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Chapter 7 & 8 ( & Hypothesis testing)
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Level of confidence c
.70, 70% invnorm (.85, 0,1) = 1.04
.75 invNorm (.75,0,1)=1.15
.80 invNorm (.80,0,1)=1.28
.85 invNorm (.85,0,1)=1.44
.90 invNorm (.90,0,1)=1.645
.95 invNorm (.95,0,1)=1.96
.98 invNorm (.98,0,1)=2.33
.99 invNorm (.99,0,1)=2.58
Use Student's t distribution
when omega is unknown (pop. stand. deviation)
Degree of Freedom d.f.
n-1
As the degrees of freedom df increase, what distribution does the students t dist. become more like?
Approaches the standard normal distribution
Student's t distributions are symmetric about a value of t. What is that t value
t=0
for a binomial experiment with x successes out of n trials, what value do we use as a point estimate for the probability of success p on a single trial?
p hat = x/n
In order to use normal distribution to compute confidence intervals for p, what condition on np & nq need to be satisfied?
np>5 nq>5
Under what conditions is it appropriate to use a normal approximation to the binomial?
np & nq >5 where q=1-p for normal approximation to the binomial
Standard error of proportion:
- mu w/ # of successes (x) =np
- omega w/ # of successes (x) = square root of pq/n
Omega 1 & Omega 2 are KNOWN
use (Z) normal distribution with margin of error
Omega 1 & Omega 2 are UNKNOWN
Use Student's t distribution w/ margin of error. d.f. = smaller of n1 -1, n2-1
When are 2 random samples independent?
if sample data drawn from 1 population are completely unrelated to the selection of sample data from the other population.
When are 2 random samples dependent?
If each data value in one sample can be paired with a corresponding data value in the other sample.
P-value (use appendix tables to get z)
- Assuming null is true, the probability that the test statistic will take on values as extreme as or more extreme than the observed test.
- The smaller the p-value computed from sample data, the stronger the evidence against null
Power of a test
1-B beta
represents the probablity of rejecing null when it is, in fact, false
What is null hypothesis? H0
being tested. Usually represents a statement of "no effect", no difference, things haven't changed.
What is an alternate hypothesis? H1
data is so strong that you reject H0. A statistical test is designed to assess the strength of the data against the null hypothesis.
What is a type I error? type II error?
We reject the null hypothesis H0, when in fact it's TRUE. (H0 is T)
We accept the null hypothesis H0, when in fact it's FALSE (H0 is F)
What is the level of significance of a test?
The probability with which we are willing to risk a type I error
What is the probability of a type II error?
B (greek beta)
If we fail to reject the null H0, have we proved it is true beyond all doubt?
No, we have failed only to find sufficient evidence to reject it.
The alternate H1 determines
Right tailed, Left tailed or 2-tailed
If we reject the null H0?
Type I error - Null is true but we REJECT IT.
If we fail to reject the null H0?
Type II error - Null is true but we FAIL to reject it.
What terminology do we use for the probability of rejecting the null hypothesis when it is true?
Level of significance, (alpha symbol), Type I error
What terminology do we use for the probability of rejecting the null hypothesis when it is false?
B beta or probability of a type II error
mean of distribution of a sample mean
mu x bar
Standard error of the means
omega x bar
Sample mean?
x bar
Population mean?
Mu
Sample proportion?
p hat
Population proportion?
p
Population Variance?
omega ^2
Sample standard Deviation
s
Sample Variance?
S^2
Sample Size?
n
Population Size?
N
Population Standard Deviation
omega
H0
Null hypothesis (=)
H1
Alternate hypothesis (<,>,not=to)
Null is true but we REJECT IT.
Type I error
Null is true but we FAIL to reject it.
Type II error
Probability of a type I error
alpha
Probability of a type II error
beta
mu >
Left tailed (p value is on the left)
mu <
Right tailed (p value is on the right)
Mu not=to
2 tailed (p values on both sides of the mean)
Interpret: p value > alpha
There is insufficient evidence at the 0.01 level to reject the null. At this point in time there's not enough data to conclude
How to test for pop. mean (mu), when pop. stand. dev (omega) is KNOWN
1. state H0 & H1
2. use know omega, sample size (n), the value x & mu from H0 to compute standardized sample test statistic. z=xbar - mu divided by omega/sq.root of sample size.
3.Use standard normal distribution (z-test) & 1 or 2 tailed to find p-value
4.Conclude the test if p-value <= alpha the reject H0, if p-value is > alpha, then do not reject H0
5. Interpret your conclusion
How to test for pop. mean (mu), when pop. stand. dev (omega) is UNKNOWN
1. state null & alternate hypothesis& set level of significance.
2. use xbar, s, & n from sample, with mu from null , to compute the sample test statistic
3.
Critical Region method of testing Mu
1. Compute the sample test using appropriate sampling distribution.
2. find critical values as determined by level of significance (alpha) & the nature of the test (l-tailed, r-tailed, 2-tailed)
3.
For sample data & null hypothesis, how does the P-value for a 2-tailed test of mu compare to that for a one-tailed test?
P-value for a 2-tailed test of mu is twice that for a one-tailed test, based on the same sample datea and null hypothesis.
not=
means a two-tailed test
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