Lecture 4, Mon Jan 27

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Lecture 4, Mon Jan 27

• Chapter 11 (some thoughts)
• Inference about the mean when sigma unknown
  (Contd.)
• JMP-IN
• Inference about a
• population variance
• population proportion (review)
• Sample size determination (review)

2
Hypothesis Testing Basic Steps

• Set up alternative and null hypotheses
• Calculate test statistic, e.g. z-score
• Find critical values and compare the test
  statistic to critical value (rejection region
  method) or find p-value (p-value method)
• Make substantive conclusions.

3
Right-, Left, Two-Sided Tests

• Right-sided
• Left-sided
• 2-sided

4
Relationship Between CIs and Hypothesis Tests

• There is a duality between confidence intervals
  and hypothesis tests
• We can construct a level hypothesis test
  based on a level confidence
  interval by rejecting if and
  only if is not in the confidence interval
• We can construct a level
  confidence interval based on a level
  hypothesis test by including in the
  confidence interval if and only if the test does
  not reject

5
Calculation of Type II error

• State alternative for which you want to find
  P(Type II error).
• Find rejection region in terms of un-standardized
  statistic (sample mean)
• Construct standardized values for the rejection
  region using the value for mean from the
  alternative hypothesis.
• Find the probability of the sample mean falling
  outside the rejection region.

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Frequent -values
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12.2 Inference About a Population Mean When the
Population Standard Deviation Is Unknown

• When the sampled population is normally
  distributed, the t statistic is Student t
  distributed with n-1 degrees of freedom.
• Confidence Interval
  where is the quantile of
  the Student t-distribution with n-1 degrees of
  freedom.

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Checking the required conditions

• In deriving the test and confidence interval, we
  have made two assumptions
• (i) the sample is a random sample from the
  population
• (ii) the distribution of the population is
  normal.
• The t test is robust the results are still
  approximately valid as long as
• (i)the population is not extremely non-normal.
  
• (ii) or, if the sample size is large.

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A rough graphical approach to examining normality
is to look at the sample histogram.
10
JMP Example

• Problem 12.45 Companies that sell groceries over
  the Internet are called e-grocers. Customers
  enter their orders, pay by credit card, and
  receive delivery by truck. A potential e-grocer
  analyzed the market and determined that to be
  profitable the average order would have to exceed
  85. To determine whether an e-grocer would be
  profitable in one large city, she offered the
  service and recorded the size of the order for a
  random sample of customers. Can we infer from
  the data than e-grocery will be profitable in
  this city at significance level 0.05?

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12.3 Inference About a Population Variance

• Sometimes we are interested in making inference
  about the variability of processes.
• Examples
• The consistency of a production process for
  quality control purposes.
• Investors use variance as a measure of risk.
• To draw inference about variability, the
  parameter of interest is s2.

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Inference About a Population Variance

• The sample variance s2 is an unbiased, consistent
  and efficient point estimator for s2.
• The statistic has a
  distribution called Chi-squared, if the
  population is normally distributed.

d.f. 5
d.f. 10
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C.I for Population Variance

• From the following probability statement P(c21-
  a/2 lt c2 lt c2a/2) 1-a
• we obtain the (by substituting c2 (n -
  1)s2/s2)
• the confidence interval
  as

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Testing the Population Variance

• Example 12.3 (operation management application)
• A container-filling machine is believed to fill 1
  liter containers so consistently, that the
  variance of the filling will be less than 1 cc
  (.001 liter).
• To test this belief a random sample of 25 1-liter
  fills was taken, and the results recorded
  (Xm12-03). s20.8659.
• Do these data support the belief that the
  variance is less than 1cc at 5 significance
  level?
• Find a 99 confidence interval for the variance
  of fills.

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Two-sided test - JMP
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12.4 Inference About a Population Proportion

• When the population consists of nominal data
  (e.g., does the customer prefer Pepsi or Coke),
  the only inference we can make is about the
  proportion of occurrence of a certain value.
• When there are two categories (success and
  failure), the parameter p describes the
  proportion of successes in the population. The
  probability of obtaining X successes in a random
  sample of size n from a large population can be
  calculated using the binomial distribution.

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12.4 Inference About a Population Proportion

• Statistic and sampling distribution
• the statistic used when making inference about p
  is

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Testing and Estimating the Proportion

• Test statistic for p

• Interval estimator for p (1-a confidence level)

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Testing the Proportion

• Example 12.5 (Predicting the winner in election
  day)
• Voters are asked by a certain network to
  participate in an exit poll in order to predict
  the winner on election day.
• The exit poll consists of 765 voters. 407 say
  that they voted for the Republican network.
• The polls close at 800. Should the network
  announce at 801 that the Republican candidate
  will win?

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Selecting the Sample Size to Estimate the
Proportion

• Recall The confidence interval for the
  proportion is
• Thus, to estimate the proportion to within W, we
  can write
• The required sample size is

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Sample Size to Estimate the Proportion

• Example
• Suppose we want to estimate the proportion of
  customers who prefer our companys brand to
  within .03 with 95 confidence.
• Find the sample size needed.
• Solution
• W .03 1 - a .95,
• therefore a/2 .025,
• so z.025 1.96

Since the sample has not yet been taken, the
sample proportion is still unknown.
We proceed using either one of the following two
methods
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Sample Size to Estimate the Proportion

• Method 1
• There is no knowledge about the value of
• Let . This results in the largest
  possible n needed for a 1-a
  confidence interval of the form .
• If the sample proportion does not equal .5, the
  actual W will be narrower than .03 with the n
  obtained by the formula below.
• Method 2
• There is some idea about what will turn out
  to be.
• Use a probable value of to calculate the
  sample size

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Practice Problems

• 12.40, 12.46, 12.58, 12.77, 12.98