Please enable JavaScript.
Coggle requires JavaScript to display documents.
Module 5 - Coggle Diagram
Module 5
Hypothesis Testing for Proportions
Establish Hypothesis: what are you testing?
The null hypothesis always takes the form H0: p = p0
Hypothesis tests are about parameters, so no p^'s in hypothesis!
Always = in H0.
P0 is placeholder; actual problems, it will be an actual proportion.
This represents historical values, claims, or base lines of comparison.
The alternative hypothesis takes one of 3 forms depending upon context:
a) Ha: p < p0 (left-tailed)
less than, decreased, diminished, gone down, below, worse, faster
b) Ha: p > p0 (right-tailed)
more than, greater than, increase, exceeds, above, more often
c) Ha: p # p0 (two-tailed)
change, different, not equal, or verifying a value
still no p^'s.
P0 should be the same value as in H0.
For example: In 2020, acceptance rate for UGA was 45.3%. We want to see if UGA has been more selective this year. What hypotheses would we test?
H0: p = .453
Ha: p < .453
more selective = not as many people admitted = lower acceptance rate.
Represents the acceptance rate in 2022. Already
For example: Your friend claims he wins 53% of the time. You don't believe him. What hypotheses would you test? What does the symbol represent?
H0: p = 53
Ha: p # .53 (use # because verifying 54% without a direction)
Prepresents proportion of the time friend actually wins.
Check Conditions - Is our sampling distribution approximately normal?
a) Unbiased and independent sample - accomplished by random sampling.
b) Sufficient sample size: np0 >= 15 and n(1-p0) >= 15
If these are met, then the sampling distribution under H0 is centered at p = p0, has SE = sqrt((p0q0)/n), and is approximately normal.
Mechanics - Does our p^ fall where we would expect?
a) Find test statistic - For hypothesis test for proportions, the test statistic is Z = (p^ - p0) / sqrt((p0q0)/n); this tells us how many SE's our sample statistic is from our hypothesized value of p.
b) Find p-value - The probability of getting a sample proportion at least this far from p0 if H0 were true.
To find p-value for H-test of proportions, we use standard normal.
For Ha: p < p0 (left-tailed), find P(x <= z) = p-value
For Ha: p > p0 (right-tailed), find P(x >= z) = p-value
For Ha: p#p0 (two-tailed), find p-value = 1-p (-z <= x <= z)
Chap 9: Hypothesis Testing
Up to this point, we have had a definite parameter and have found probabilities associated with hypothetical sample values.
Now, our statistics are definite, and our parameters are hypothetical.
Proof by contradiction/contrapositive
In this type of proof, we establish an initial claim. We then determine the necessary implications of that claim. If those necessary implications are absent, the initial claim must be false.
e.g., if it is
raining outside
(claim),
incoming students should be wet or have an umbrella
(implication). If
students enter dry
(lack of observation of that implication), we can say
it is not raining
(conclude initial claim is false).
A hypothesis test is just a probabilities version of a "proof" by contrapositive.
The initial claim is given by the null hypothesis (H0).
Usually a historical value, widely believed number, or baseline of comparison.
In opposition to H0, we have an alternative hypothesis (Ha) that represents another possibility - oftentimes what we aim to prove.
We then take a sample and find a statistic. We try to determine if this statistic is feasible/probable in cases where H0 is true.
If H0 true, then we would expect this statistic to be close to whatever H0 says is the parameter.
We quantify how well or results align with the expectation under H0 using a p-value
P-value
: the probability of finding sample results at least as extreme as what we found if the H0 were true.
We compare this p-value to our standard of evidence, which is the significance level (alpha)
If our statistic is too unlikely under the assumptions of H0, we reject H0 and go with the conclusions of Ha.
If the statistic aligns with our expectations under H0, we do not reject H0.
Suppose we consider H0: mu = 15 vs. Ha: mu >15. Sampling distribution of x bar under H0
What is the most likely, that we got a strange sample from a population centered at mu = 15, or that we got a normal sample from a population centered at mu > 15?
