Lecture 6 Hypothesis Tests for Proportions

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Lecture 6Hypothesis Tests for Proportions Two
PopulationsSlides available from Statistics
SPSS page of www.gpryce.com

• Social Science Statistics Module I
• Gwilym Pryce

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Notices

• Register
• Class Reps and Staff Student committee.

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Aims Objectives

• Aim
• the aim of this lecture is to continue with our
  introduction of the method of hypothesis testing
  and to demonstrate a number of applications
• Objectives
• by the end of this lecture students should be
  able to carry out hypothesis tests on
• two population means
• one population proportion
• two population proportions

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Plan

• 1. Review of Significance
• 2. Review of one sample tests on the mean
• 3. Hypothesis tests about Two population means
• Homogenous variances
• Heterogeneous variances
• 4. Deciding on whether variances are equal
• 5. Hypothesis tests about proportions
• One population
• Two populations

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Macro commands
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Review of Significance

• P significance level chances of our observed
  sample mean occurring given that our assumption
  about the population (denoted by H0) is true.
• So if we find that this probability is small, it
  might lead us to question our assumption about
  the population mean.
• I.e. if our sample mean is a long way from our
  assumed population mean then it is
• either a freak sample
• or our assumption about the population mean is
  wrong.

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• If we draw the conclusion that it is our
  assumption re m that is wrong and reject H0 then
  we have to bear in mind that there is a chance
  that H0 was in fact true.
• In other words
• when P 0.05, for every twenty times we reject
  H0, then on one of those occasions we would have
  rejected H0 when it was in fact true.

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2. Review of one sample tests on the mean

• We introduced a common framework for hypothesis
  testing

4 Steps of Hypothesis testing Step (1) state H0
and H1 Step (2) state a and formula Step (3)
state decision rule Step (4) compute P decide
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We also looked at 2 specific tests

• Large sample sig. Test on one mean
• Formula
• Macro syntax
• H_L1M n(?) x_bar(?) m(?) s(?).
• Small sample sig. Test on one mean
• Formula
• Macro syntax
• H_S1M n(?) x_bar(?) m(?) s(?).

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3. Hypothesis tests about two population means

• In SPSS this is called the Independent Sample
  t-test
• go to Analyse, Compare Means...
• Two different formulas for computing t

Equal Variances (formula has an exact
t-distribution)
Unequal Variances (does not have an exact
t-distribution)
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Example where variances are different

• As part of your PhD, you want to test whether the
  new Fun Phonics reading method is better than
  the Letterland method. You examine the reading
  power of 6 year old children from two similar
  schools.
• The first used the FP method and you found that
  this produced an average reading proficiency
  score of 53.7 (based on a sample of 22 children
  s.d. 11.5).
• The second school used the Letterland method and
  you found that this produced an average reading
  proficiency score of 42.51 (sample 24 s.d.
  16.9).
• Test whether the FP method produces higher
  results at the 1 significance level.

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• Use the 4 steps and the following formula to test
  whether the FP method produces higher results at
  the 1 significance level.
• Can you use the canned SPSS procedure to do this
  problem?

4 Steps of Hypothesis testing Step (1) state H0
and H1 Step (2) state a and formula Step (3)
state decision rule Step (4) compute P decide
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• (1) H0 mFP mL (means are equal)
• H1 mFP gt mL (upper tail test)
• (2) a 0.01 (implies critical t value of
  2.528),
• (3) Reject H0 iff P lt a, I.e. if P lt 0.01
• (4) P Prob(t gt 2.644) 0.0076, so reject H0

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Doing the calculation in SPSS

• You cannot use the canned SPSS procedure unless
  you have the original data.
• But you can use the following macro commands
• Homogenous variances
• H_S2Mp n1(?) n2(?) x_bar1(?)
  x_bar2(?) s1(?) s2(?).
• Heterogeneous variances
• H_S2Md n1(?) n2(?) x_bar1(?)
  x_bar2(?) s1(?) s2(?).

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For the Letterland/FP example we would use the
diff. Variances syntax
H_S2Md n1(22) n2(24) x_bar1(53.7)
x_bar2(42.51) s1(11.5) s2(16.9).

• The upper tail sig. 0.007588
• I.e. less than 1 chance of false rejection,
  therefore reject H0 of equal means in favour of
  the alternative hypothesis that Fun Phonics
  results in higher reading scores on average than
  Letterland.

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4. How do we decide on whether the variances are
similar?

• Where variances are hugely different or exactly
  the same, the decision is simple.
• When there is any ambiguity, we can use one of
  two tests to help us
• Simple Ratio of Variances Test
• Levenes Test

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Simple Ratio of Variances test

• If we divide the ratio of variances of samples
  from two independent populations we find that
  that ratio has an F distribution in repeated
  samples
• where the denominator degrees of freedom
  calculated as n11 and the numerator degrees of
  freedom calculated as n21.
• NB Because the critical values for the F
  distribution are only calculated for the upper
  tail, if the F value you are have calculated is
  less than one, you need to invert it
• i.e. swap round the numerator and denominator.

F s12 / s22
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• This is the formula behind the following command
• H8_S2VF n1(?) n2(?) s1(?) s2(?)
• E.g. For the Letterland/FP example
• H8_S2VF n1(22) n2(24) s1(11.5) s2(16.9).
• Which tells us that there is less than a 5
  chance of false rejection if we reject the null
  of equal variances. So reject the null
• I.e. we can be sure that the population variances
  are indeed different.

