Inferential Statistics Part 1

1
Inferential Statistics Part 1

• Chapter 8
• P. 253- 278

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Collecting a random sample

• Goal to understand characteristics about a
  population
• Examples
• Whats the average commuting time for city
  residents?
• Whats the average household income of the
  patrons of a particular grocery store?
• Whats the average leaf size size of birch trees
  on August 1 in a particular state park?
• What proportion of people in a particular
  tropical city have had malaria?

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Estimating the mean

• One of the most common goals of statistical
  inference is estimating a population mean with a
  sample mean

4
Central Limit Theorem

• When we have n independent, identically
  distributed (X1..Xn) random variables, the mean
  of those random variables approaches a normal
  distribution with mean µ and variance ,
  as n gets large.
• Independence of random variables means that the
  value of one observation has no effect on the
  value of another observation.
• Identical distribution of random variables means
  that each random variable comes from the same
  population (e.g., roll of a die, coin flip).

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Simple random sampling

• Each observation drawn does not depend on others
  drawn
• Thus observations are independent
• Each observation (i.e., each random variable) is
  identically distributed
• The population has a distribution that doesnt
  change (each observation is randomly drawn from
  an identical distribution the distribution of
  the population).
• So the Central Limit Theorem applies! (when n is
  large)

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What does this mean?
Suppose we take a sample of n50 observations
from a population that has this distribution
frequency
0
10
20
30
Mean (µ) 20
2
Variance ( ) 100 Std. dev ( ) 10
We then find the mean of this sample (suppose
this mean 19). Take another sample of 50
observations and find the mean (suppose its 24).
Do this many times, and well come up with a
distribution of means. The Central Limit Theorem
tells us this distribution will always look like
the next slide (as long as n is large, and 50
is large enough)
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The normal curve
20
24
16
18
22
Mean (µ) 20 Sample size (n) 50
variance of sample mean 2
8
Symbols

• Population Parameter
• Estimate
• Expected

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Basic Types of Inference

• Point Inference
• The value of a population parameter is
  estimated using a single value
• Examples mean, standard deviation, etc.
• Interval Inference
• Attaching a probability to an estimate (i.e.,
  making a confidence interval)
• Example we are 95 confident that µ is between
  10 and 20

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Judging the Quality of the Estimator

• Bias the difference between and
  (i.e., )
• Bias may be positive or negative (e.g., a
  positively biased estimator would indicate the
  population parameter is higher than it actually
  is)
• Efficiency how clustered the distribution of
  is (i.e., how peaked is its distribution)

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Judging the Quality of the Estimator

• Best case scenario to have an unbiased
  estimator, with a high level of efficiency
• We can measure the quality of the estimator using
  the Mean Squared Error (MSE) or its counterpart
  RMSE (the square root of the MSE)
• Remember that the variance in this case it the
  variance of a random variable so we use the
  equation

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Point Estimates (inferring population parameters
from samples)

• Population Mean
• Population Proportions
• Population Variance
• Population Standard Deviation

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Confidence Intervals

• The degree of confidence we have in our estimates
  defined by a percentage
• Common examples 90, 95, or 99 confident
• The confidence interval is defined with the a
  symbol
• In confidence intervals, alpha (a) is the
  proportion of time your confidence interval is
  wrong
• The typical usage is
• Why do we divide by 2?

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Confidence Interval Example

• What is the 95 confidence interval for a
  normally distributed variable?
• a 1 - desired confidence interval
• a 1 0.95 0.05
• Remember that we divide a by 2 since we have
  uncertainty both above and below the mean (i.e.,
  2 tails)
• Therefore we use z0.025 for the 95 confidence
  interval
• From the z-table we find that z0.025 1.96
• What does this mean?

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Interval Estimation (making confidence intervals
for population parameters estimated from samples)

• Case 1 estimating an interval for µ when X is
  normally distributed and we know s
• This is the simplest case because normality
  allows us to use the z-table
• This is also unlikely since it requires knowing
  the distribution and the s (which implies knowing
  µ already)

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Example 1 Create a confidence interval for µ

• A town is considering building a new bridge over
  a river. The primary goal is to reduce workers
  commute times from a particular community. A
  random sample of workers in that community are
  asked to estimate their reduction in commute time
  if the bridge were built. Our goal is to
  estimate the mean reduction in commute time for
  the whole community if the bridge were built.
  Create a 95 confidence interval for this mean.

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Example 1 Data

• n 100 workers are sampled
• x 17 minutes
• s 30 minutes
• What is the 95 confidence interval for the mean?

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Constructing a confidence interval

• Construct a 95 confidence interval around the
  sample mean
• So we can say that the 95 C.I. is 17 /- 5.88 or
  11.12, 22.88

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Example 1 Questions

• What would happen to our interval if we used a
  99 confidence interval instead?
• What would happen to our confidence interval if
  we sampled 200 people instead of 100 people?

