Lecture 5 Introduction to Hypothesis tests

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Lecture 5Introduction to Hypothesis tests

• Quantitative Methods Module I
• Gwilym Pryce
• g.pryce_at_socsci.gla.ac.uk

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Notices

• Register
• Class Reps and Staff Student committee.

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Aims Objectives

• Aim
• To introduce hypothesis testing
• Objectives 
• By the end of this session, students should be
  able to
• Understand the 4 steps of hypothesis testing
• Run hypothesis test on a mean from a large
  sample
• Run hypothesis test on a mean from a small
  sample

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Plan

• 1. Statistical Significance
• 2. The four steps of hypothesis testing
• 3. Hypotheses about the population mean
• 3.1 when you have large samples
• 3.2 when you have small samples

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1. Significance

• Does not refer to importance but to real
  differences in fact between our observed sample
  mean and our assumption about the population mean
• 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.

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• 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.
• 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 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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• Obviously, as the sample mean moves further away
  from our assumption (H0) about the population
  mean, we have stronger evidence that H0 is false.
• If P is very small, say 0.001, then there is only
  1 chance in a thousand of our observed sample
  mean occurring if H0 is true.
• This also means that if we reject H0 when P
  0.001, then there is only one in a thousand
  chance that we have made a mistake (I.e. that we
  have been guilty of a Type I error)

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• There is a tradition (initiated by English
  scientist R. A. Fisher 1860-1962) of rejecting H0
  if the probability of incorrectly rejecting it is
  ? 0.05.
• If P ? 0.05 then we say that H0 can be rejected
  at the 5 significance level.
• If P gt 0.05, then, argued Fisher, the chances of
  incorrectly rejecting H0 are too high to allow us
  to do so.
• the probability of a sample mean at least as
  extreme as our observed value occurring, will be
  determined not just by the difference between our
  assumed value of m, but also by the standard
  deviation of the distribution and the size of our
  sample.

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Type I and Type II errors

• P significance level chances of incorrectly
  rejecting H0 when it is in fact true.
• Called a Type I error
• So sig Pr(Type I error) Pr(false rejection)
• If we accept H0 when in fact the alternative
  hypothesis is true
• Called a Type II error.
• On this course we shall be concerned only with
  Type I errors.

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2. The four steps of hypothesis testing

• Last week we looked at confidence intervals
• establish the range of values of the population
  mean for a given level of confidence
• e.g. we are 90 confident that population mean
  age of HoHs in repossessed dwellings in the Great
  Depression lay between 32.17 and 36.83 years (s
  20).
• Based on a sample of 200 with mean 34.5yrs.
• But what if we want to use our sample to test a
  specific hypothesis we may have about the
  population mean?
• E.g. does m 30 years?
• If m does 30 years, then how likely are we to
  select a sample with a mean as extreme as 34.5
  years?
• I.e. 4.5 years more or 4.5 years less than the
  pop mean?

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One tailed test P how likely we are to select
a sample with mean age at least as great as 34.5?
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How do we find the proportion of sample means
greater than 34.5?

• Because all sampling distributions for the mean
  (assuming large n) are normal, we can convert
  points on them to the standard normal curve
• e.g. for 34.5
• z (34.5 - 30)/(20/?200)
• 4.5 / 1.4
• 3.2

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Upper tailed test
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Two tailed test
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3. Steps to Hypothesis tests

• 1. Specify null and alternative hypotheses and
  say whether its a two, lower, or upper tailed
  test.
• 2. Specify threshold significance level a and
  appropriate test statistic formula
• 3. Specify decision rule (reject H0 if P lt a)
• 4. Compute P and state conclusion.

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P values for one and two tailed tests

• Use diagrams to explain how we know the following
  are true
• Upper Tail Test population mean gt specified
  value
• H1 m gt m0 then P Prob(z gt zi)
• Lower Tail Test population mean lt specified
  value
• H1 m lt m0 then P Prob(z lt zi)
• Two Tail Test population mean ? specified value
• H1 m ? m0 then P 2xProb(z gt zi)

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E.g. The obesity threshold for men of a
particular height is defined as weighing over
187lbs mean weight of men in your sample with
this height is 190.5lbs, sd 13.7lbs, n 94.
Are the men in your sample typically obese?

• Test the hypothesis that the average man in the
  population is obese.
• How do we write Step 1?
• Because H1 m gt m0 then P Prob(z gt zi)
• So this is an Upper tailed test we write
• H0 m 187lbs
• H1 m gt 187lbs

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How do we write Step 2? (a and appropriate test
statistic formula)

• Large sample

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How do we write Step 3?
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How do we write Step 4?
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• The upper tail significance level is given by
  SIGZ_UTL 0.00663
• What can we conclude from this?

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eg Test the hypothesis that male super
heroes/villains tend to be c. six foot tall.

• 1st you need to convert scale 6ft 182.88cm
• 2nd you need to run descriptive stats on height
  to get the n, x-bar, and s
• n 29
• xbar 181.72cm
• s 8.701

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H_L1M n(29) x_bar(181.72) m(182.88)
s(8.701).

• Compare this output with that of the large sample
  95 confidence interval interpret

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Hypotheses about the population mean when you
have small samples

• This is exactly the same as the large sample
  case, except that one uses the t-distribution
  provided that x is normally distributed.
• Many statisticians use t rather than z even when
  the sample size is large since
• (i) strictly speaking our approximation for the
  SE of the mean has a t rather than z distribution
• (ii) t tends towards the z distribution when n is
  large

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E.g. re-run the hypothesis test on height of
super heroes using a t testH_S1M n(29)
x_bar(181.72) m(182.88) s(8.701).

• How do the results differ, if at all?
• N.B. the t-distribution tends to have fatter
  tails smaller the sample, fatter the tails
  become.

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Reading Exercises

• Confidence Intervals
• MM section 6.1 and exercises for 6.1 (odd
  numbers have answers at the back)
• Tests of Significance
• MM section 6.2 and exercises for 6.2
• Use and Abuse of Tests
• MM section 6.3 and exercises for 6.3
• Power and inference as a Decision
• Type I II errors etc.
• optional