION
Two-Sample Inference Procedures with Means
•
•
The goal of these inference procedures is to compare the responses to
characteristics of
We have
samples from each treatment or population
or to compare the
Formula for Mean:
Formula for Standard Deviation:
Example 1
Suppose we have a population of adult men with a mean height of 71 inches and standard deviation of 2.6 inches.
We also have a population of adult women with a mean height of 65 inches and standard deviation of 2.3 inches.
Assume heights are normally distributed.
a. Describe the distribution of the difference in heights between males and females (male-female).
b.
What is the probability that the height of a randomly selected man is at most 5 inches taller than the height
of a randomly selected woman?
c.
What is the 70th percentile for the difference (male-female) in heights of a randomly selected man &
woman?
d.
What is the probability that the mean height of 30 men is at most 5 inches taller than the mean height of 30
women?
e.
What is the 70th percentile for the difference (male-female) in mean heights of 30 men and 30 women?
Assumptions: (Use when)
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AP STATISTICS CHAPTER 11: INFERENCE FOR A DISTRIBUTION
Degrees of Freedom
Option 1:
Option 2:
Confidence Interval:
Hypothesis Test:
Example 2: Two competing headache remedies claim to give fast-acting relief. An experiment was performed to
compare the mean lengths of time required for bodily absorption of brand A and brand B. Assume the absorption
time is normally distributed. Twelve people were randomly selected and given an oral dosage of brand A. Another
12 were randomly selected and given an equal dosage of brand B. The length of time in minutes for the drugs to
reach a specified level in the blood was recorded. The results follow:
mean
SD
n
Brand A
20.1
8.7
12
Brand B
18.9
7.5
12
Describe the shape & standard error for sampling distribution of the differences in the mean speed of absorption.
Find a 95% confidence interval difference in mean lengths of time required for bodily absorption of each brand.
Is there sufficient evidence that these drugs differ in the speed at which they enter the blood stream?
Suppose that the sample mean of Brand B is 16.5, then is Brand B faster?
Confidence interval statements:
Hypothesis Statements:
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AP STATISTICS CHAPTER 11: INFERENCE FOR A DISTRIBUTION
Example 3:
The length of time in minutes for the drugs to reach a specified level in the blood was recorded. The results follow:
mean
SD
n
Brand A
20.1
8.7
12
Brand B
18.9
7.5
12
Is there sufficient evidence that these drugs differ in the speed at which they enter the blood stream?
Example 4: An article in a professional journal examines the relationship between attitudes towards success at
college-level mathematics. Twenty men and thirty-eight women selected at random from those identified at being
high-risk of failure participated in the study. Each student was asked to respond to a series of questions, and the
answers were combined to obtain a math anxiety score. For this particular scale, the higher the score, the lower the
level of anxiety towards mathematics. Here are the summary values.
n
x
s
Males
20
35.9
11.9
Females
38
36.6
12.3
Does this data provide evidence that the mean anxiety score is different for women that it is for men?
Robustness:
Example 5: A modification has been made to the process for producing a certain type of time-zero film (film that
begins to develop as soon as the picture is taken). Because the modification involves extra cost, it will be
incorporated only if sample data indicate that the modification decreases true average development time by more
than 1 second. Should the company incorporate the modification?
Original
8.6
5.1
4.5
5.4
6.3
6.6
5.7
8.5
Modified
5.5
4.0
3.8
6.0
5.8
4.9
7.0
5.7
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AP STATISTICS CHAPTER 11: INFERENCE FOR A DISTRIBUTION
SUMMARY OF INFERENCE PROCEDURES
FOR DISTRIBUTIONS – t TESTS
ONE-SAMPLE t-TEST
TWO-SAMPLE t-TEST
Use when the sample is random and the population
standard deviation is unknown. Verify normality, or
the absence of outliers and strong skewness, based
upon sample size.
Use when
ONE-SAMPLE t-TEST FOR DIFFERENCES
Use in a matched pairs setting, when population
standard deviation is unknown. Verify normality, or
the absence of outliers and strong skewness, based
upon sample size.
STANDARD ERROR:
s
SEM 
n
TEST STATISTIC:
t
STANDARD ERROR:
TEST STATISTIC:
x
s
n
CONFIDENCE INTERVAL:
x  t*
CONFIDENCE INTERVAL:
s
n
DEGREES OF FREEDOM:
df  n  1
SUMMARY/QUESTIONS TO ASK IN CLASS
DEGREES OF FREEDOM: