PSYB07H3 Study Guide - Final Guide: Confidence Interval, Interval Estimation, Bias Of An Estimator

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18 Apr 2012
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AFTER MIDTERM EXAM NOTES
T-TEST
ONE-SAMPLE T-TEST
-if is known and μ σ2 is unknown, we must estimate σ2 with s2 i.e. if you
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 (s2) is an unbiased estimator of the population
variance (σ2) but s2 is not a normal distribution s2 tends to be
positively skewed (if z is used) we more often underestimate
population variance (σ2) 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
than z because s2 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:
- 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
we use s2
-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 , no need for n-1 just nμ
-1 df is lost because we used (sample mean) to calculate s2
(x-) = 0, all x can vary save for 1Σ
Example 1: the effect of statistic tutorials
Last 100 years: = 76.0 (no tutorials)μ
This year’s: = 79.3 (tutorials)
N = 20
s = 6.4
1. State the null and the alternative hypothesis
H0: = 76 Hμ1: ≠ 76μ
2. Plug information into formula
N
x
N
xx
z
x
2
σ
µ
σ
µ
σ
µ
=
=
=
vs. (substitute s2 for σ2)
N
S
x
N
S
x
S
x
t
x
2
µµµ
=
=
=
error standard
meann Pop' -Mean Sample
=
=
x
S
x
t
µ
N
s
x
µ
=
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3. 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 = 0.05,α
T0.05(19) is ± 2.093 (critical value)
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 the t-value effect size or observed difference
2. Size of s2 as n increases, denominator decreases, t increases
3. Size of n as n increases, denominator decreases, t increases
4. levelα
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 could be without rejecting Hμ0 if we ran a t-test on
the obtained sample mean
Confidence Limits (C.I.)
Using Example 1 data,
-we know , s and N and we know the critical value for t at = 0.05:α
-solve for by rearranging the equation:μ
20
4.6
763.79
=
43.1
3.3
=
31.2=
N
S
x
S
x
t
x
µµ
=
=
093.2
)19(
05.
== t
3.79)43.1(093.2 +±=
µ
3.79993.2 +±=
µ
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Using +2.993 and –2.993
More on Confidence Limits (from Internet)
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
31.763.79993.2 =+=
lower
µ
29.8231.76.
95.
=
µ
IC
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