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Stats: Modeling the World Chapter 19 Confidence Intervals

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Title: Stats: Modeling the World Chapter 19 Confidence Intervals


1
Stats Modeling the World
  • Chapter 19
  • Confidence Intervals for Proportions

2
Making an educated guess
  • Rarely do we actually know information about our
    population.
  • Usually we - take a sample
  • - find a sample statistic
  • - make a guess about the true parameter
    value
  • Our guess will be a bit off, this is the idea of
    a confidence interval.

3
The sample proportion ?
  • Recall from Chapter 18 that the sampling
    distribution model of ? is centered at p, with
    standard deviation .
  • Since we dont know p, we cant find the true
    standard deviation of the sampling distribution
    model, so we need to find the standard error

4
The Empirical Rule Revisited
  • By the 68-95-99.7 Rule, we know
  • - about 68 of all samples will have ? s within
    1 SE of p
  • - about 95 of all samples will have ? s within
    2 SEs of p
  • - about 99.7 of all samples will have ? s
    within 3 SEs of p

5
So then, what is a Confidence Interval?
  • Consider the 95 level
  • - Theres a 95 chance that p is no more than 2
    SEs away from ? .
  • - So, if we reach out 2 SEs, we are 95 sure
    that p will be in that interval. In other words,
    if we reach out 2 SEs in either direction of ? ,
    we can be 95 confident that this interval
    captures the true proportion.
  • This is called a 95 confidence interval.

6
Margin of Error
  • In general, the formula for a confidence
    interval is
  • estimate /- margin of error
  • The margin of error (ME) is the extent of the
    interval on either side of ?.
  • The more confident we want to be, the larger our
    ME needs to be.

7
Example pg 378 1 (WWP/HW)
  • A TV newsman reports the results of a poll of
    voters, and then says, The margin of error is
    plus or minus 4. Explain carefully what that
    means.

8
Common Levels and Critical Values
  • The most commonly chosen confidence levels are
    90, 95, and 99.
  • Critical Values 90 1.645
  • 95 1.96
  • 99 2.576

9
So what is the Confidence Level ?
  • The confidence level (typically 90, 95, or
    99) tells us the proportion of intervals that
    in the long run will capture our true
    population parameter by using this method
  • Heres an applet to help explain
  • http//bcs.whfreeman.com/ips4e/cat_010/applets/co
    nfidenceinterval.html

10
Those pesky conditions!
  • As with Chapter 18, we must still check our
    conditions to make sure the normal model applies
    before we use it.
  • - Randomization
  • - 10 condition
  • - Success/failure condition

11
WAP pg 378 3a
  • Consider each situation described below.
    Identify the population and the sample, explain
    what and represent, and tell whether the
    methods of this chapter can be used to create a
    confidence interval.
  • Police set up an auto checkpoint at which drivers
    are stopped and their cars inspected for safety
    problems. They find that 14 of 134 cars stopped
    have at least one safety violation. They want to
    estimate the percentage of all cars that may be
    unsafe.

12
WAP pg 378 3b
  • Consider each situation described below.
    Identify the population and the sample, explain
    what and represent, and tell whether the
    methods of this chapter can be used to create a
    confidence interval.
  • b. A TV talk show asks viewers to register
    their opinions on prayer in schools by logging on
    to a Web site. Of the 602 people who voted, 488
    favored prayer in schools. We want to estimate
    the level of support among the general public.

13
WAP pg 378 3c
  • Consider each situation described below.
    Identify the population and the sample, explain
    what and represent, and tell whether the
    methods of this chapter can be used to create a
    confidence interval.
  • A school is considering requiring students to
    wear uniforms. The PTA surveys parent opinion by
    sending a questionnaire home with all 1245
    students 380 surveys are returned, with 228
    families in favor of the change.

