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Confidence Intervals

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... claims that the true mean pulse rate for adults is 72 beats per minute. ... The 95% confidence interval contains the claim of 72 beats per minute. ... – PowerPoint PPT presentation

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Title: Confidence Intervals


1
Confidence Intervals
  • Chapter 10.1

2
Rate your confidence0 - 100
  • Guess Mrs. Pughs age within 10 years?
  • within 5 years?
  • within 1 year?
  • Shooting a basketball at a wading pool, will make
    basket?
  • Shooting the ball at a large trash can, will make
    basket?
  • Shooting the ball at a carnival, will make basket?

3
What happens to your confidence as the interval
gets smaller?
The smaller the interval, the lower your
confidence.




4
Point Estimate
  • Use a single statistic based on sample data to
    estimate a population parameter
  • Simplest approach
  • But not always very precise due to variation in
    the sampling distribution

5
Confidence intervals
  • Are used to estimate the unknown population mean
  • Formula
  • estimate margin of error

6
Margin of error
  • Shows how accurate we believe our estimate is
  • The smaller the margin of error, the more precise
    our estimate of the true parameter
  • Formula

7
Confidence level
  • Is the success rate of the method used to
    construct the interval
  • Using this method, ____ of the time the
    intervals constructed will contain the true
    population parameter

8
Critical value (z)
  • Found from the confidence level
  • The upper z-score with probability p lying to its
    right under the standard normal curve
  • Confidence level tail area z
  • .05 1.645
  • .025 1.96
  • .005 2.576

z1.645
z1.96
z2.576
90
95
99
9
Confidence interval for a population mean
Standard deviation of the statistic
Critical value
estimate
Margin of error
10
What does it mean to be 95 confident?
  • 95 chance that m is contained in the confidence
    interval
  • The probability that the interval contains m is
    95
  • The method used to construct the interval will
    produce intervals that contain m 95 of the time.

11
  • A level C confidence interval for a parameter has
    two parts
  • An interval calculated from the data, usually of
    the form
  • A confidence level C, which gives the
    probability that the interval will capture the
    true parameter value in repeated samples.

12
Note you have no way of knowing whether your
sample is one of the 95 for which the interval
captures ? or one of the unlucky 5. The
statement that we are 95 confident that the
unknown ? lies between 452 and 470 is shorthand
for saying We have arrived at these numbers by a
method that gives correct results 95 of the
time. To sum it up Confidence interval gives
us a range of values that should contain the true
(unknown) value of the population parameter with
a high level of confidence
13
Steps for doing a confidence interval
  • Conditions
  • SRS from population
  • Sampling distribution is normal (or approximately
    normal)
  • Given (normal)
  • Large sample size- CLT (approximately normal)
  • Graph data (approximately normal)
  • s is known
  • Calculate the interval
  • Write a statement about the interval in the
    context of the problem.

14
Statement (memorize!!)
  • We are ________ confident that the true mean
    context lies within the interval ______ and
    ______.

15
A test for the level of potassium in the blood is
not perfectly precise. Suppose that repeated
measurements for the same person on different
days vary normally with s 0.2. A random sample
of three has a mean of 3.2. What is a 90
confidence interval for the mean potassium
level?
Assumptions Have an SRS of blood
measurements Potassium level is normally
distributed (given) s known We are 90
confident that the true mean potassium level is
between 3.01 and 3.39.
16
  • 95 confidence interval?

Assumptions Have an SRS of blood
measurements Potassium level is normally
distributed (given) s known We are 95
confident that the true mean potassium level is
between 2.97 and 3.43.
17
99 confidence interval?
Assumptions Have an SRS of blood
measurements Potassium level is normally
distributed (given) s known We are 99
confident that the true mean potassium level is
between 2.90 and 3.50.
18
  • What happens to the interval as the confidence
    level increases?

the interval gets wider as the confidence level
increases
19
How can you make the margin of error smaller?
  • z smaller
  • (lower confidence level)
  • s smaller
  • (less variation in the population)
  • n larger
  • (to cut the margin of error in half, n must
    be 4 times as big)

