Confidence Intervals: Admitting that Estimates Are Not Exact

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Lesson 2

• Confidence Intervals Admitting that Estimates
  Are Not Exact

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Chapter 8Interval Estimation

• Population Mean s Known

• Population Mean s Unknown

• Determining the Sample Size

• Population Proportion

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Overview

• Confidence Interval
• Computed from data
• Has a known probability of including the unknown
  population parameter being estimated
• Statistical Inference
• An exact probability statement about the
  population, based on sample data
• Confidence Level (Confidence coefficient)
• The probability of including the population
  parameter within the confidence interval
• 95 is the usual standard. Also 99, 99.9, 90

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Interval Estimation of a Population
MeanLarge-Sample Case

• Sampling Error
• Probability Statements about the Sampling Error
• Constructing an Interval Estimate
• Large-Sample Case with ?? Known
• Calculating an Interval Estimate
• Large-Sample Case with ?? Unknown

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Sampling Error

• The absolute value of the difference between an
  unbiased point estimate and the population
  parameter it estimates is called the sampling
  error.
• For the case of a sample mean estimating a
  population mean, the sampling error is
• Sampling Error

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Probability StatementsAbout the Sampling Error

• Knowledge of the sampling distribution of
  enables us to make probability statements about
  the sampling error even though the population
  mean ? is not known.
• A probability statement about the sampling error
  is a precision statement.
• ? is the level of significance

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Margin of Error and the Interval Estimate
A point estimator cannot be expected to provide
the exact value of the population parameter.
An interval estimate can be computed by adding
and subtracting a margin of error to the point
estimate.
Point Estimate /- Margin of Error
The purpose of an interval estimate is to
provide information about how close the point
estimate is to the value of the parameter.
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Margin of Error and the Interval Estimate
The general form of an interval estimate of a
population mean is
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Confidence Interval

• Estimate of Margin
• the Parameter of Error

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Confidence Interval

• Estimate of t or z
  standard error
• the Parameter of the
  estimate

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Probability StatementsAbout the Sampling Error

• Precision Statement
• There is a 1 - ? probability that the value of
  a sample mean will provide a sampling error of
  or less.

1 - ? of all values
?/2
?/2
?
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Interval Estimation of a Population Means Known

• With ? ?Known
• where is the sample mean
• 1 -? is the confidence coefficient
• z?/2 is the z value providing an area of
• ?/2 in the upper tail of the
  standard
• normal probability distribution
• s is the population standard deviation
• n is the sample size

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Interval Estimation of a Population Means Known

• There is a 1 - ? probability that the
  value of a
• sample mean will provide a margin of error of
• or less.

?/2
?/2
?
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Interval Estimate of a Population Means Known
?/2
?/2
interval does not include m
?
interval includes m
interval includes m
-------------------------
-------------------------
-------------------------
-------------------------
-------------------------
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Interval Estimate of a Population Mean s Known

• Adequate Sample Size

In most applications, a sample size of n 30
is adequate.
If the population distribution is highly skewed
or contains outliers, a sample size of 50 or
more is recommended.
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Interval Estimate of a Population Mean s Known

• Adequate Sample Size (continued)

If the population is not normally distributed
but is roughly symmetric, a sample size as small
as 15 will suffice.
If the population is believed to be at least
approximately normal, a sample size of less than
15 can be used.
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Example National Discount, Inc.

• National Discount has 260 retail outlets
  throughout the United States. National evaluates
  each potential location for a new retail outlet
  in part on the mean annual income of the
  individuals in the marketing area of the new
  location.
• Sampling can be used to develop an interval
  estimate of the mean annual income for
  individuals in a potential marketing area for
  National Discount.
• A sample of size n 36 was taken. The sample
  mean, , is 21,100 and the population standard
  deviation, ?, is 4,500. We will use .95 as the
  confidence coefficient in our interval estimate.

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Example National Discount, Inc.

