Statistics and Quantitative Analysis U4320

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Statistics and Quantitative Analysis U4320

• Segment 7
• Hypothesis Testing
• Prof. Sharyn OHalloran

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Hypothesis Testing

• I. Introduction
• A. Review of Confidence Intervals

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Introduction (cont.)

• B. Hypothesis Testing Basic Definitions
• 1. A Hypotheses is a statement about the
  population
• 2. Null Hypothesis
• The Null Hypothesis (Ho)- the statement about
  our data that we want to test.
• It is always stated as an equality. For
  instance
• Ho m 82, where m is the average test score
• Or, H0 D 0, where D is the difference between
  men's and women' salaries is zero.

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Introduction (cont.)

• 3. Alternative Hypothesis
• Every Null Hypothesis has an associated
  Alternative Hypothesis, denoted Ha.
• This is always stated as an inequality either ?,
  gt, or lt.
• For instances, the alternative hypothesis to the
  test scores having a mean of 82 might be Ha m ?
  82.
• The alternative hypothesis to men's and women's'
  salaries being equal might be Ha D gt 0.

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Introduction (cont.)

• 4. One Tail vs. Two Tail Tests
• If the alternative hypothesis is in terms of a ?
  sign, it is called a two-tailed test.
• If the alternative hypothesis is in terms of a lt
  or gt sign, it is called a one-tailed test.

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Introduction (cont.)

• C. Three Methods for Testing Hypothesis
• 1. Method I Testing hypotheses using confidence
  intervals.
• 2. Method II Testing hypotheses using p-values.
  
• 3. Method III Testing hypotheses using critical
  values.

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Hypothesis Testing Using Confidence Intervals

• II. Method I Hypothesis Testing Using
  Confidence Intervals
• Note This method works only for two-tail tests

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Hypothesis Testing Using Confidence Intervals
(cont.)

• A. Example Differences in Means
• In a large university, 10 male professors and 5
  female professors were randomly sampled. Their
  salaries were

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Hypothesis Testing Using Confidence Intervals
(cont.)

• 1. Step 1 Define Hypothesis
• We are interested in the difference between the
  means of men's and women's salaries. Call this
  difference D (m1-m2),
• The males state that D 0,
• The females say that D 7,
• Do the data support both of these hypotheses, one
  of them, or neither?
• We will test these hypotheses at the 5
  a-level.

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Hypothesis Testing Using Confidence Intervals
(cont.)

• 2. Step 2 Calculate a Confidence Interval
• Form a 95 confidence interval
• Notice that our data are two samples, one of men
  and other of women, from the same larger
  population of university professors. So we can
  pool our sample variances.

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Hypothesis Testing Using Confidence Intervals
(cont.)

• (cont.)
• So the 95 confidence interval is from 1 to 9
  thousand dollars.

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Hypothesis Testing Using Confidence Intervals
(cont.)

• 3. Step 3 Accept or Reject the Hypothesis
• According to these data, is the claim that D 0
  plausible?
• We must reject the hypothesis that D 0 because
  it falls outside the 95 confidence interval
• What about the hypothesis that D 7?

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Hypothesis Testing Using Confidence Intervals
(cont.)

• 4. Summary Step by Step Procedure
• 1. Step 1 Define Hypothesis
• Pick a significance level the usual one is 5.
• 2. Step 2 Construct confidence interval
• Formula depends on type of data, (matched or
  pooled variance) and how confident you want to
  be.
• 3. Step 3 Accept or Reject
• If falls within this interval, then we fail
  to reject the null, otherwise we reject it.

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Hypothesis Testing Using Confidence Intervals
(cont.)

• B. Another Example Matched Data
• A firm producing plate glass has developed a less
  expensive tempering process to allow glass for
  fireplaces to rise to a higher temperature
  without breaking. To test it, five different
  plates of glass were drawn randomly from a
  production run, then cut in half, with one half
  tempered by the new process and one half tempered
  by the old. The two halves were then heated until
  they broke. The results of the experiment look
  like this (next slide)

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Hypothesis Testing Using Confidence Intervals
(cont.)