Really cannot change!
20
A random sample of 50 SWVGS students was taken
and their mean SAT score was 1250. (Assume s
105) What is a 95 confidence interval for the
mean SAT scores of SWVGS students?
We are 95 confident that the true mean SAT score
for SWVGS students is between 1220.9 and 1279.1
21
  • Suppose that we have this random sample of SAT
    scores
  • 1130 1260 1090 1310 1420 1190
  • What is a 95 confidence interval for the true
    mean SAT score? (Assume s 105)

We are 95 confident that the true mean SAT
score for SWVGS students is between 1115.1 and
1270.6.
22
Find a sample size
  • If a certain margin of error is wanted, then to
    find the sample size necessary for that margin of
    error use

Always round up to the nearest person!
23
The heights of SWVGS male students is normally
distributed with s 2.5 inches. How large a
sample is necessary to be accurate within .75
inches with a 95 confidence interval?
n 43
24
  • In a randomized comparative experiment on the
    effects of calcium on blood pressure, researchers
    divided 54 healthy, white males at random into
    two groups, takes calcium or placebo. The paper
    reports a mean seated systolic blood pressure of
    114.9 with standard deviation of 9.3 for the
    placebo group. Assume systolic blood pressure is
    normally distributed.
  • Can you find a z-interval for this problem? Why
    or why not?

25
Students t- distribution
  • Developed by William Gosset
  • Continuous distribution
  • Unimodal, symmetrical, bell-shaped density curve
  • Above the horizontal axis
  • Area under the curve equals 1
  • Based on degrees of freedom

26
  • Graph examples of t- curves vs normal curve

27
How does t compare to normal?
  • Shorter more spread out
  • More area under the tails
  • As n increases, t-distributions become more like
    a standard normal distribution

28
How to find t
Can also use invT on the calculator! Need upper
t value with 5 is above so 95 is
below invT(p,df)
  • Use Table C for t distributions
  • Look up confidence level at bottom df on the
    sides
  • df n 1
  • Find these t
  • 90 confidence when n 5
  • 95 confidence when n 15

t 2.132
t 2.145
29
Formula
Standard deviation of statistic
Critical value
estimate
Margin of error
30
Assumptions for t-inference
  • Have an SRS from population
  • s unknown
  • Normal distribution
  • Given
  • Large sample size
  • Check graph of data

31
  • For the Ex. 4 Find a 95 confidence interval for
    the true mean systolic blood pressure of the
    placebo group.
  • Assumptions
  • Have an SRS of healthy, white males
  • Systolic blood pressure is normally distributed
    (given).
  • s is unknown
  • We are 95 confident that the true mean systolic
    blood pressure is between 111.22 and 118.58.

32
Robust
  • An inference procedure is ROBUST if the
    confidence level or p-value doesnt change much
    if the assumptions are violated.
  • t-procedures can be used with some skewness, as
    long as there are no outliers.
  • Larger n can have more skewness.

33
  • Ex. 5 A medical researcher measured the pulse
    rate of a random sample of 20 adults and found a
    mean pulse rate of 72.69 beats per minute with a
    standard deviation of 3.86 beats per minute.
    Assume pulse rate is normally distributed.
    Compute a 95 confidence interval for the true
    mean pulse rates of adults.

(70.883, 74.497)
34
  • Another medical researcher claims that the true
    mean pulse rate for adults is 72 beats per
    minute. Does the evidence support or refute
    this? Explain.

The 95 confidence interval contains the claim of
72 beats per minute. Therefore, there is no
evidence to doubt the claim.
35
  • Ex. 6 Consumer Reports tested 14 randomly
    selected brands of vanilla yogurt and found the
    following numbers of calories per serving
  • 160 200 220 230 120 180 140
  • 130 170 190 80 120 100 170
  • Compute a 98 confidence interval for the
    average calorie content per serving of vanilla
    yogurt.

(126.16, 189.56)
36
  • A diet guide claims that you will get 120
    calories from a serving of vanilla yogurt. What
    does this evidence indicate?

Note confidence intervals tell us if something
is NOT EQUAL never less or greater than!
Since 120 calories is not contained within the
98 confidence interval, the evidence suggest
that the average calories per serving does not
equal 120 calories.
37
Some Cautions
  • The data MUST be a SRS from the population
  • The formula is not correct for more complex
    sampling designs, i.e., stratified, etc.
  • No way to correct for bias in data

38
Cautions continued
  • Outliers can have a large effect on confidence
    interval
  • Must know s to do a z-interval which is
    unrealistic in practice
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