14
WAP pg 378 3d
  • Consider each situation described below.
    Identify the population and the sample, explain
    what and represent, and tell whether the
    methods of this chapter can be used to create a
    confidence interval.
  • d. A college admits 1632 freshmen one year,
    and four years later 1388 of them graduate on
    time. The college wants to estimate the
    percentage of all their freshman enrollees who
    graduate on time.

15
What Not to Say
  • Confidence Intervals are powerful tools because
    they allow us to make good estimates, however
    they are often misinterpreted.
  • Dont suggest the parameter varies (There is a
    95 chance that the true proportion is between
    42.7 and 51.3)
  • Dont claim that other samples will agree with
    yours (In 95 of samples of US adults, the
    proportion who think x is better than y will be
    between 42.7 and 51.3)
  • Dont be certain about the parameter (Between
    42.1 and 61.7 of persons are infected)

16
What Not to Say
  • Confidence Intervals are powerful tools because
    they allow us to make good estimates, however
    they are often misinterpreted.
  • Dont forget, its the parameter we are
    estimating! (Im 95 confident that my samples
    proportion is between 42.1 and 61.7)
  • Dont claim to know too much (Im 95 confident
    that between 42.1 and 61.7 of all the people in
    the world are infected)

17
WAP pg 378 5a-c
  • A catalog sales company promises to deliver
    orders placed on the
  • Internet within 3 days. Follow-up calls to a few
    randomly selected
  • customers show that a 95 confidence interval for
    the proportion of all
  • orders that arrive on time is 88 6. What
    does this mean? Are
  • these conclusions correct? Explain.
  • a. Between 82 and 94 of all orders arrive
    on time.
  • b. 95 of all random samples of customers
    will show that 88 of orders arrive on time.
  • c. 95 of all random samples of customers
    will show that 82 to 94 of orders arrive on
    time.

18
WAP pg 378 5d-e
  • A catalog sales company promises to deliver
    orders placed on the
  • Internet within 3 days. Follow-up calls to a few
    randomly selected
  • customers show that a 95 confidence interval for
    the proportion of all
  • orders that arrive on time is 88 6. What
    does this mean? Are
  • these conclusions correct? Explain.
  • d. We are 95 sure that between 82 and 94
    of the orders placed by the customers in this
    sample arrived on time.
  • e. On 95 of the days, between 82 and 94 of
    the orders will arrive on time.

19
One-proportion z-interval
  • When the conditions are met, we can find the
    confidence interval for the population proportion
    p.

20
WAP pg 378 7
  • Several factors are involved in the
    creation of a confidence interval. Among them
    are the sample size, level of confidence, and
    margin of error. Which statements are true?
  • a. For a given sample size, higher confidence
    means a smaller margin of error.
  • b. For a specified confidence level, larger
    samples provide smaller margins of error.

21
WAP pg 379 11
  • A May 2000 Gallup Poll found that 38 of a
    random sample of 1012 adults said that they
    believe in ghosts.
  • Find the margin of error for this poll if we want
    90 confidence in our estimate of the percent of
    American adults who believe in ghosts.
  • Explain what that margin of error means.

22
WAP pg 379 11
  • A May 2000 Gallup Poll found that 38 of a
    random sample of 1012 adults said that they
    believe in ghosts.
  • If we want to be 99 confident, will the margin
    of error be larger or smaller? Explain.
  • Find that margin of error.
  • In general, if all other aspects of the situation
    remain the same, will smaller margins of error
    involve greater or less confidence in the
    interval?

23
Procedure for CIs PANIC!!!
  • Use the PANIC acronym to help you remember all
    of the steps in a confidence interval.
  • P define your Parameter
  • A state your Assumptions/conditions
  • N Name your interval
  • I find your Interval
  • C write your Conclusion in Context.

24
How do you write a conclusion?
  • Heres the general recipe for writing a
    conclusion for a confidence interval
  • I am ___ confident that the true population
    _______ lies between ____ and _____.

