Transcript Lecture 2 Calculating z Scores

Quantitative Methods Module I
Gwilym Pryce
[email protected]
Lecture 5
Introduction to
Hypothesis tests
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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;
3
Plan:
1. Statistical Significance
x
 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 > 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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14
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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 it’s 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 < 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 > specified value
H1: m > m0 then P = Prob(z > zi)
– Lower Tail Test: population mean < specified value
H1: m < m0 then P = Prob(z < zi)
– Two Tail Test: population mean  specified value
H1: m  m0 then P = 2xProb(z > |zi|)
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Confidence Interval
Find the 90% confidence interval of the
population mean age
1.
Choose the appropriate test
statistic:
Hypothesis Tests
Test the hypothesis that the population mean age = 30
using a significance level of 0.1
1.
Specify null and alternative hypothesis:
2.
Specify the level of significance and the test
statistic
xi  m
* s
zi 
 m  xi  z
s/ n
n
H0: m = 30
H1: m  30
Significance level:
a = likelihood of Type I error that you are
prepared to tolerate
= Prob(Reject H0 when it is true) = 0.1
Test Statistic:
n > 30, therefore we can us z:
zc 
xi  m
s/ n
= zc 
xi  30
s/ n
i.e. we write the zc formula assuming that H0 is correct
2.
Establish the value of z*:
3.
Prob(-z*<z<z*) = 0.9
Area of tails = (0.1)/2 = 0.05
 z* = 1.65
3.
Calculate the confidence interval:
m  34.5  1.65
20
200
= 45.5 2.33
Specify the decision rule:
Reject H0 iff P (the calculated level of Type I error)
is no greater than the tolerated level:
i.e. Reject H0 iff P  a
(the smaller is P, the less risk involved in rejecting H0)
4.
Compute P and state your conclusion:
zc = 3.18 ;
PProb(z < 3.18)
Since P < a(i.e. , its safe to reject H0
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Lower Tail Hypothesis Tests
Test the hypothesis that the population mean age
< ? using a significance level of a
Upper Tail Hypothesis Tests
Test the hypothesis that the population mean age
> ? using a significance level of 0.1
1.
1.
Specify null and alternative hypothesis:
H0: m = ?
H1: m < ?
2. Specify the level of significance and the
test statistic
Significance level:
a = likelihood of Type I error that you
are prepared to tolerate
= Prob(Reject H0 when it is true)
Test Statistic:
If n > 30, we can us z:
xi  m
s/ n
H0: m = ?
H1: m > ?
2. Specify the level of significance and the
test statistic
zc 
Specify null and alternative hypothesis:
= zc 
Significance level:
a = likelihood of Type I error that you
are prepared to tolerate
= Prob(Reject H0 when it is true)
Test Statistic:
If n > 30, we can us z:
xi  ?
zc 
s/ n
xi  m
s/ n
= zc 
xi  ?
s/ n
i.e. we write the zc formula assuming that H0 is
correct
i.e. we write the zc formula assuming that H0 is
correct
3.
3.
Specify the decision rule:
Reject H0 iff P (the calculated level of Type I
error) is no greater than the tolerated level:
i.e. Reject H0 iff P  a
4.
Compute P and state your conclusion:
PProb(z < zc)
x
Note that zc will be negative if
m
Specify the decision rule:
Reject H0 iff P (the calculated level of Type I
error) is no greater than the tolerated level:
i.e. Reject H0 iff P  a
4.
Compute P and state your conclusion:
PProb(z > zc) m  x
Note that zc will be positive if
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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 >
m0 then P = Prob(z > zi)
• So this is an Upper tailed test & we write:
H0: m = 187lbs
H1: m > 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:

Descriptive Statistics
N
Height in cm
Valid N (listwise)

n
 xbar
 s
29
29
Minimum
165
Maximum
205
= 29
= 181.72cm
= 8.701
Mean
181.72
Std. Deviation
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 test:
H_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:
– M&M section 6.1 and exercises for 6.1 (odd
numbers have answers at the back)

Tests of Significance:
– M&M section 6.2 and exercises for 6.2

Use and Abuse of Tests:
– M&M section 6.3 and exercises for 6.3

*Power and inference as a Decision
– Type I & II errors etc.
– *optional
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