Chapter 18 Part 1

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Chapter 18 -- Part 1

• Sampling Distribution Models for

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Sampling Distribution Models
Population Parameter?
Population
Inference
Sample Statistic
Sample
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Objectives

• Describe the sampling distribution of a sample
  proportion
• Understand that the variability of a statistic
  depends on the size of the sample
• Statistics based on larger samples are less
  variable

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Review

• Chapter 12 Sample Surveys
• Parameter (Population Characteristics)
• m (mean)
• p (proportion)
• Statistic (Sample Characteristics)
• (sample mean)
• (sample proportion)

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Review

• Chapter 12
• Statistics will be different for each sample.
  These differences obey certain laws of
  probability (but only for random samples).
• Chapter 14
• Taking a sample from a population is a random
  phenomena. That means
• The outcome is unknown before the event occurs
• The long term behavior is predictable

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Example

• Who? Stat 101 students in Sections G and H.
• What? Number of siblings.
• When? Today.
• Where? In class.
• Why? To find out what proportion of students
  have exactly one sibling.

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Example

• Population
• Stat 101 students in sections G and H.
• Population Parameter
• Proportion of all Stat 101 students in sections G
  and H who have exactly one sibling.

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Example

• Sample
• 4 randomly selected students.
• Sample Statistic
• The proportion of the 4 students who have exactly
  one sibling.

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Example

• Sample 1
• Sample 2
• Sample 3

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What Have We Learned

• Different samples produce different sample
  proportions.
• There is variation among sample proportions.
• Can we model this variation?

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Example

• Senators Population Characteristics
• p proportion of Democratic Senators
• Take SRS of size n 10
• Calculate Sample Characteristics
• sample proportion of Democratic Senators

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Example
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SRS characteristics

• Values of and are random
• Change from sample to sample
• Different from population characteristics
• p 0.50

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Imagine

• Repeat taking SRS of size n 10
• Collection of values for and ARE DATA
• Summarize data make a histogram
• Shape, Center and Spread
• Sampling distribution for

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Sampling Distribution for

• Mean (Center)
• We would expect on average to get p.
• Say is unbiased for p.

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Sampling Distribution for

• Standard deviation (Spread)
• As sample size n gets larger, gets smaller
• Larger samples are more accurate

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Example

• 50 of people on campus favor current academic
  calendar.
• 1. Select n people.
• 2. Find sample proportion of people favoring
  current academic calendar.
• 3. Repeat sampling.
• 4. What does sampling distribution of sample
  proportion look like?

• n2
• n5
• n10
• n25

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Example

• 10 of all people are left handed.
• 1. Select n people.
• 2. Find sample proportion of left handed people.
• 3. Repeat sampling.
• 4. What does sampling distribution of sample
  proportion look like?

• n2
• n10
• n50
• n100

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Sampling Distribution for

• Shape
• Normal Distribution
• Two assumptions must hold in order for us to be
  able to use the normal distribution
• The sampled values must be independent of each
  other
• The sample size, n, must be large enough

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Sampling Distribution for

• It is hard to check that these assumptions hold,
  so we will settle for checking the following
  conditions
• 10 Condition the sample size, n, is less than
  10 of the population size
• Success/Failure Condition np gt 10, n(1-p) gt 10
• These conditions seem to contradict one another,
  but they don't!

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Sampling Distribution for

• Assuming the two conditions are true (must be
  checked for each problem), then the sampling
  distribution for is

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Sampling Distribution for

• But the sampling distribution has a center (mean)
  of p (a population proportion) often times we
  dont know p.
• Let be the center.

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Example

• Senators
• Check assumptions (p 0.50)
• 10(0.50) 5 and 10(0.50) 5
• n 10 is 10 of the population size.
• Assumption 1 does not hold.
• Sampling Distribution of ????

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Example 1

• Public health statistics indicate that 26.4 of
  the U.S. adult population smoked cigarettes in
  2002. Use the 68-95-99.7 Rule to describe the
  sampling distribution for the sample proportion
  of smokers among 50 adults.

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Example 1

• Check assumptions
• np (50)(0.264) 13.2 gt 10
• nq (50)(0.736) 36.8 gt 10
• n 50, less than 10 of population
• Therefore, the sampling distribution for the
  proportion of smokers is

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Example 1

• About 68 of samples have a sample proportion
  between 20.2 and 32.6
• About 95 of samples have a sample proportion
  between 14 and 38.8
• About 99.7 of samples have a sample proportion
  between 7.8 and 45

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Example 2

• Information on a packet of seeds claims that the
  germination rate is 92. What's the probability
  that more than 95 of the 160 seeds in the packet
  will germinate?
• Check assumptions 1. np (160)(0.92) 147.2 gt
  10 nq (160)(0.08) 12.8 gt 10 2. n 160,
  less than 10 of all seeds?

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Review - Standardizing

• You can standardize using the formula

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Review

• Chapter 6 The Normal Distribution
• Y N(70,3)
• Do you remember the 68-95-99.7 Rule?

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Example 2

• Therefore, the sampling distribution for the
  proportion of seeds that will germinate is

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Big Picture
Population Parameter?
Population
Inference
Sample Statistic
Sample
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Big Picture

• Before we would take one random sample and
  compute our sample statistic. Presently we are
  focusing on
• This is an estimate of the population parameter
  p.
• But we realized that if we took a second random
  sample that from sample 1 could possibly be
  different from the we would get from sample
  2. But from sample 2 is also an estimate of
  the population parameter p.
• If we take a third sample then the for third
  sample could possibly be different from the first
  and second s. Etc.

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Big Picture

• So there is variability in the sample statistic
  .
• If we randomized correctly we can consider as
  random (like rolling a die) so even though the
  variability is unavoidable it is understandable
  and predictable!!! (This is the absolutely
  amazing part).

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Big Picture

• So for a sufficiently large sample size (n) we
  can model the variability in with a normal
  model so

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Big Picture

• The hard part is trying to visualize what is
  going on behind the scenes. The sampling
  distribution of is what a histogram would
  look like if we had every possible sample
  available to us. (This is very abstract because
  we will never see these other samples).
• So lets just focus on two things

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Take Home Message

• 1. Check to see that
• A. the sample size, n, is less than 10 of the
  population size
• B. np gt 10, n(1-p) gt 10
• 2. If these hold then can be modeled with a
  normal distribution that is

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Example 3

• When a truckload of apples arrives at a packing
  plant, a random sample of 150 apples is selected
  and examined for bruises, discoloration, and
  other defects. The whole truckload will be
  rejected if more than 5 of the sample is
  unsatisfactory (i.e. damaged). Suppose that
  actually 8 of the apples in the truck do not
  meet the desired standard. What is the
  probability of accepting the truck anyway?

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Example 3

• What is the sampling distribution?
• np (150)(0.08) 12gt10nq (150)(0.92)
  138gt10
• n 150 gt 10 of all applesSo, the sampling
  distribution is N(0.08,0.022).What is the
  probability of accepting the truck anyway?