• Interval Estimate of the Population Mean ?
  known
• Interval Estimate of ? is
• 21,100 1,470
• or 19,630 to 22,570
• We are 95 confident that the interval contains
  the
• population mean.

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Example National Discount, Inc.
Precision Statement There is a .95 probability
that the value of a sample mean for National
Discount will provide a sampling error of 1,470
or less. determined as follows 95 of the
sample means that can be observed are within
1.96 of the population mean ?. If
, then 1.96 1,470.
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Example Restaurant Survey

• n 100 residents
• 23.91 average expenditures
• ? 11.49 variability of individuals
• 1.149 variability of the sample average
• z 1.960 for 2-sided 95 confidence,
• From
• To
• We are 95 sure that the unknown population mean
  expenditure ? is between 21.66 and 26.16 for
  all N 77,386 residents in the population
• Even though we only observed 100 people!

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MOST COMMONLY USED CONFIDENCE LEVELS FOR Z
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Interval Estimation of a Population Means
Unknown

• If an estimate of the population standard
  deviation s cannot be developed prior to
  sampling, we use the sample standard deviation s
  to estimate s .

• This is the s unknown case.

• In this case, the interval estimate for m is
  based on the t distribution.

• (Well assume for now that the population is
  normally distributed.)

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t Distribution

• The t distribution is a family of similar
  probability distributions.
• A specific t distribution depends on a parameter
  known as the degrees of freedom.
• As the number of degrees of freedom increases,
  the difference between the t distribution and
  the standard normal probability distribution
  becomes smaller and smaller.
• A t distribution with more degrees of freedom
  has less dispersion.
• The mean of the t distribution is zero.

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t Distribution
Standard normal z values
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Interval Estimation of a Population Meanwith ?
Unknown

• Interval Estimate
• where 1 -? the confidence coefficient
• t?/2 the t value providing an
  area of ?/2 in the upper
  tail of a t distribution
• with n - 1 degrees of freedom
• s the sample standard deviation

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Example Apartment Rents

• Interval Estimation of a Population Mean
• with ? Unknown
• A reporter for a student newspaper is writing
  an
• article on the cost of off-campus housing. A
  sample of 16 one-bedroom units within a half-mile
  of campus resulted in a sample mean of 650 per
  month and a sample standard deviation of 55.
• Let us provide a 95 confidence interval
  estimate of the mean rent per month for the
  population of one-bedroom units within a
  half-mile of campus. Well assume this
  population to be normally distributed.

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Example Apartment Rents

• At 95 confidence, ? .05, and ?/2 .025.

t.025 is based on n - 1 16 - 1 15 degrees of
freedom.
In the t distribution table we see that t.025
2.131.
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Example Apartment Rents

• Interval Estimation of a Population
  MeanSmall-Sample Case (n lt 30) with ? Unknown
• We are 95 confident that the mean rent per month
  for the population of efficiency apartments
  within a half-mile of campus is between 620.70
  and 679.30.

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Excel Example

• Lets suppose we took a random sample of of 40
  students from a population of 44,000 students at
  UCF. One of the variables the professor is
  interested in is AGE. The following is the
  distribution of the random sample for the
  variable age.

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Descriptive Statistics to Summarize a Variable

• Variable Name
• Number of Observations
• Lowest Value
• Mean
• Median
• Standard Deviation
• Standard Error
• Maximum Value
• 1st Quartile
• 3rd Quartile.

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(No Transcript)
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Select tools from the menu bar.
Then select Data Analysis from the pull down menu
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From the Data Analysis box select Descriptive
Statistics and then click on the button OK.
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Input the range on the Input Range Box
Select the Summary Statistics and Confidence
level for Means Box
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Excel
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Descriptive Statistics to Summarize a Variable

• Variable Name AGE
• Number of Observations 40
• Lowest Value 17
• Mean 24.475
• Median 22.5
• Standard Deviation 6.11
• Standard Error 0.96
• Maximum Value 45
• 1st Quartile QUARTILE(A2A41,1)19.75
• 3rd Quartile QUARTILE(A2A41,3)28

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Detecting Outliers Using IQR

• IQR 3rd Quartile 1st Quartile
• The lower limit is located 1.5(IQR) below Q1.
• The upper limit is located 1.5(IQR) above Q3.
• Data outside these limits are considered outliers.