25
WAP pg 378 9 (WWP/HW)
  • What fraction of cars are made in Japan? The
    computer output below summarizes the results of a
    random sample of 50 autos. Explain carefully
    what it tells you.
  • z-interval for proportion
  • With 90.00 confidence
  • 0.29938661 lt p(japan) lt 0.46984416

26
WAP pg 379 13
  • An insurance company checks police records on
    582 accidents selected at random and notes that
    teenagers were at the wheel in 91 of them.
  • Create a 95 confidence interval for the
    percentage of all auto accidents that involve
    teenage drivers.
  • Explain what your interval means.

27
WAP pg 379 13
  • An insurance company checks police records on
    582 accidents selected at random and notes that
    teenagers were at the wheel in 91 of them.
  • Explain what 95 confidence means.
  • A politician urging tighter restrictions on
    drivers licenses issued to teens says, In one
    of every 5 auto accidents a teenager is behind
    the wheel. Does your confidence interval
    support or contradict this statement? Explain.

28
WAP pg 379 15 (WWP/HW)
  • Some food retailers propose subjecting food to a
    low level of radiation in order to improve
    safety, but sale of such irradiated food is
    opposed by many people. Suppose a grocer wants
    to find out what his customers think. He has
    cashiers distribute surveys at checkout and ask
    customers to fill them out and drop them in a box
    near the front door. He gets responses from 122
    customers, of whom 78 oppose the radiation
    treatments. What can the grocer conclude about
    the opinions of all his customers?

29
WAP pg 379 21
  • Vitamin D, whether ingested as a dietary
    supplement or produced naturally when sunlight
    falls upon the skin, is essential for strong,
    healthy bones. The bone disease rickets was
    largely eliminated in England during the 1950s,
    but now there is concern that a generation of
    children more likely to watch TV or play computer
    games than spend time outdoors is at increased
    risk. A recent study of 2700 children randomly
    selected from all parts of England found 20 of
    them deficient in vitamin D.
  • Find a 98 confidence interval.

30
WAP pg 379 21
  • Vitamin D, whether ingested as a dietary
    supplement or produced naturally when sunlight
    falls upon the skin, is essential for strong,
    healthy bones. The bone disease rickets was
    largely eliminated in England during the 1950s,
    but now there is concern that a generation of
    children more likely to watch TV or play computer
    games than spend time outdoors is at increased
    risk. A recent study of 2700 children randomly
    selected from all parts of England found 20 of
    them deficient in vitamin D.
  • Explain carefully what your interval means.
  • Explain what 98 confidence means.

31
Margin of Error and Sample Size
  • By knowing how precise we want our answer to be,
    we can find out how many people we should ask.
  • If you dont have information about ?, use .5 as
    a conservative estimate
  • In order to cut the ME in half, you have to
    quadruple the sample size.

32
WAP pg 379 23
  • Wildlife biologists inspect 153 deer taken by
    hunters and find 32 of them carrying ticks that
    test positive for Lyme disease.
  • Create a 90 confidence interval for the
    percentage of deer that may carry such ticks.

33
WAP pg 379 23
  • Wildlife biologists inspect 153 deer taken by
    hunters and find 32 of them carrying ticks that
    test positive for Lyme disease.
  • If the scientists want to cut the margin of error
    in half, how many deer must they inspect?
  • What concerns do you have about this sample?

34
WAP pg 380 25
  • Its believed that as many as 25 of adults over
    50 never graduated from high school. We wish to
    see if this percentage is the same among the 25
    to 30 age group.
  • How many of this younger age group must we survey
    in order to estimate the proportion of non-grads
    to within 6 with 90 confidence?

35
WAP pg 380 25
  • Its believed that as many as 25 of adults over
    50 never graduated from high school. We wish to
    see if this percentage is the same among the 25
    to 30 age group.
  • Suppose we want to cut the margin of error to 4.
    Whats the necessary sample size?
  • What sample size would produce a margin of error
    of 3?
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