Statistics and Quantitative Analysis U4320

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Statistics and Quantitative Analysis U4320

• Segment 6
• Prof. Sharyn OHalloran

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I. Introduction

• A. Review of Population and Sample Estimates 

• B. Sampling 
• 1. Samples
• The sample mean is a random variable with a
  normal distribution.
• If you select many samples of a certain size then
  on average you will probably get close to the
  true population mean.

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I. Introduction (cont.)

• 2. Central Limit Theorem
• Definition
• If a Random Sample is taken from any population
  with mean_ and standard deviation_, then the
  sampling distribution of the sample means will be
  normally distributed with
• 1) Sample Mean E() _, and
• 2) Standard Error SE() _/_n
• As n increases, the sampling distribution of
  tends toward the true population mean_.

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I. Introduction (cont.)

• C. Making Inferences
• To make inferences about the population from a
  given sample, we have to make one correction,
  instead of dividing by the standard deviation, we
  divided by the standard error of the sampling
  process
• Today, we want to develop a tool to determine how
  confident we are that our estimates lie within a
  certain range.

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II. Confidence Interval

• A. Definition of 95 Confidence Interval
  (_ known)
• 1. Motivation
• We know that, on average, is equal to_.
• We want some way to express how confident we are
  that a given is near the actual_ of the
  population.
• We do this by constructing a confidence interval,
  which is some range around that most probably
  contains_.
• The standard error is a measure of how much error
  there is in the sampling process. So we can say
  that is equal to__ the standard error
• ? X_Standard Error

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II. Confidence Interval (cont.)

• 2. Constructing a 95 Confidence Interval
• a. Graph
• First, we know that the sample mean is
  distributed normally, with mean_ and standard
  error

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II. Confidence Interval (cont.)

• b. Second, we determine how confident we want to
  be in our estimate of_.
• Defining how confident you want to be is called
  the ?-level.
• So a 95 confidence interval has an associate ?-
  level of .05.
• Because we are concerned with both higher and
  lower values, the relevant range is _/2
  probability in each tail.

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II. Confidence Interval (cont.)

• C. Define a 95 Confidence Range
• Now let's take an interval around ? that contains
  95 of the area under the curve.
• So if we take a random sample of size n from the
  population, 95 of the time the population mean _
  will be within the range

• What is a z value associated with a .025
  probability?
• From the z-table, we find the z-value associated
  with a .025 probability is 1.96.
• So our range will be bounded by

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II. Confidence Interval (cont.)

• Now, let's take this interval of size -1.96 SE
  , 1.96 SE and use it as a measuring rod.
• d. Interpreting Confidence Intervals

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II. Confidence Interval (cont.)

• What's the probability that the population mean ?
  will fall within the interval ? 1.96 SE?
• e. In General
• We get the actual interval as 1.96 SE on either
  side of the sample mean .
• We then know that 95 of the time, this interval
  will contain ?. This interval is defined by

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II. Confidence Interval (cont.)

• For a 95 confidence interval
• Examples
• 1. Example 1 Calculate a 95 confidence Interval
• Say we sample n180 people and see how many times
  they ate at a fast-food restaurant in a given
  week. The sample had a mean of 0.82 and the
  population standard deviation ? is 0.48.
  Calculate the 95 confidence interval for these
  data.

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II. Confidence Interval (cont.)

• Answer
• Why 95?

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II. Confidence Interval (cont.)

• 2. Example 2 Calculating a 90 confidence
  interval
• A random sample of 16 observations was drawn from
  a normal population with s 6 and 25. Find a
  90 (a .10) confidence interval for the
  population mean, _.
• First, find Z.10/2 in the standard normal tables

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II. Confidence Interval (cont.)

• Second, calculate the 90 confidence interval
• 90 of the time, the mean will lie with in this
  range.

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II. Confidence Interval (cont.)

• What if we wanted to be 99 of the time sure that
  the mean falls with in the interval?
• What happens when we move from a 90 to a 99
  confidence interval?

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II. Confidence Interval (cont.)