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Example Apartment for Rent

• IQR 28 19.75 8.25
• Lower Limit Q1 - 1.5(IQR) 19.75 - 1.5(8.25)
  7.375
• Upper Limit Q3 1.5(IQR) 28 1.5(8.25)
  40.375
• There might be 1 outlier (value greater than
  40.374 ) AGE 45.

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Confidence Interval

• Excel is providing us with the information we
  need to build a 95 confidence interval estimate
  of the mean.
• Margin of Error
• Therefore, we are 95 Confident that the
  population mean for the variable age is between
• 26.429 and 22.521.

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Sample Size for an Interval Estimateof a
Population Mean

• Let E the maximum sampling error
• E is the amount added to and subtracted from the
  point estimate to obtain an interval estimate.
• E is often referred to as the margin of error.
• We have
• Solving for n we have

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Continued

• In case we Sigma is unknown we should use the
  variance of the sample.

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Example National Discount, Inc.

• Sample Size for an Interval Estimate of a
  Population Mean
• Suppose that Nationals management team wants
  an estimate of the population mean such that
  there is a .95 probability that the sampling
  error is 500.
• How large a sample size is needed to meet the
  required precision?

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Example National Discount, Inc.

• Sample Size for Interval Estimate of a Population
  Mean
• At 95 confidence, z.025 1.96.
• Recall that ?? 4,500.
• Solving for n we have
• We need to sample 312 to reach a desired
  precision of
• 500 at 95 confidence.

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Interval Estimationof a Population Proportion
The general form of an interval estimate of a
population proportion is
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Interval Estimationof a Population Proportion
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Interval Estimationof a Population Proportion
?/2
?/2
p
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Interval Estimationof a Population Proportion

• Interval Estimate

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Interval Estimation of a Population Proportion

• Example Political Science, Inc.

• Political Science, Inc. (PSI)
• specializes in voter polls and
• surveys designed to keep
• political office seekers informed
• of their position in a race.
• Using telephone surveys, PSI interviewers ask
• registered voters who they would vote for if the
• election were held that day.

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Interval Estimation of a Population Proportion

• Example Political Science, Inc.

In a current election campaign, PSI has just
found that 220 registered voters, out of
500 contacted, favor a particular
candidate. PSI wants to develop a 95
confidence interval estimate for the proportion
of the population of registered voters that
favor the candidate.
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Interval Estimation of a Population Proportion
PSI is 95 confident that the proportion of all
voters that favor the candidate is between .3965
and .4835.
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Sample Size for an Interval Estimateof a
Population Proportion

• Margin of Error

Solving for the necessary sample size, we get
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Sample Size for an Interval Estimateof a
Population Proportion

• Necessary Sample Size

The planning value p can be chosen by 1. Using
the sample proportion from a previous sample of
the same or similar units, or 2. Selecting a
preliminary sample and using the sample
proportion from this sample.
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Sample Size for an Interval Estimateof a
Population Proportion

• Suppose that PSI would like a .99 probability
• that the sample proportion is within .03 of
  the population proportion.
• How large a sample size is needed to meet the
  required precision? (A previous sample of
  similar units yielded .44 for the sample
  proportion.)

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Sample Size for an Interval Estimateof a
Population Proportion
A sample of size 1817 is needed to reach a
desired precision of .03 at 99 confidence.
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Sample Size for an Interval Estimateof a
Population Proportion
Note We used .44 as the best estimate of p
in the preceding expression. If no information
is available about p, then .5 is often assumed
because it provides the highest possible sample
size. If we had used p .5, the recommended n
would have been 1843.
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End of Lesson 2