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The Levenes test

• If we have the original data (rather than just
  the summary statistics) we can use Levenes test
  which is a canned routine in SPSS.
• The Levenes test is more sophisticated robust
  than the simple ratio of variances test
• If P (I.e. sig.) from the Levenes test is
  small reject the H0 of equal variances use the
  1st t-formula.
• If P from the Levenes test is large, accept H1
  use the 2nd t-formula to compute the test
  statistic.

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SPSS Output from test equal purchase prices
between Cumberland and Durham (Nationwide)
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Two tails from one

• Along with the Levenes test results, SPSS
  automatically supplies t-test results for both
  the equal and different variances formulas.
• One problem with the SPSS t-test, however, is
  that it only gives the 2 tail sig., but you can
  work out the one tail sigs as follows
• The two tailed significance is twice that of the
  smallest one tailed significance
•   2 tailed sig. 2 ? minlower tail sig., upper
  tail sig.
• But it can be a bit confusing working out which
  one tail significance level is the one you want
  (see notes).

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Testing for 2 means summary

• If youve got the original data,
• First do the Levenes test in SPSS
• Analyze, Compare Means, Independent Samples
• Then do the appropriate macro t-test to avoid
  confusion.
• H_S2Mp for equal variances or H_S2Md for
  different variances
• If you dont have the original data,
• First do the ratio of variances test
• H8_S2VF
• Then do the appropriate macro t-test
• H_S2Mp for equal variances or H_S2Md for
  different variances

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5.1 Hypothesis tests on proportions 1 population
(large samples only)

• So far looked at
• how to make inferences about the population mean
  from our sample mean.
• But sometimes the variable of interest is
  categorical
• household has or has not insurance
• person is homosexual or not homosexual
• a person has Aids or does not have Aids

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• In such cases, what we are interested in is the
  proportion of cases that fall into a particular
  category
• the proportion of households with insurance
• the proportion of people who are homosexual
• the proportion of people with Aids

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• Calculating the sample proportion
• p x / n
• where
• x cases with the attribute of interest
• e.g. the number of households with insurance
• n sample size

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CLT and Proportions

• Q/ Does the Central Limit Theorem apply to sample
  proportions?
• A/ Yes.
• Proportions from repeated random samples will be
  normally distributed around the population
  proportion p.
• We can then translate any sample proportion onto
  the standard normal curve by calculating its z
  score

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Example

• E.g. 1 As a historian, you want to find the
  proportion of citizens in medieval Scotland that
  contracted the plague. From a sample of 400
  parish records, you find that 22 died of the
  plague. The assumption in the literature has
  been that 10 of the population had died. Test
  whether this assumption is valid using both 2 and
  1 tailed tests.

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Summary of data n 400 x 22 p0 0.1

• (1) H0 p 10
• H1 p ? 10 (2-tailed test)
• (2) a 0.02, for example.

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• (3) Reject H0 iff P lt a, I.e. if P lt 0.02
• (this will happen if zc lt - 2.33 or if zc gt 2.33,
  where 2.33 is the z value associated with a
  0.02. Since zc -3.948, we know we can safely
  reject H0).
• (4) Calculate z
• P 2x(Prob(z lt -3.00))
• 2x 0.0013 0.0026
• since P lt 0.02 (I.e. less than one in 50 chance
  of type I error) we can reject H0.
• In fact, the chances of incorrect rejection of H0
  are less than one in 3,000.
• I.e. the chances of observing p (our sample
  proportion) assuming H0 (p 10) to be true are
  so small that we are forced to question this
  assumption about p

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One tailed test

• (1) H0 p 10
• H1 p lt 10 (lower tail test)
• (2) a 0.02
• (3) Reject H0 iff P lt a, I.e. if P lt 0.02
• (4) Lower tail sig. P Prob(z lt -3.00)
  .001350
• since P lt 0.02 we can reject H0 knowing that the
  chances of incorrect rejection of H0 are less
  than one in 740
• our cut-off rule for rejecting H0 was no more
  than a one in 50 chance
• one in 740 is a lot less than one in 50 so we can
  reject H0 with confidence.

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• The macro syntax for one proportion tests is as
  follows
• H6_L1P n(400) x(22) pi(0.1).
• Which comes to the same result.

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5.2 Hypothesis tests about Two population
proportions

• To test the hypothesis that the population
  proportions are equal
• H0 p1 p2
• compute the z statistic

where SEDp is the pooled standard error
and
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Example

• Two surveys of mortgage payment protection
  insurance (MPPI) are carried out, one on single
  parents with 1 child and one on single parents
  with 3 children. Amongst the first group, 67 out
  of a sample of 300 were found to have taken out
  MPPI, compared with 15 out of a sample of 101 in
  the second group. Is take-up significantly lower
  amongst the HHs with three children?
• p1 67/300 0.2233
• p2 15/101 0.1485
• p (300 101)/(6715) 0.2045

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• (1) H0 p1 p2
• H1 p1 gt p2
• (2) a 0.01 (z ?2.33)
• (3) Reject H0 if P lt 0.01
• (4) P 0.053.
• Take-up is not significantly lower amongst HHs
  with 3 children at the1 sig. level or even at
  5 significance level.
• I.e. we cannot say that the difference in
  proportions is anything more than the effect of
  sampling variation.

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• H7_L2P n1(300) n2(101) x1(67) x2(15)
  .

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Summary

• 1. Review of Significance
• 2. Review of one sample tests on the mean
• 3. Hypothesis tests about Two population means
• Homogenous variances
• Heterogeneous variances
• 4. Deciding on whether variances are equal
• 5. Hypothesis tests about proportions
• One population
• Two populations