Statistics Exam 2 Notes
Chapter 5 – The one sample z test
We use one sample z tests when:



We know the population variance (2), or
We do not know the population variance but our sample size is large n ≥ 40
We want to compare a sample mean with the population mean
If we have a sample size of less than 40 and do not know the population variance, then we must
use a t-test.
Alpha ():




Level of significance
The amount of risk you decide to take > when the chance of being wrong is low enough
(lower than alpha) you can reject the null hypothesis > results are statistically significant
Probability of making a type I error
Alpha can be defined as the percentage of null (ineffective) experiments that still attain
statistical significance. Specifically, when alpha is 5%, it means that only 5% of the null
experiments will be declared statistically significant
P value/ p level:
 The probability of obtaining test results at least as extreme as the results actually observed,
under the assumption that the null hypothesis is correct.
 A measure of the strength of the evidence against the null hypothesis
 The smaller the p value, the greater the evidence against the null hypothesis
 Probability that your results are due to chance provided that the null hypothesis is true
If p level is smaller than alpha … p<0.05 – reject null
If p level is larger than alpha…p>0.05 – fail to reject null
If t -calculated is smaller than t-critical – fail to reject null
If t-calculate is larger than t-critical – reject null
Sampling distribution of the mean:
Gives us probabilities of means, given that the null hypothesis is true
Tell us something about the probability of a particular mean from a particular sample
The standard error of the mean tells you:
How far sample means are from population mean
How precise an estimate your sample mean is
Null hypothesis:
Experimental results merely come from sampling the original population at random (not a
different population)
μsampling dist = μpopulation
The null hypothesis distribution is a “map” of what results are or are not likely by chance. For
the one sample case, the null hypothesis distribution is the sampling distribution of the mean.
standard error (deviation) equal to
Type I error:
 False positive (p=)
 Rejecting the null hypothesis when it’s true
 Saying that there is a difference when there is none.
 To reduce, lower alpha (e.g. from 0.05 to 0.01). this will narrow down rejection region
making it hard to make a mistake when rejecting the null hypothesis BUT will increase
chances of making type II error
Type II error:
 False negative (p=)
 Failing to reject null hypothesis when it’s false
 Saying there is no difference when if fact there is one.
 To reduce, use a one-tailed test instead of a two-tailed one (but you rule out possibility of
testing results in the other tail). This means alpha doesn’t need to be divided by 2, increasing
rejection region on one side
Ineffective (null) experiments (α) + Effective experiments (1 – β) = total number of sig results

Null Hypothesis Testing as a Spam Filter
– Bayes’s Theorem
p(H 0 S) =
p(H 0 ) p(S H 0 )
p(H 0 ) p(S H 0 ) + p(H A ) p(S H A )
Researcher’s
Decision
Accept the Null
Hypothesis
Reject the Null
Hypothesis
Actual Situation
Null Hypothesis is Null Hypothesis is
True
False
Correct Decision Type II Error
(p = 1 – α)
(p = β)
Type I Error
Correct Decision
(p = α)
(p = 1 – β)
(power)
Assumptions Underlying the One-sample z Test
 The DV was interval or ratio scale
 The sample was drawn randomly. This affects the generalization of your conclusions.
 The variable measured has a normal distribution in the population. Because of the
Central Limit Theorem, this assumption is not critical when the size of the group is about 30
or more. There is reason for concern if there is reason to believe that the population is very
far from normal, and fairly small sample sizes are being used.
 The standard deviation for the sampled population is the same as that of the
comparison population.
Chapter 6 - Interval estimation and the t-distribution
t-tests are a statistical way of testing a hypothesis when:


