Chapter 9: Testing Claims about Proportions Flashcards
Define Significance Test/Hypothesis Test
an inference procedure that uses data to decide between two competing claims about a parameter
Note:
confidence interval = gives us an estimate for out parameter
hypothesis test = don’t give an estimate; they accept/reject a claim about the parameter based on sample data.
Define Hypothesis
always refers to a population, not to a sample.
Be sure to state Ho and Ha in terms of population parameter.
Ho = parameter = value
Ha = P </> value = one sided
Ha = P ≠ value = two sided
Stating Hypothesis
Null Hypothesis Ho - hypothesis of “no difference.” we want to find evidence against this hypothesis
Alternative Hypothesis Ha - hypothesis we are trying to find evidence for
Define P-Value
The definition, assuming Ho is true, that the statistic would take a value as extreme as or more extreme than the one actually observed is called the P-value of the test. The smaller the P-value, the stronger the evidence against Ho provided by the data.
Interpretation of P-value
Assuming __Ho in context (Ho)__, there is a __p-value__ probability of getting the __observed result__ or __less/greater/more extreme__, purely by chance.
ex. Assuming __mean body temperature is 98.6 F (Ho: µ = 98.6__, there is a __0.023__ probability of getting the __sample mean of 97.9 F__ or __less__, purely by chance.
Conclusion in a Significance Test
P- value < ∂ => reject Ho => conclude Ha (in context)
P-value > ∂ => fail to reject Ho => cannot conclude Ha (in context).
Significance Level (∂)
The significance Level, ∂, is the value we use as a boundary to decide if we reject Ho or fail to reject Ho.
Type I Error
We reject Ho, when Ho is true. The data gives convincing evidence Ha is true when it isn’t.
Interpretation of Type I Error
The __Ho context__ is true, but we find convincing evidence for __Ha context__
Type II Error
We fail to reject Ho, when Ha is true. The data does NOT give convincing evidence that Ha is true when it is.
Interpretation of Type II Error
The __Ha context__ is true, but we don’t find convincing evidence for __Ha context__.
Type I Error Probability
The significance level [∂] is the probability of a Type I Error. That is, ∂ is the probability that the test will reject the null hypothesis Ho when Ho is in fact true.
Type II Error Probability
ß is the probability of a Type II Error. the two probabilities (∂ and ß) are inversely related. Decreasing one increases the other in a fixed sample size.
Test Statistic
a test statistic measures how far a sample statistic diverges from what we would expect if the null hypothesis Ho were true, in standardized units.
(statistic - parameter (Ho)) / standard error of the statistic
One-Sample z Test for a Proportion
state:
1. hypothesis (Ho and Ha)
2. ∂
3. define p = the (true) proportion of _______.
plan:
1. random
2. 10 %
3. large count
do:
1. find p hat
2. calculate test statistic
3. find p-value
conclude:
using p-value and ∂
Power
The probability of rejecting Ho correctly.
Power = 1 - ß = 1 - P(Type II error)
Interpretation of Power
If __Ha context is true at a specific value__ there is a __power__ probability the significance test will correctly reject __Ho__.
Increasing the Power of a Significance Test
- increase sample size
- increase ∂ (significance level)
- the Ho and true value of parameter are farther apart.
Tests about a Difference in Proportions
(Two sample z Test for P1-P2)
State:
1. hypothesis with defined p1 and p2
2. ∂
plan:
random, 10%, and large count (with weird p hat)
do:
find p-value using the equation
conclude:
same as 1-sample size.
