Unit IV: Introduction To Inferential Statistics

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Unit IV Introduction To Inferential Statistics

• Sampling Distributions
• Confidence Intervals (Mean Proportion)
• Hypothesis Testing

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What is Inferential Statistics?

• The branch of Statistics which allows us to draw
  conclusions and/or make decisions concerning a
  population based only on sample data

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Inferential Statistics

• Sample statistics Population
  parameters
• (known) Inference
  (unknown, but can
• be estimated from
• sample evidence)

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SAMPLING DISTRIBUTIONS

• What is a Sampling Distribution?
• Sampling Distribution of the Mean
• Central Limit Theorem
• Sampling Distribution of Proportions

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What is a Sampling Distribution? I

• Recall
• A statistic is defined as a numerical quantity
  calculated in a sample
• Some points to remember
• A random sample should represent the population
  well, so sample statistics from a random sample
  should provide reasonable estimates of population
  parameters
• All sample statistics have some error in
  estimating population parameters
• Because sample measurements are observed values
  of random variables, the value for a sample
  statistic will vary in a random manner from
  sample to sample. In other words, since sample
  statistics are random variables, they possess
  probability distributions
• A larger sample provides more information than a
  smaller sample so a statistic from a large sample
  should have less error than a statistic from a
  small sample

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What is a Sampling Distribution? II

• The sampling distribution of a sample statistic
  calculated from a sample of n measurements is the
  probability distribution of the statistic.
• The probability distribution of a statistic is
  called its sampling distribution.

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Definitions

• An estimator of a population parameter is a
  sample statistic used to estimate or predict the
  population parameter.
• An estimate of a parameter is a particular
  numerical value of a sample statistic obtained
  through sampling.
• A point estimate is a single value used as an
  estimate of a population parameter.

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Estimators
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Exercise

• Random samples of size 2 are drawn, without
  replacement, from the finite population which
  consists of the numbers 4, 5, 6 and 7.
• Find the mean and variance of this population.
• Find the probability distribution for the mean
  for random samples of size 2 drawn without
  replacement.
• Find the mean of the probability distribution.
• Find the variance of the probability distribution
  of means.

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Sampling Distribution of the Mean

• For the sampling distribution of
• The mean
• Is denoted by
• Is always equal to the population mean
• Since E( ) µ, is called an unbiased
  estimator of µ
• The standard deviation
• Is denoted by
• Is equal to

Sampling Without Replacement if sample size is
relatively small
Sampling With Replacement
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The Central Limit Theorem

• Sometimes the population or the sample size may
  be too large thus making it extremely tedious to
  list out all the possible samples.
• When many samples are taken from the same
  population, the distribution of values for the
  sample mean are centred around the population
  mean (regardless of sample size)
• As the sample size increases the mean of the
  means are closer to the population mean
• The standard deviation of the sample mean
  decreases as the sample size increases
• The distribution of the sample mean becomes more
  symmetrical as the sample size gets larger and
  becomes approximately normal for large sample
  sizes

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The Central Limit Theorem (contd)

• The central limit theorem therefore states that
• If the sample size (n) is large enough, the
  sample mean ( ) has a normal distribution
  with mean µ and standard deviation
  regardless of the population distribution.
• A large enough sample is where n 30

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CLT The Effect of Sample Size I
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CLT The Effect of Sample Size II
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Finding Probabilities for Sampling Distributions

• Step 1 Standardize the values to be found using
• or
• Step 2 Find probabilities as usual using the
  standardized values

For the Mean
For the Proportion
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Example I Sampling Distribution of the Mean

• A manufacturer of automobile batteries claims
  that the distribution of the lengths of life of
  its battery has a mean of 54 months and a
  standard deviation of 6 months. Suppose a
  consumer group decides to check the claim by
  purchasing a sample of 50 of these batteries and
  subjecting them to tests that determine battery
  life.
• Assuming that the manufacturers claim is true,
  describe the sampling distribution of the mean
  lifetime of a sample of 50 batteries.
• Assuming that the manufacturers claim is true,
  what is the probability that the consumer groups
  sample has a mean life of 52 or fewer months?

