Sampling Distribution of the Difference Between Two statistics

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Sampling Distribution of the Difference Between
Two statistics
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5.4 Sampling Distribution of the Difference
Between Two sample means

• Independent random samples of size n1 and n2 are
  selected from two populations that have means ?1
  and ?2 and standard deviations ?1 and ?1,
  respectively. If n1 and n2 are each larger than
  30, then

• is approximately normally distributed.
• is centered at ?1-?2
• has a standard deviation of

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Note

• If the two populations are normally distributed,
  then the sample distribution of is
  also normally distributed regardless of the
  sample size

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Sampling Distribution of the Difference Between
Two Means
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Example 5.4

• The starting salaries of MBA students from two
  universities (MSU and UM) are 62,000 (stand.dev.
  14,500), and 60,000 (stand. dev. 18,300).
• What is the probability that a sample mean of MSU
  students will exceed the sample mean of UM
  students? (nMSU 50 nUM 60)

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Example 5.4 Solution

• We need to determine

m1 - m2 62,000 - 60,000 2,000
1 - .2611
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Sampling Distribution of Difference Between Two
Proportions

• If n1 and n2 are sufficient large, then the
  sampling distribution of P1-P2
• is approximately normally distributed.
• is centered at ?1-?2.
• has a standard deviation of

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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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Confidence Intervals (CI)

• A confidence interval for a population
  parameter is
• an interval of possible values for the unknown
  parameter.
• The interval is computed from sample data in such
  a way that we have a high degree of confidence
  that the interval contains the true value of the
  parameter.
• The degree of confidence, stated as a percent, is
  the confidence level.

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The form of confidence interval

• estimate margin of error (ME)

• Three factors must be made to develop a CI
• a good point estimator of the parameter
• The sampling distribution or approximate
  distribution of the point estimate (standard
  deviation of the estimate).
• The desired confidence level (1- ? )100.

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7.1 Confidence Interval for population
proportion ?

• Standard error of the estimate p
• SE(p)
• In general , the large sample (1- ?
  )100confidence interval for ? of based on p is
  given by
• pME p z

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The critical value for the standard normal
distribution

• Find z and z such that

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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 Z-table
  corresponding to the level (1-?)
• ME z SE
• CIpME

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Example 7.1

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

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• (23.65,36.35)

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Confidence intervals for ? Interpretation

• The (for example) 95 confidence interval
  procedure has a 0.95probability of working
  (giving an interval containing ?) before the data
  are collected. (Before the data are collected, is
  a random variable .)

• If samples of the same size are drawn repeatedly
  from a population, and a confidence interval is
  calculated from each sample, then 95 of these
  intervals should contain the population
  proportion

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Question1

• How much data do we need to get a confidence
  interval with margin of error B or less?
• Sample Size Determination for Estimating ?

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Finding the desired sample size for estimating ?
B
B
B
B
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 estimate p from a smaller
sample size.use p in the old studies
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Example 7.1 (continued)

• 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?
• Answer 897, for B0.03

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Exercise 7.1

• Of 900 people treated with new drug, 180 showed
  an allergic reaction. Estimate with a 90
  confidence interval the proportion of the
  proportion who will show an allergic reaction.
• How large a sample is necessary to ensure that
  the margin of error is no greater than .03

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• Answer
• (.178,.222)
• 482

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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
Y

• The margin of error (ME ) 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 to estimate ?.

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

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Example 7.2

• 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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Exercise 7.2

• A consumer research group sampled 100 hand-held
  video games, all of the same make and model. The
  sample mean life was 560 hours. Assuming the
  standard deviation is 35 hours, construct 90
  confidence interval estimate of the true mean
  life span of the video games.
• Is a 95 confidence interval wider or narrower
  than the interval you got?

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• Answer
• (554.24, 565.76)
• A 95 CI would be wider because the margin of
  error would be multiplied by 1.96 instead of 1.645

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Exercise 7.3

• 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?
  Standard error?
• Whats the width of the interval below?

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• Answer7.3
• Sample mean4.5
• Width of CI0.6
• ME0.3
• SE0.153

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Example7.2(continue)

• Earlier we obtained a confidence interval
• for the mean fuel capacity of a certain
• model of car. In that example, we were
• given the sample size of 35. This time,
• 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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Exercise 7.4

• To determine the diameter of Venus, an astronomer
  makes 36 measurements of the diameter and finds
• miles. Assuming
  miles, find a 95 confidence interval
  estimate of the diameter of Venus. What sample
  size is required so that the margin of error in
  determining the diameter of Venus is only 50
  miles.

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• Answer 7.4
• (7746.73, 7949.27)
• 148