AP Stat Ch 9-10 Flashcards
Hypothesis test
A method for using data from a sample to decide between two competing hypotheses about a population characteristic.
Hypothesis
A hypothesis is a claim or statement about the value of a single population parameter.
Null hypothesis
A statistical test starts with a careful statement of the claims we want to compare. We initially assume that one of the hypothesis is true. This is called the null hypothesis. Ho
This is the claim, the population characteristic.
Alternative hypothesis
The other hypothesis, Ha. This is the alternative to the claim
Not equal to, bigger, or smaller than the null hypothesis
Proportion hypothesis test steps
- State the hypothesis, alpha level, define parameters, one sample z test for prop
Ho and Ha and alpha and define p = … - Conditions
A. SRS
B. Normal? np>10 and n(1-p)>10. Use the null hypothesis p value, not p hat
C. Indep? As long as population size is at least 10 times sample size - Calculate
If not equal to, 2*P (p hat <= sample proportion) = 2 * P (Z<= (p hat-p)/SQRT (p(1-p)/n)
Get the P-value - Conclusion
Either: since my p value is bigger than my alpha value, I fail to reject Ho and cannot conclude Ha in context
Or since my alpha value is bigger than my p value, I reject Ho in favor in favor of Ha and conclude Ha in context
Type I error
Rejecting Ho, when Ho is actually true.
Convicting an innocent person
Type II error
Failing to reject Ho when Ho is false
Letting guilty go free
Ex. A machine is set to produce bolts with an average diameter of 1 cm. Every hour a sample is inspected and the machine is adjusted if there is convincing evidence that the average diameter is not 1 cm.
State Ho, Ha, type I and type II errors
Ho: mu = 1
Ha: mu not equal 1
Type 1: mean actually is 1, but you say it’s not – adjust the machine but shouldn’t have
Type 2: shouldn’t have adjusted the machine, but did
Prob of type I error
Alpha, the level of significance.
The probability that the test will reject the null hypothesis when Ho is actually true.
Probability of type II error
Denotes by Beta. Ha is true, but you fail to reject Ho.
Power of a test
Power is 1-Beta, the probability of correctly rejecting Ho. The probability of rejecting Ho when it is false.
Prob that Ha is true and you reject Ho
How to increase the power
- Increase significance level (alpha):
Alpha bigger, beta smaller, so power bigger - Consider alternative farther away from mu not. Larger effect size
- Increase the sample size
More data, more information about X bar, so better chance of distinguishing values of mu. n increases, x bar –> mu. More confident about p value, more confident about decision to reject Ho. - Decrease sigma
Key: choose as high an alpha level you’re willing to risk and large a sample size as you can afford
One sample z test for a population mean (assume sigma is known)
Standardize and use test statistic z = x bar - mu not / (sigma / SQRT (n))
Steps:
1. State Ho, Ha, alpha level, 1 sample z test for mu = true mean…
2. Conditions
A. SRS?
B. Norm? n>30 or make a box plot
C. Indep? Assume more than 10 times population
3. Calculate
P (x bar > sample stat) = P (Z > (x bar - mu not)/ (sigma/ SQRT (n)))
Gives the p value
4. Conclude:
Either since my p value is less than my alpha value, I reject Ho in favor of Ha and conclude Ha in context
Or since my p value is greater than my alpha value, I fail to reject Ho and cannot conclude Ha in context.
One sample t stat
t = (x bar - mu)/ (s/ SQRT (n))
Four step process for one sample t test
- Hypotheses:
State Ho, Ha, alpha level, 1 sample t test for mu = true mean… - A. SRS?
B. Large sample size? (n>30 or pop is approx normal (look at data))
C. Independence? Pop ten times sample - Test:
n-1 degrees of freedom–>
P (x bar < value) = P (t<= (x bar - mu not)/(s/SQRT(n))
P (t< value)
Use the table, get df = n-1 and round down. Look for t value and go to top, find what p value it is between - Conclude:
Since p value is bigger than alpha, I fail to reject the null hypothesis and cannot conclude the alternative hypothesis in context.
Or since alpha value is bigger than p value, I reject null hypothesis and conclude alternative hypothesis in context
Paired t test
Still a one sample t test, but data is paired
Pair subjects together, look at the difference in each pair
- Hypothesis:
Ho: mu difference, usually = 0
Ha: mu not equal to zero or bigger or smaller
Five alpha value, name the test, one sample matched pair t test
Conditions:
A. SRS? Randomly assigned to groups A and B
B. Normality? (n>30 or look at the data)
C. Indep? More than ten times sample size in pop - Calculate:
P (x bar > value) = P (t> (x bar - mu)/(s/SQRT(n)) - Conclude in context
Confidence interval for difference of two proportions
Remember statistic +- (critical value)(standard deviation of stat)
4 step:
1. ___% z interval for p1 - p2 where p1 = and p2 =
2. Conditions:
A. Random given to each group and random treatments assigned
B. Normality? FOUR TESTS n1(p1), n1(1-p1), n2(p2), n2(1-p2) all need to be greater than 10
C. N>10n
3. Calculate:
Use p hats–>
(P1-p2) +- z and then the formula is on back page of reference sheet
Get the interval
4. Conclude:
C% confident that the interval from ___ to ___ less if neg or more if positive will result from product one than 2
If all of interval is pos or neg, then can conclude which is more effective
Ho and Ha for significance tests for p1-p2
An observed difference between two sample proportions can reflect an actual difference in the parameters or it may just be due to random chance variation in sampling. Significance tests help us decide which explanation makes more sense
We assume hypothesized difference between parameters is 0, so
Ho: p1-p2=0 or p1=p2
Ha: whatever difference we expect
P hat c
Weighted average of p hat 1 and p hat 2 for a two proportion significance test
(n1(p hat 1) + n2(p hat 2))/(n1 + n2)
Total number of successes / total number of samples
For example, if p1 = 7/10 and p2 = 12/20 (p hat)
P hat c = 7+12 / 10+20
4 step procedure for a two sample z test for difference in proportions
- 2 prop z test for alpha = … for p1-p2 where p1= and p2 =
Ho: p1=p2
Ha: either p1>p2, less than, or not equal to - Conditions:
A. Random sample
B. Normality? FOUR TESTS n1(p1), n1(1-p1), n2(p2), n2(1-p2) all need to be greater than 10
C. Greater than two times population - Calc:
STAT, TEST, 6
WRITE DOWN Z SCORE - If p value is less than alpha, reject null in favor of alternative and conclude alternative in context
If p value is bigger than alpha, fail to reject null, cannot conclude alternative in context
4 step procedure for two sample means
- 2 sample t test for alpha =, for mu1-mu2 where mu1= and mu2=
Ho: mu1=mu2
Ha: some variation (less than, greater than, not equal to) - Conditions:
A. Random
B. Sample sizes are large (both greater than 30)
Or both populations are approx normal
C. Indep–> greater than 10 times population - Calculate:
t= (observed difference - expected difference) / standard deviation
Much easier to use calculator– STAT, TEST, 4
Never pool - Conclude based on alpha and p value
4 step procedure for t two sample confidence interval for means
- ___ % confidence t interval for mu1 - mu2 where mu1= and mu2 =
- Conditions:
A. Random
B. Greater than 30 or normal
C. Indep– N>10n - Calc:
(x bar1 - x bar 2) +- t* STNDRD DEV on form sheet
Note: df is given by smaller of n1-1 or n2-1 - Conclude
