Chapter 11 some thoughts

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Lecture 2

• Chapter 11 (some thoughts)
• Type II error calculation
• Additional example
• Some more information on Hypothesis Testing
• Type II error
• Practical significance vs. Statistical
  significance
• Inference about mean when sigma is unknown
  (Chapter 12).

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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.

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Right-, Left, Two-Sided Tests

• Right-sided
• Left-sided
• 2-sided

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Frequent -values
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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

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Calculating the Probability of a Type II Error

• To properly interpret the results of a test of
  hypothesis, we need to
• specify an appropriate significance level or
  judge the p-value of a test
• understand the relationship between Type I and
  Type II errors.
• How do we compute a type II error?

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Calculation of the Probability of a Type II
Error

• A Type II error occurs when a false H0 is not
  rejected.
• To calculate Type II error we need to
• express the rejection region directly, in terms
  of the parameter hypothesized (not standardized).
• specify the alternative value under H1.
• Let us revisit Example 11.1

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Calculation of the Probability of a Type II
Error

• Let the alternative value be m 180 (rather than
  just mgt170)

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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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Judging the Test

• A hypothesis test is effectively defined by the
  significance level a and by the sample size n.
• A measures of effectiveness is the probability of
  Type II error. Typically we want to keep the
  probability of Type II error as small as
  possible.
• If the probability of a Type II error b is judged
  to be too large, we can reduce it by
• increasing a, and/or
• increasing the sample size.

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Judging the Test

• Increasing the sample size reduces b

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Judging the Test

• In Example 11.1, suppose n increases from 400 to
  1000.

• a remains 5, but the probability of a Type II
  drops dramatically.

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Judging the Test

• An alternate measure of effectiveness of tests
  that is often used is the Power of a test
• The power of a test is defined as 1 - b.
• It represents the probability of rejecting the
  null hypothesis when it is false.

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Planning Studies

• Power calculations are important in planning
  studies.
• Using a hypothesis test with low power makes it
  unlikely that you will reject H0 even if the
  truth is far from the null hypothesis.
• Operating characteristic curve is a plot of
  versus values of from the alternative for
  a fixed sample size n and a fixed significance
  level

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Operating Characteristic Curve for Example 11.1
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Problem 11.54

• Many Alpine ski centers base their projections of
  revenues
• and profits on the assumption that the average
  Alpine skier
• skis 4 times per year.
• To investigate the validity of this assumption, a
  random
• sample of 63 skiers is drawn and each is asked to
  report the
• number of times they skied the previous year.
• Assume that the population standard deviation is
  2, and the
• sample mean is 4.84. Can we infer at the 10
  level that the
• assumption is wrong?

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Problem 11.54 (Contd.)

• What is the probability of making a Type II error
  if the average Alpine skier skis 4.2 times per
  year?

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Problem Effects of SAT Coaching

• Suppose that SAT mathematics scores in the
  absence of coaching have a normal distribution
  with 475 and standard deviation 100. Suppose
  further that coaching may change the mean but not
  the standard deviation. Calculate the p-value
  for the test of versus
  
• for each of the following three situations
• (a) A coaching service coaches 100 students
  their SAT-M scores average
• (b) By the next year, the coaching service
  has coached 1000 students their SAT-M scores
  average
• (c) An advertising campaign brings the total
  number of students coached to 10,000 their
  average score is still
• 
  
  
  
  

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Practical Significance vs. Statistical
Significance

• An increase in the average SAT-M score from 475
  to 478 is of little importance in seeking
  admission to college, but remember a large sample
  size will always declare very small effects
  statistically significant.
• A confidence interval provides information about
  the size of the effect and should always be
  reported. The two-sided 95 confidence intervals
  for the SAT coaching problem are
  . Thus, for
• (a) (458.4,497.6)
• (b) (471.8,484.2)
• (c) (476.04,479.96).
• For large samples, the CI says Yes, the mean
  score is higher after coaching but only by a
  small amount.

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Chapter 12

• In this chapter we utilize the approach developed
  before to describe a population.
• Identify the parameter to be estimated or tested.
• Specify the parameters estimator and its
  sampling distribution.
• Construct a confidence interval estimator or
  perform a hypothesis test.

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Inference for Population Mean When Sigma is
Unknown

• Recall that when s is known we use the
  following
• statistic to estimate and test a population
  mean
• When s is unknown, we use its point estimator s,
  and the z-statistic is replaced then by the
  t-statistic

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t-Statistic

• When the sampled population is normally
  distributed, the t statistic has a Student t
  distribution with n-1 degrees of freedom.
• Then a 100 (1- ) confidence interval is
  given by
• where is the quantile of
  the Student t-distribution with n-1 degrees of
  freedom.

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The t - Statistic
t
s
The degrees of freedom, (a function of the
sample size) determine how spread
the distribution is (compared to the normal
distribution)
The t distribution is mound-shaped, and
symmetrical around zero.
d.f. v2
d.f. v1
v1 lt v2
0
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tA
t.100
t.05
t.025
t.01
t.005
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Example 12.1

• In order to determine the number of workers
  required to meet demand, the productivity of
  newly hired trainees is studied.
• It is believed that trainees can process and
  distribute more than 450 packages per hour within
  one week of hiring.
• Fifty trainees were observed for one hour. In
  this sample of 50 trainees, the mean number of
  packages processed is 460.38 and s38.82.
• Can we conclude that the belief is correct, based
  on the productivity observation of 50 trainees?

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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.
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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