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Example II Sampling Distribution of the Mean

• Certain light bulbs manufactured by a company
  have a mean lifetime of 800 hours and a standard
  deviation of 60 hours. Find the probability that
  a random sample of 64 light bulbs taken from a
  production batch will have a mean lifetime of
• Less than 785 hours
• More than 820 hours
• Between 800 and 810 hours
• Between 770 and 830 hours

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Sampling Distribution of the Proportion

• The sample proportion is the percentage of
  successes in n binomial trials. It is the number
  of successes, X, divided by the number of trials,
  n.
• Sample Proportion
• As the sample size, n, increases, the sampling
  distribution of approaches a normal
  distribution with mean p and standard deviation

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Example Sampling Distribution of the Proportion

• In recent years, convertible sports coupes have
  become very popular in Japan. Toyota is
  currently shipping Celicas to Los Angeles, where
  a customiser does a roof lift and ships them back
  to Japan. Suppose that 25 of all Japanese in a
  given income and lifestyle category are
  interested in buying Celica convertibles. A
  random sample of 100 Japanese consumers in the
  category of interest is to be selected. What is
  the probability that at least 20 of those in the
  sample will express an interest in a Celica
  convertible?

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CONFIDENCE INTERVALS

• Estimation Estimators
• What is a Confidence Interval?
• Confidence Intervals for Means
• Confidence Intervals for Proportions

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Estimation

• There are two types
• Point Estimation
• Interval Estimation
• Estimation act of estimating a specific value
  in the population from the sample
• Estimate a specific statistic to estimate the
  value of the parameter
• Estimator a specific value calculated from the
  sample which is used to find the estimate

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Types of Estimates

• A point estimate is a single number,
• A confidence interval provides a range of values
  for estimating a particular population parameter,
  it therefore provides additional information
  about variability

Upper Confidence Limit
Lower Confidence Limit
Point Estimate
Width of confidence interval
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Deficiencies of Point Estimation

• A specific point estimate is not likely to be
  exact because it is one among many possible point
  estimators
• It provides no assessment of the probability that
  a sample point estimate value is reasonably close
  to the parameter being estimated
• An interval estimate provides more information
  about a population characteristic than does a
  point estimate

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What is a Confidence Interval?

• A confidence interval is a range of guesses at a
  population value
• Means
• Proportions
• It is a type of interval estimation
• The confidence level is that chance (probability)
  that the range of values captures the true
  population value (or will contain the unknown
  population parameter)
• The general formula for a confidence interval is

Point Estimate (Critical Value)(Standard
Deviation of the Point Estimate)
Aka the Standard Error
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Confidence Level

• Confidence
• A number between 0 and 100 that reflects the
  probability that the interval estimate will
  include the parameter
• A high confidence is desired (between 90 and
  99.7)
• Higher confidence levels require wider intervals

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Estimation Process
Random Sample
Population
Mean X 50
(mean, µ, is unknown)
Sample
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Confidence Interval for µ

• Confidence Interval Estimate
• 
  
• 
  or
• where is the point estimate
• Z is the normal distribution critical value
  for a probability of ?/2 in each tail
• is the standard error

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Finding the Critical Value, Z

• Consider a 95 confidence interval

Z -1.96
Z 1.96
Z units
0
Lower Confidence Limit
Upper Confidence Limit
X units
Point Estimate
Point Estimate
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Common Levels of Confidence

• Commonly used confidence levels are 90, 95, and
  99

Confidence Coefficient,
Confidence Level
Z value
1.28 1.645 1.96 2.33 2.58 3.08 3.27
0.80 0.90 0.95 0.98 0.99 0.998 0.999
80 90 95 98 99 99.8 99.9
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Steps in Calculating Confidence Intervals for
Means

• Step 1 Calculate the mean for the sample
• Step 2 Calculate the square root of the variance
  divided by the sample size
• Step 3Calculate the critical value
• Step 4 Apply the formula

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Example - Confidence Interval for the Population
Means I

• A sample of 35 circuits has a mean resistance of
  2.20 ohms. We know from past testing that the
  population standard deviation is 0.35 ohms.
• Determine a 95 confidence interval for the true
  mean resistance of the population.

