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Inferential Statistics 2. Maarten Buis January 11, 2006. outline. Student Recap Sampling distribution Hypotheses Type I and II errors and power testing means testing correlations. Sampling distribution. PrdV example from last lecture. - PowerPoint PPT Presentation
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Inferential Statistics 2
Maarten Buis
January 11, 2006
outline
• Student Recap
• Sampling distribution
• Hypotheses
• Type I and II errors and power
• testing means
• testing correlations
Sampling distribution
• PrdV example from last lecture.
• If H0 is true, than the population consists of 16 million persons of which 41% (=6.56 million persons) supports de PrdV.
• I have drawn 100,000 random samples of 2,598 persons each and compute the average support in each sample.
Sampling distribution
• 5% or 50,000 samples have a mean of 39% or less.
• So if we reject H0 when we find a support of 39% or less than we will have a 5% chance of making an error.
• Notice: We assume that the only reason we would make an error is random sampling error.
0
2000
4000
6000
8000
1.0e+04
Fre
quen
cy
.36 .38 .4 .42 .44 .46% support for PrdV
sampling distribution of support for PrdV
More precise approach
• We want to know the score below which only 5% of the samples lie.
• Drawing lots of random samples is a rather rough approach, an alternative approach is to use the theoretical sampling distribution.
• The proportion is a mean and the sampling distribution of a mean is the normal distribution with a mean equal to the H0 and a standard deviation (called standard error) of N
More precise approach
• For a standard normal distribution we know the z-score below which 5% of the samples lie (Appendix 2, table A): -1.68
• So if we compute a z-score for the observed value (.31) and it is below -1.68 we can reject the H0, and we will do so wrongly in only 5% of the cases
• se
xz
More precise approach
• is the mean of the sampling distribution, so .41 (H0)
• se is , of a proportion is
• so the se is
• so the z-score is
• -10.4 is less than -1.68, so we reject the H0
pp 1N
0096.2598
)41.1(41.
4.100096.
41.31.
se
xz
Null Hypothesis
• A sampling distribution requires you to imagine what the population would look like if H0 is true.
• This is possible if H0 is one value (41%)
• This is impossible if H0 is a range (<41%)
• So H0 should always contain a equal sign (either = or ≤ or ≥)
Null hypothesis
• In practice the H0 is almost always 0, e.g.:
– difference between two means is 0– correlation between two variables is 0– regression coefficient is 0
• This is so common that SPSS always assumes that this is the H0.
Undirected Alternative Hypotheses
• Often we have an undirected alternative hypothesis, e.g.:– the difference between two means is not zero
(could be either positive or negative)– the correlation between two variables is not
zero (could be either positive or negative)– the regression coefficient is not zero (could be
either
Directed alternative hypothesis
• In the PrdV example we had a directed alternative hypothesis: Support for PrdV is less than 41%, since PrdV would have still participated if his support were more than 41%.
Type I and Type II errors
actual situation
decision H0 is True H0 is False
reject H0Type I error
probability = correct decision
probability = 1-(power)
do not reject H0
correct decision
probability = 1-Type II error
probability =
Type I error rate
• You choose the type I error rate ()
• It is independent of sample size, type of alternative hypothesis, or model assumptions.
Type I versus type II error rate
• a low probability of rejecting H0 when H0 is true (type I error), is obtained by:
• rejecting the H0 less often, • Which also means a higher probability of not
rejecting H0 when H0 is false (type II error),• In other words: a lower probability of finding
a significant result when you should (power).
How to increase your power:
• Lower type I error rate
• Larger sample size
• Use directed instead of undirected alternative hypothesis
• Use more assumptions in your model (non-parametric tests make less assumptions, but are also have less power)
Testing means
• What kind of hypotheses might we want to test:– Average rent of a room in Amsterdam is 300
euros– Average income of males is equal to the
average income of females
Z versus t
• In the PrdV example we knew everything about the sampling distribution with only an hypothesis about the mean.
• In the rent example we don’t: we have to estimate the standard deviation.
• This adds uncertainty, which is why we use the t distribution instead of the normal• Uncertainty declines when sample size becomes
larger.• In large samples (N>30) we can use the normal.
t-distribution
• It has a mean and standard error like the normal distribution.
• It also has a degrees of freedom, which depends on the sample size
• The larger the degrees of freedom the closer the t-distribution is to the normal distribution.
Data: rents of rooms
rent rent
room 1 175 room 11 240
room 2 180 room 12 250
room 3 185 room 13 250
room 4 190 room 14 280
room 5 200 room 15 300
room 6 210 room 16 300
room 7 210 room 17 310
room 8 210 room 18 325
room 9 230 room 19 620
room 10 240
Rent example
• H0: =300, HA: ≠ 300
• We choose to be 5%
• N = 19, so df= 18
• We reject H0 if we find a t less than -2.101 or more than 2.101 (appendix B, table 2)
• We do not reject H0 if we find a t between
-2.101 and 2.101 .
Rent example
•
• We use s2 as an estimate of 2
• So
• -1.85 is between -2.101 and 2.101, so we do not reject H0
nsese
xt
,
19,99,300,258 Nsx
85.11999
300258
t
Compare means in SPSS
Independent Samples Test
19,012 ,000 10,365 2250 ,000 633,95876 61,16368 514,01564 753,90189
10,370 2225,825 ,000 633,95876 61,13297 514,07516 753,84236
Equal variancesassumed
Equal variancesnot assumed
incmid householdincome in guilders
F Sig.
Levene's Test forEquality of Variances
t df Sig. (2-tailed)Mean
DifferenceStd. ErrorDifference Lower Upper
95% ConfidenceInterval of the
Difference
t-test for Equality of Means
Group Statistics
1131 2833,2228 1530,70376 45,51556
1121 2199,2640 1366,42170 40,81144
sex sex respondent1 male
2 female
incmid householdincome in guilders
N Mean Std. DeviationStd. Error
Mean
2121
222
211
21
2121
222
211
21
21
222
211
21
21
222
211
21
222
2112
2
21
21212121
112
11
11
2
11211
211
2
11
0
21
21
21
212121
NNNNsNsN
xxt
NNNN
sNsN
NN
NNsNsN
se
NN
NNsNsN
se
NN
sNsNs
ss
NN
sse
se
xx
se
xx
se
xxt
xx
xx
pool
poolpool
poolxx
xxxxxx
Do before Monday
• Read Chapter 9 and 10
• Do the “For solving Problems”
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