• Matched Data (cont.)
• We want to test the hypothesis that the two
  processes are equal at the 95 confidence level
  or at the a .05 significance level.

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Hypothesis Testing Using Confidence Intervals
(cont.)

• 1. Step 1 Define Hypothesis
• H0 D 0
• Ha D ¹ 0
• Significance level a 5.
• 2. Step 2 Calculate a 95 Confidence interval.
  (s2unknown)

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Hypothesis Testing Using Confidence Intervals
(cont.)

• Step 2 (cont.)

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Hypothesis Testing Using Confidence Intervals
(cont.)

• 3. Step 3 Accept or reject null hypothesis?
• So we do not reject the hypothesis that H0 D 0
  because 0 falls within that range. The two
  processes are seen as indistinguishable.

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p-Values

• III. Method II p-Values
• P-values are essentially the significance level.
• In essence, we are calculating the probability
  that the hypothesis is true. It summarizes the
  credibility of the null hypothesis.

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p-Values

• A. s known
• 1. Step 1 State the Hypothesis
• A manufacturing process produces TV. tubes with
  an average lifem1200 hours and s 300 hours.
  A new process is thought to give tubes a higher
  average life. And out of a sample of 100 tubes
  we find that they have an average life 1265
  hours. Is the new process really any better
  than the old?

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p-Values

• Step 1 (cont.)
• H0 m 1200
• Ha m gt 1200
• a .05 or 5 significance-level
• This is a one-tailed test because we have put all
  the area in one-tail of the distribution. We are
  interested in those values that are greater than
  the mean.

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p-Values

• 2. Step 2 Calculate p-value
• We know s and n is large so we can use the normal
  distribution.
• m0 1200, and s 300 and n 100
• Standard error s/Ön 300/ Ö100 30.
• The observed value 1265.
• a. Standardize
• We then standardize (get the z-value )

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p-Values

• b. Find z-score (probability of the event
  occurring)

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p-Values

• 3. Step 3 Accept or Reject the Hypothesis
• This suggests that if the null hypothesis was
  true that there would be only a 1.5 probability
  of observing as larger as 1265.
• Since 1.5 lies to the right of our initial 5
  significance level, we can reject the null
  hypothesis.

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p-Values

• 4. Two-Tailed Test
• H0 m 1200
• Ha m ¹ 1200
• a .05 or 5 significance-level

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p-Values

• Accept or Reject
• Since the area to the right of 1265 is only 1.5,
  we can again reject H0.

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p-Values

• B. s unknown
• Usually s is unknown and has to be estimated with
  the sample standard deviation s. The test
  statistic is then t instead of Z.

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p-Values

• 1. Step 1 State Hypothesis (e.g., difference
  in men's and women's salaries)
• We know from the above example, ( - ) 5
• Standard Error 1.84
• Is this a one or a two tailed test?

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p-Values

• 2. Step 2 Calculate p-value
• a. Standardize

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p-Values

• b. Find probability of event from t-table
• Degrees of freedom (n-1) 13
• So the probability of observing a t-value of 2.72
  lies beyond
• This means that the tail probability is smaller
  than .01. That is, p-value lt .01.

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p-Values

• 3. Step 3 Accept or Reject Hypothesis
• Since the p-value is a measure of the credibility
  of H0, such a low value (below a 5) leads us
  to conclude that H0 is implausible.
• Therefore, we reject the null hypothesis.

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p-Values

• C. Getting t-values from Computers (Review of
  Homework)
• 1. Calculate t-values
• How does the computer calculate the t-value?

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p-Values

• 2. Calculate p-value
• The 2-tail probability gives the area to the
  right of the t-value times two.
• If this value is less than your significance
  level for a 2-tail test, then reject your null
  hypothesis.

