Confidence Intervals

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Chapter 7

• Confidence Intervals

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Estimation

• There are 2 types of estimation in statistics
  point estimation and interval estimation.

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Point Estimation

• With point estimation, a single statistic such as
  the sample mean, sample median, sample proportion
  is given as an estimate of the parameter of
  interest.
• Sometimes there are various choices when choosing
  a point estimate (ie sample mean vs. sample
  median)

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Illustration of CI

• The center is the target (the parameter of
  interest).
• The blue points are statistics. Each one is
  based on a different sample. On average, they
  equal the target.

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• The larger the diameter, the more confident we
  are. Confidence level is given by a percentage
  number.

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• The confidence interval is given in the form
• estimate margin of error
• Three factors must be made to develop a CI
• a good estimator of the parameter
• The sampling distribution or approximate
  distribution of the estimate (standard deviation
  of the estimate).
• The desired confidence level.

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7.1 Confidence Interval for population
proportion ?

• In general , confidence interval of ? of a
  sample with a large sample size n, is
• p z

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z ?
The blue area is 1-?
Standard Normal Distn
z
-z
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Select values of z

• (1-?) is how confident we want to be that the
  confidence interval WILL contain the parameter of
  interest. Well refer to these as the level of
  confidence.

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Steps to calculate CI for ?

• Calculate estimate p and SE(p)
• Find the critical value z from the Table
  corresponding to the level (1-?)
• ME z SE
• CIp ME

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Example 1

• To find the proportion of all students who study
  on weekends, survey 200 students and find out 60
  students study on weekends, find out
• 80 confidence interval for ?
• 95 confidence interval for ?
• 98 confidence interval for ?

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Finding the desired sample size for proportions
In this case, 0.5 is used in place of p because
this is where is largest. This is
a worst case scenario.
Here, you have to calculate p.
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Example 1 (continued)

• To find the proportion of all students who study
  on weekends, survey 200 students and find out 60
  students study on weekends, the estimate margin
  of error for the 95 confidence interval is
  .0635, find the sample size necessary to reduce
  the margin to .03? How about .01?

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7.2 Confidence interval for ? based on
when ? is known the z-interval.
The standard deviation of the estimate.
An estimate for µ
Depends on how confident you want to be.
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Margin of Error

• The margin of error (E ) is half the width of the
  confidence interval.

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Steps to calculate CI for ?

• Calculate estimate and SE( )
• Find the critical value z from the Table
  corresponding to the level (1-?)
• ME z SE
• CI ME

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Finding the sample size.

• For a given bound B on the margin of error, the
  sample size

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An example

• Based on a sample of 35 cars of a particular
  model, the fuel tank capacity is calculated for
  each. Based on this data, the sample mean is
  18.99 gallons. The population standard deviation
  is believed to be 3.5. Obtain a 90 confidence
  interval for the mean fuel capacity of this model
  of car.

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Cont.

• We are 90 confident that the mean fuel
• capacity is between 18.01 and 18.96
• gallons.

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Example

• Suppose a 95 confidence interval for µ is (4.2,
  4.8). What is the sample mean? Whats the width
  of this interval? What is the margin of error?
• Whats the width of the interval below?

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Finding the sample size. (cont.)

• Based on a sample of 35 cars of a particular
  model, the fuel tank capacity is calculated for
  each. Based on this data, the sample mean is
  18.99 gallons. The population standard deviation
  is believed to be 3.5. suppose that we want to
  obtain a 90 confidence interval for µ and we
  want the margin of error to be 0.2.

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We want margin of error (ME) to be 0.2.
Calculate the sample size?
This means we need at least a sample of size 829
to achieve this margin of error.
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What if s is unknown?
When you dont know s, you estimate it with s but
the distribution becomes a t with n-1 degrees of
freedom.
Z is used because the statistic below is
standard normal when s is known.
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7.2 Confidence interval for ? based on
when ? is unknown the t-interval.
The standard deviation of the estimate.
An estimate for µ
Depends on how confident you want to be.
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Margin of Error

• The margin of error (E ) is half the width of the
  confidence interval.

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Steps to calculate CI for ?

• Calculate estimate and SE( )
• where s is the sample standard deviation
• Find the critical value t from the Table. Its
  the upper ?/2 critical value with dfn-1
• ME t SE
• (1-?)100 CI for ? is given by
• CI ME

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Finding the sample size.

• For a given bound B on the margin of error, the
  sample size

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Exercise

• For the following cases, to find the
  corresponding critical value
• a) 90 confidence level, n20
• b) 98 confidence level, n30
• c) 95 confidence level, n51

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Example

• Based on a sample of 35 cars of a particular
  model, the fuel tank capacity is calculated for
  each. Based on this data, the sample mean is
  18.99 gallons. The sample standard deviation is
  believed to be 3.5.
• Obtain a 90 confidence interval for the mean
  fuel capacity of this model of car.
• suppose that we want to obtain a 90 confidence
  interval for µ and we want the margin of error to
  be 0.2. Calculate the sample size?

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Exercise

• To find out the weight of a particle, it was
  measured 41 times, and sample mean 174 units
  with sample standard deviation s1.1 units. Give
  a 95 CI of its true weight.

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A real estate agent needs to estimate the
average value of a residential property of a
given size in a given area. He believes that the
standard deviation of the property values is s
5,500, and that property values are
approximately normally distributed. A random
sample of 16 units gives a sample mean of
89,673.12. What is the 95 confidence interval
for the average value of all properties of this
kind?
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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

•   Exercise
• A manufacturer of pharmaceutical products
  analyzes a specimen from each batch of a product
  to verify the concentration of the active
  ingredient. The chemical analysis is not
  perfectly precise. Repeated measurements on the
  same specimen give slightly different results.
  The results of repeated measurements follow a
  normal distribution. The analysis procedure has
  no bias, so the mean of the population of all
  measurements is the true concentration in the
  specimen. The standard deviation of this
  distribution is known to be 0.0068 g/l. Three
  analyses of one specimen give the following
  concentrations 0.8403 0.8363 0.8447
• Calculate the 99 confidence interval for the
  true concentration.