We do not know the population variance
Our sample size is small, n < 40
The t-distribution has father tails than the normal distribution, which means that the critical value
for t will be larger than the critical value for z for the same alpha level (and ofc the same number
of tails)
Df= n-1
As the df increases (as n increases) the t distribution more closely resembles the normal
distribution. By the time df reaches about 100, the difference between the two distributions is
considered negligible.
Larger sample sizes tend to improve the likelihood of significant results in two ways:
By increasing the df and thus reducing the critical value of t
By causing the formula for t to be multiplied by a larger number (if all else stays the same, the
calculated t is doubled when the sample size is multiplied by 4)
Assumptions for one-sample t test are the same as those for the one-sample z test:
Independent random sampling
Normally distributed variable
Same SD for both the sample and population
N and the one sample test
As n increases, so does t
As n increases, the SD decreases, therefore t increases
As n increases, chance of significance increases
Dividing n by a constant c results in t (calc) being divided by square root of c, all else being
equal
If you divide s(sample sd) by a constant c, t(cacl) will be multiplied by c, all else being equal
If you multiply sample size by c, you divide width of CI by square root of c
Sampling biases creep in more when there is:
Small samples
Lack of control group
Width of a confidence interval is based on:
Level of confidence
Sample size
SD of sample
The standard error of the mean is the standard deviation of the sampling distribution
CI is good for:
Elimination
Determining precision or errors
Hypothesis testing
When performing a one sample test, we should do a t-test when we have the sample standard
deviation (not population SD) and a sample size less than 40
Chapter 7 - The t Test for Two Independent Sample Means
Null hypothesis: 1=2
Alternative hypothesis: 12
Df= n1 + n2 -2
Homogeneity of variance:
If one variance is not more than double the other variance, we have HOV
If there is NO HOV > use separate variances t-test
t=
(X
1
)
- X 2 - (µ1 - µ2 )
s12 s22
+
n1 n2
OR
t=
(X
1
)
- X 2 - (µ1 - µ2 )
sX 1 - X 2
If there IS HOV > use pooled-variances estimate
2
p
s
(
n1 - 1)s12 + (n2 - 1)s22
=
n1 + n2 - 2
Or
s2p =
SS1 + SS2
n1 + n2 - 2
(X
)
Plug these into pooled variances t-test formula:
t=
1
- X 2 - (µ1 - µ2 )
÷1 1 ÷
s2p ÷
÷n + n ÷
÷
2 ÷
÷ 1
If there is equal sample sizes > use pooled variances t test
s12 + s22
s =
2
• This is the same as the separate variances
t-test for 2 groups with equal n’s
2
p
Therefore:
t
X
1

 X 2  1   2 
 s12  s 22

 n



• This means that, with equal n’s, it doesn’t
matter whether a pooled or a separate ttest is more appropriate
• tcv will depend on which test you use
Confidence Intervals for the difference between two population means:
(
)
µ1 - µ2 = X 1 - X 2 ± t crit sX
1-X 2
If 0 is NOT in the interval > reject H0
If 0 is in the interval > fail to reject H0
Assumptions:
 Independent random sampling
o Both groups should be random samples
o Each individual selected for one sample should be independent of the individuals in
the other sample
o Random assignment
 Normal Distributions
o Central Limit Theorem
Central Limit Theorem:
The central limit theorem states that if you have a population with mean μ and
standard deviation σ and take sufficiently large random samples from the population
with replacement , then the distribution of the sample means will be approximately
normally distributed.
Two sample tests are used to analyze:
Quasi-experiment
True experiment
Appropriate only when the dependent variable is interval or ratio scale (if ordinal or categorical
> use nonparametric statistics)
HOV tests:
F test
Levene’s
Finding the right null hypothesis distribution is known as the Behrens- fisher problem
To perform a hypothesis test with a confidence interval we need to compare the null hypothesis
value with the interval
If t(calc) is larger than t(crit) > reject null
If t(calc) is smaller than t(crit) > fail to reject null
Chapter 8 – Power and effect size
Power:
The probability of rejecting a null hypothesis when it is false
Acceptable power is 0.8 or higher (0.2 type II error rate)
Powe = 1 - 
 = probability of making a type II error
Power is a function of:
Alpha
Sample size (n)
Effect size
How to improve power:
Increase alpha ( if you made alpha smaller, beta will get bigger and lose powe)
Increase n
As n increases, power increases
As alpha increases, power increases
As sd decreases, power increases
Expelted t value = delta
If the sample size (n) is multiplied by a constant (k), delta is multiplied by square root of k
kn= (square root of k) (delta)
If d is doubled, delta is also doubled
Kd=k(delta)
Significance tells us about probability + effect size tells us about size of the result
d=
(µ1 - µ2 )
s
d = .2 small effect size
d = .5 medium effect size
d = .8 large effect size
Although certain the effect size is not 0, a large t value does not imply that effect size is
large
Statistical significance does not imply that the effect size is large enough to be interesting or
of any practical importance