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Example - Confidence Interval for the Population
Means II

• The real estate assessor for Kingston wants to
  study various characteristics of single-family
  houses in the parish. A random sample of 70
  houses reveals the following
• Area of the house in square feet x-bar 1759, s
  380.
• Construct a 99 confidence interval estimate of
  the population mean area of the house.

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Example Confidence Interval for Means III

• A publishing company has just published a new
  college textbook. Before the company decides the
  price at which to sell this textbook, it wants to
  know the average price of all such textbooks in
  the market. The research department at the
  company took a sample of 36 comparable textbooks
  and collected information on their prices. This
  information produces a mean price of 70.50 for
  this sample. It is known that the standard
  deviation of the prices of all such textbooks is
  4.50.
• What is the point estimate of the mean price of
  all such textbooks?
• Construct a 90 confidence interval for the mean
  price of all such college textbooks.

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Confidence Intervals for the Population
Proportion, p

• An interval estimate for the population
  proportion (p) can be calculated by adding an
  allowance for uncertainty to the sample
  proportion

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Confidence Intervals for the Population
Proportion, p
(continued)

• Recall that the distribution of the sample
  proportion is approximately normal if the sample
  size is large, with standard deviation
• We will estimate this with sample data

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Confidence Intervals for the Population
Proportion, p
(continued)

• Upper and lower confidence limits for the
  population proportion are calculated with the
  formula
• where
• is the sample proportion
• n is the sample size
• Z is the normal distribution critical value for
  a probability of ?/2 in each tail

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Example Confidence Interval for Proportions I

• A random sample of 100 people shows that 25 are
  left-handed.
• Form a 95 confidence interval for the true
  proportion of left-handers

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Example Confidence Interval for Proportions II

• The real estate assessor for Kingston wants to
  study various characteristics of single-family
  houses in the parish. A random sample of 70
  houses reveals the following
• 42 houses have central air-conditioning
• Set up a 95 confidence interval estimate of the
  population proportion of houses that have central
  air-conditioning

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Sampling Error

• The required sample size can be found to reach a
  desired margin of error (e) with a specified
  level of confidence (1 - ?)
• The margin of error is also called sampling error
• the amount of imprecision in the estimate of the
  population parameter
• the amount added and subtracted to the point
  estimate to form the confidence interval

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Determining Sample Size
Determining
Sample Size
For the Mean
Sampling error (margin of error)
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Determining Sample Size
(continued)
Determining
Sample Size
For the Mean
Now solve for n to get
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Determining Sample Size
(continued)

• To determine the required sample size for the
  mean, you must know
• The desired level of confidence (1 - ?), which
  determines the critical Z value
• The acceptable sampling error, e
• The standard deviation, s

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Example - Sample Size Determination (Mean) I

• If ? 45, what sample size is needed to estimate
  the mean within 5 with 90 confidence?

So the required sample size is n 220
(Always round up)
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Example - Sample Size Determination (Mean) II

• A department store wishes to estimate, with a
  confidence level of 98 and a maximum error of
  5, the true mean value of purchases per month of
  its customers. Determine the minimum size of
  the sample that is required to ensure this, given
  that the standard deviation is 15.

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Example - Sample Size Determination (Mean) III

• An alumni association wants to estimate the mean
  debt of this years university graduates. It is
  known that the population standard deviation of
  debts of this years college graduates is
  11,800. How large a sample should be selected
  so that the estimate with a 99 confidence level
  is within 800 of the population mean?

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Determining Sample Size
(continued)
Determining
Sample Size
For the Proportion
Now solve for n to get
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Example - Sample Size Determination (Proportion) I

• How large a sample would be necessary to estimate
  the true proportion of defectives in a large
  population within 3, with 95 confidence?
• (Assume a pilot sample yields 0.12)

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Example - Sample Size Determination (Proportion)
II

• A consumer agency wants to estimate the
  proportion of all drivers who wear seatbelts
  while driving. Assume that a preliminary study
  has shown that 76 of drivers wear seatbelts
  while driving. How large should the sample be so
  that the 99 confidence interval for the
  population proportion has a maximum error of 0.03?

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Example - Sample Size Determination (Proportion)
III

• A preliminary sample of 200 parts produced by a
  new machine showed that 7 of them are defective.
  How large a sample should the company select so
  that the 95 confidence interval for p is within
  0.02 of the population proportion.