Confidence Intervals

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Confidence Intervals

• Chapter 10.1

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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?

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




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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

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Confidence intervals

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

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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

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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

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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
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Confidence interval for a population mean
Standard deviation of the statistic
Critical value
estimate
Margin of error
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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.

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• 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.

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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
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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.

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Statement (memorize!!)

• We are ________ confident that the true mean
  context lies within the interval ______ and
  ______.

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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.
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• 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.
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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.
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• What happens to the interval as the confidence
  level increases?

the interval gets wider as the confidence level
increases
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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!
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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
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• 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.
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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!
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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
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• 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?

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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

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• Graph examples of t- curves vs normal curve

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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

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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
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Formula
Standard deviation of statistic
Critical value
estimate
Margin of error
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Assumptions for t-inference

• Have an SRS from population
• s unknown
• Normal distribution
• Given
• Large sample size
• Check graph of data

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• 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.

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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.

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• 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)
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• 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.
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• 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)
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• 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.
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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

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Cautions continued

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