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Interval Estimation (making confidence intervals
for population parameters estimated from samples)

• Case 2 estimating an interval for µ when X is
  not normally distributed and we know s
• In this case the n matters a lot, why?
• This is also unlikely since it requires knowing
  the distribution and the s (which implies knowing
  µ already)

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Interval Estimation (making confidence intervals
for population parameters estimated from samples)

• Case 3 estimating an interval for µ when s and
  the distribution are unknown
• What should we used instead of s?
• Can we use the z-table in this case?
• This case is what we see most commonly

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t-distribution vs. z-distribution

• When we only have s (and not s) we use the
  t-distribution rather than the z-distribution
• To do so we use the t-table
• How are they different?
• The t-distribution changes depending on the
  degrees of freedom (n-1)
• This is reflected in the table and in the symbol
• The t-distribution accounts for more uncertainty
  (i.e., wider confidence intervals) since s is
  just an estimate for s

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t-distribution vs. z-distribution

• As n approaches infinity t and z become equal
• This means that even when we have s instead of s
  we can use the z-distribution if n is large
• Central Limit Theorem as n gets large.
• What is large?
• Rule of thumb 30
• For n less than 30, the distribution of x does
  not follow the normal distribution accurately
  enough.
• But the distribution of x does closely follow a
  t-distribution for sample sizes of less than 30.
• For this class use the t-distribution any time
  you have s instead of s

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Example 2

• n 16
• x 30
• s2 1600
• What is the 95 C.I. for the mean?

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Example 2

• s 40
• Degrees of freedom n 1 15
• 
  (from the t-table)
• The 95 confidence interval for the mean is
  (8.69, 51.31)

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Interval Estimation (making confidence intervals
for population parameters estimated from samples)

• Case 4 estimating an interval for a proportion p
  based on a sample proportion p
• Remember that p x/n
• In other word, p the number of successes
  divided by the number of samples
• For example the proportion of people over 6ft
  tall
• In this case we dont need s or s, but we do need
  the standard deviation of p
  
• Which we estimate as

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Interval Estimation (making confidence intervals
for population parameters estimated from samples)

• Case 4 continued
• Equation
• We use the z-distribution for estimating an
  interval for a proportion p based on a sample
  proportion p
• This also limits us to using only large samples
  (in this case n gt 100)
• For smaller samples, we calculate the entire
  distribution using the binomial mass function
  (i.e., solve for all
  x values)

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Example 3

• n 150 people at a convention
• 63 people sampled were over 6 feet tall
• What is the 99 C.I. for the true proportion of
  all people 6 ft tall at the convention?

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Example 3

• p 63/150 0.42
• 99 C.I. -gt (from the z-table)
• The 99 confidence interval for p 0.42 is
  (0.316, 0.524)

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Sample Size Determination

• Often, before we conduct a sample, we want to
  know how large of a sample we need
• Required sample sizes can be determined for
  population parameters (mean, proportions, etc.)
  by modifying the equations weve been going
  through
• An additional component is the error (E)
• This is basically the term that defines how far
  off we are willing to be (i.e., the margin of
  acceptable error)
• Strictly speaking, E is one-half the difference
  between the upper and lower values for an
  interval for a given C.I.
• Note that E is not the same as C.I.

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Sample Size Determination

• Equation for µ
• Equation for p
• What obvious flaw do you see?

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Example 4

• A movie theatre wants to know the mean number of
  tickets sold per day. How many days must they
  count to know the mean daily ticket sales within
  100 tickets with a 95 confidence interval?
• From previous sales reports, it is determined
  that s 175

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Example 4

• What numbers do we plug into our equation?
• What should zalpha/2 be?
• What should E be?
• Why dont we multiply this by 2?
• What should s be?

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Example 4

• z 1.96
• E 100
• s 175
• n number of days we should sample

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Example 5

• A city council election is being held with
  several candidates expecting reasonably large
  returns.
• To avoid a run-off between the top 2 vote
  getters, the leading candidate must receive at
  least 45 of the vote
• How many people do we need to sample using exit
  polls to determine with 99 confidence and an
  acceptable error of 0.005 whether there will be a
  run-off vote?

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Example 5

• z 2.58
• E 0.005
• p 0.45
• n number of people we should sample

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Class Problem

• Given this sample of middle school kid heights
  (in inches)
• 56, 64, 52, 69, 66, 64, 63, 46, 46, 49, 47, 60,
  54, 45, 45, 69, 62, 67, 49, 43, 59
• What is the 99 confidence interval for the
  population mean (µ)?

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Solution

• n 21
• x 1175/21 55.95
• s 8.96
• talpha/2 , n-1 2.845
• So the 99 C.I. for the population mean (µ) is
  50.387, 61.513

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For Friday

• Come with questions about homework 6

For Monday

• Read chapter 9 pages 280-306