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p-Values

• 3. Example Sample Homework
• For example, the difference of means test between
  men and women's incomes, produced a t-value
  6.60 and an associated p-value of .00.
• Therefore, I can reject the hypothesis that m1-m2
  0 because .00 is less than .025.

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p-Values

• D. Summary
• 1. Step 1 Define Hypothesis
• Choose H0, Ha and a significance level a (default
  is 5).
• 2. Step 2 Calculate p-value
• Calculate your p-value from the statistics
• if s known
• if s is unknown

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p-Values

• 3. Step 3 Accept or Reject hypothesis
• Reject H0 if p-value a
• For a One-Tailed Test
• Reject H0 if the p-value is less than the
  significance level a.
• Accept H0 otherwise.
• For a Two-tailed Test
• Reject H0 if the p-value is less than 1/2 the
  significance level. (i.e., 1/2a .025)
• Accept H0 otherwise.

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Critical Values

• IV. Method III Critical Values
• Classical hypothesis testing is very similar to
  the p-value approach.
• A. Example Manufacturing of TV tubes
• 1. State the Hypothesis
• H0 m 1200 n100
• Ha m gt 1200 m01200
• a 5. s300

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Critical Values

• 2. Test Hypothesis Find the Critical Values
• A. In General
• What z-value is associated with 5 of the area
  under the curve?
• From the z-tables we see that the area of 5 is
  associated with a z-value of 1.64.
• The question is what value on the x-axis
  corresponds to a z-value of 1.64?

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Critical Values

• B. Critical Value
• The critical value is the X-value that
  corresponds to a Z-value.
• We obtain the critical value by arbitrarily
  setting a 5 and calculating
• C. Calculating the Critical Value for
  Manufacturing TV Tubes
• We know that the m01200, and SE300/Ö10030.
• The Critical Value then is

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Critical Values

• 3. Step 3 Reject or Accept the Hypothesis
• To accept or reject our hypothesis we collect
  data and see if our sample mean is greater then
  this critical value.
• From the above example we observed a sample mean
  1265.
• Therefore we reject H0 m1200 because 1265gt1249.
  
• So we once again conclude that the new process is
  better than the old.

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Critical Values

• B. Example of 2-tailed test
• How do we construct a two-tailed test at the 5
  significance value?
• 1. Step 1 State Hypothesis
• H0 m 1200
• Ha m ¹ 1200
• a 5.

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Critical Values

• 2. Step 2 Calculate Critical Value
• We use Z.025 instead of Z.05.
• In this case, we would get c m0 Z.025SE.
• c 1200 1.9630 1141 and 1259.

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Critical Values

• 3 Step 3 Accept or reject null Hypothesis
• We would reject H0 if the observed fell below
  1141 or above 1259.
• Again 1265 exceeds the critical value so we still
  reject H0.

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Critical Values

• C. Summary
• 1. Step 1 Define Hypothesis
• State H0
• State Ha and
• Choose a significance level a.

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Critical Values

• 2. Step 2 Calculate Critical Value
• Draw a normal curve and find the critical values
  at the level of significance you arbitrarily set.
  Usually at the .05 significance-level.
• For two-tailed test
• s known c m0 Z.025SE.
• s unknown c m0 t.025SE(estimated)
• For one-tailed test
• s known c m0 Z.05SE.
• s unknown c m0 t.05SE(estimated)

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Critical Values

• 3. Step 3 Accept or Reject
• Then collect sample data.
• If the sample mean exceeds the critical value,
  then reject H0 otherwise accept H0.

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Notes About the Exam

• V. Notes About the Exam
• 1. Hand in your homework at the beginning of
  class
• 2. The exam will cover the material through
  today's lecture.
• 3. Problems, no definitions.
• 4. You may bring a calculator and one 3 X 5
  index card with whatever you want written on it.
• 5. Z-tables and t-tables will be supplied.

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Review Session

• Review Session Saturday March 8
• 11 to 1 PM
• Room 411 IAB