false

Department

Psychology

Course Code

PSYB07H3

Professor

Douglas Bors

Description

AFTER MIDTERM EXAM NOTES
T-TEST
ONE-SAMPLE T-TEST
-if μ is known and σ is unknown, we must estimate σ with s i.e. if you2
don’t know the population variance use the sample variance to estimate
-because we use s, we can no longer declare the answer to be a
z but rather it is a t
WHY?
-the sample variance (s ) is an unbiased estimator of the population
2 2 2
variance (σ ) but s is not a normal distribution s tends to be
positively skewed (if z is used) we more often underestimate
2
population variance (σ ) than overestimate esp. with a small n
(underestimating is more of a bigger mistake)
-THEREFORE the z table cannot be used as the t is likely to be larger
2
than z because s is in the denominator z table can be only used in
this situation if we have an infinity observations but this is never
-t table has critical values instead of t-values
Differences in formulas:
x − μ x − μ x − μ vs. (substitute s for σ ) 2 x − μ x − μ x − μ
z = σ = σ = 2 t = S = S = 2
x σ x S
N N N N
- To treat t as a Z would give us too many significant results
Therefore, for the student’s t distribution, we switch to the t-table when
2
we use s
-unlike the z, the distribution is a function of the degrees of freedom
with n = infinity and t = z z = 1.96 but for t value is dependent of
degrees of freedom with an infinity number of df, it approaches 1.96
-for one-sample cases, degrees of freedom: df = n-1 for sample but if
given μ, no need for n-1 just n
-1 df is lost because we used (sample mean) to calculate s 2
Σ(x-) = 0, all x can vary save for 1
Example 1: the effect of statistic tutorials
Last 100 years: μ = 76.0 (no tutorials)
This year’s: = 79.3 (tutorials)
N = 20
s = 6.4
1. State the null and the alternative hypothesis
H 0 μ = 76 H : 1 ≠ 76
2. Plug information into formula
x − μ SampleMean -Pop'n mean
t = =
S x standard error
x − μ
= s3. Check up on t-table
-not area (p) above or below value of 1
-fives t values that cut off at critical areas i.e. 0.05
-t also defined for each degree of freedom
THEREFORE, at α = 0.05,
T0.0519) is ± 2.093 (critical value)
79.3−76 3.3
= 6.4 = 1.43 = 2.31
20
2.31>2.093 reject null hypothesis evidence that this year’s group
is different from previous years’ (but there is still a 5% chance of making
a Type I error)
Factors Affecting the Magnitude of t and Decision
1. Difference between and μ ( - μ) the larger the numerator, the
larger th2 t-value effect size or observed difference
2. Size of s as n increases, denominator decreases, t increases
3. Size of n as n increases, denominator decreases, t increases
4. α level
5. one- or two-tailed test (always do two tailed test)
Confidence Limits on Mean
-point estimate a specific value taken as estimator of a parameter
-interval estimates a range of values estimated to include parameter
-confidence limits a range of values that has a specific (p) of
bracketing the parameter (i.e. 95%) End Points = confidence limits
-How large or small μ could be without rejecting H if we0ran a t-test on
the obtained sample mean
Confidence Limits (C.I.)
Using Example 1 data,
x −μ x −μ
t = =
S x S
N
-we know , s and N and we know the critical value for t at α = 0.05:
= t (19)= 2.093
.05
-solve for μ by rearranging the equation:
79.3−μ 79.3−μ
±2.093 = =
6.4 1.43
20
μ = ±2.093(1.43)+79.3
μ = ±2.993+79.3Using +2.993 and –2.993
More on Confidence Limits (from Internet)
μ upp= −2.993+79.3 = 76.31
lower
C.I .95 76.31≤ μ ≤ 82.29
Confidence limits for the mean are an interval estimate for the mean.
Interval estimates are often desirable because the estimate of the mean
varies from sample to sample. Instead of a single estimate for the mean,
a confidence interval generates a lower and upper limit for the mean.
The interval estimate gives an indication of how much uncertainty there
is in our estimate of the true mean. The narrower the interval, the more
precise is our estimate.
Confidence limits are expressed in terms of a confidence coefficient.
Although the choice of confidence coefficient is somewhat arbitrary, in
practice 90%, 95%, and 99% intervals are often used, with 95% being
the most commonly used.
As a technical note, a 95% confidence interval does not mean that there
is a 95% probability that the interval contains the true mean. The
interval computed from a given sample either contains the true mean or
it does not. Instead, the level of confidence is associated with the
method of calculating the interval. The confidence coefficient is simply
the proportion of samples of a given size that may be expected to
contain the true mean. That is, for a 95% confidence interval, if many
samples are collected and the confidence interval computed, in the long
run about 95% of these intervals would contain the true mean.
The width of the interval is controlled by two factors:
1. As N increases, the interval gets narrower from the term.
That is, one way to obtain more precise estimates for the mean is
to increase the sample size.
2. The larger the sample standard deviation, the larger the
confidence interval. This simply means that noisy data, i.e., data
with a large standard deviation, are going to generate wider
intervals than data with a smaller standard deviation.
To test whether the population mean has a specific value, , against
the two-sided alternative that it does not have a value , the confidence interval is converted to hypothesis-test form. The test is a one-sample t-
test.
TWO RELATED SAMPLE T TEST
- design in which the same subject is observed under more than one
condition (therefore also known as repeated measures, matched
samples)
-each subject will have 2 measures x and x 1hat wil2 be correlated
this must be taken into account
Example 1: Promoting social skills in adolescents before and after
intervention
1. State the hypotheses
DIFFERENCE SCORES: Set of scores representing the difference between
the subject’s performance on two occasion (x and x ) 1 2
x1 x2 Difference( D )
18 12 6 The difference scores are the new data
and the standard deviation and mean of
5 4 1
19 17 2 the difference scores are the ones that
13 11 2 are used therefore we are testing the
12 8

More
Less
Unlock Document

Related notes for PSYB07H3

Only pages 1,2 and half of page 3 are available for preview. Some parts have been intentionally blurred.

Unlock DocumentJoin OneClass

Access over 10 million pages of study

documents for 1.3 million courses.

Sign up

Join to view

Continue

Continue
OR

By registering, I agree to the
Terms
and
Privacy Policies

Already have an account?
Log in

Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.