• C. Confidence Intervals (_ unknown)
• 1.Characteristics of a Student-t distribution
• a.Shape the student t-distribution
• The t-distribution changes shape as the sample
  size gets larger, and in the limit it becomes
  identical to the normal.

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II. Confidence Interval (cont.)

• b. When to use t-distribution
• i. s is unknown
• ii. Sample size n is small
• 2. Constructing Confidence Intervals Using
  t-Distribution
• A. Confidence Interval
• 95 confidence interval is
• B. Using t-tables
• Say our sample size is n and we want to know
  what's the cutoff value to get 95 of the area
  under the curve.

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II. Confidence Interval (cont.)

• i) Find Degrees of Freedom
• Degree of freedom is the amount of information
  used to calculate the standard deviation, s. We
  denote it as d.f. _ d.f. n-1
• ii) Look up in the t-table
• Now we go down the side of the table to the
  degrees of freedom and across to the appropriate
  t-value.
• That's the cutoff value that gives you area of
  .025 in each tail, leaving 95 under the middle
  of the curve.
• iii) Example
• Suppose we have sample size n15 and t.025 What
  is the critical value? 2.13

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II. Confidence Interval (cont.)

• Answer

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II. Confidence Interval (cont.)

• III. Differences of Means
• A. Population Variance Known
• Now we are interested in estimating the value (?1
  - ?2) by the sample means, using 1 - 2.
• Say we take samples of the size n1 and n2 from
  the two populations. And we want to estimate the
  differences in two population means.
• To tell how accurate these estimates are, we can
  construct the familiar confidence interval around
  their difference

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II. Confidence Interval

• This would be the formula if the sample size were
  large and we knew both ?1 and ?2.
• B. Population Variance Unknown
• 1. If, as usual, we do not know ?1 and ?2, then
  we use the sample standard deviations instead.
  When the variances of populations are not equal
  (s1 ? s2)
• Example Test scores of two classes where one is
  from an inner city school and the other is from
  an affluent suburb.

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II. Confidence Interval (cont.)

• 2. Pooled Sample Variances, s1 s2 (s2 is
  unknown)
• If both samples come from the same population
  (e.g., test scores for two classes in the same
  school), we can assume that they have the same
  population variance . Then the formula becomes
• or just
• In this case, we say that the sample variances
  are pooled. The formula for is

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II. Confidence Interval (cont.)

• The degrees of freedom are (n1-1) (n2-1), or
  (n1n2-2).
• 3. Example
• Two classes from the same school take a test.
  Calculate the 95 confidence interval for the
  difference between the two class means.

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II. Confidence Interval (cont.)

• Answer

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II. Confidence Interval (cont.)
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II. Confidence Interval (cont.)

• C. Matched Samples
• 1. Definition
• Matched samples are ones where you take a single
  individual and measure him or her at two
  different points and then calculate the
  difference.
• 2. Advantage
• One advantage of matched samples is that it
  reduces the variance because it allows the
  experimenter to control for many other variables
  which may influence the outcome.

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II. Confidence Interval (cont.)

• 3. Calculating a Confidence Interval
• Now for each individual we can calculate their
  difference D from one time to the next.
• We then use these D's as the data set to estimate
  ?, the population difference.
• The sample mean of the differences will be
  denoted .
• The standard error will just be
• Use the t-distribution to construct 95
  confidence interval

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II. Confidence Interval (cont.)

• 4. Example

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II. Confidence Interval (cont.)

• Notice that the standard error here is much
  smaller than in most of our unmatched pair
  examples.

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IV. Confidence Intervals for Proportions

• Just before the 1996 presidential election, a
  Gallup poll of about 1500 voters showed 840 for
  Clinton and 660 for Dole. Calculate the 95
  confidence interval for the population proportion
  ? of Clinton supporters.
• Answer n 1500
• Sample proportion P

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IV. Confidence Intervals for Proportions (cont.)

• Create a 95 confidence interval
• where ? and P are the population and sample
  proportions, and n is the sample size.
• That is, with 95 confidence, the proportion for
  Clinton in the whole population of voters was
  between 53 and 59.