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AP Stats: Unit 5 (Chapters 18- 22)
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Terms in this set (21)
Conditions for sampling distribution of categorical data to be Normal
sampling distribution is called p-hat
1. independence assumption: success is independent for different members of sample
2. SRS
3. 10% condition: < 10% of the population
4. success/failure condition
np > 10
nq> 10
z - score
(p-hat - p)/SD(p-hat)
Conditions for sampling distribution of quantitative data to be Normal
sampling distribution is called y-bar
1. independence assumption
2. SRS
3. 10% condition
4. Large enough condition
--- unimodal and symmetric, then sample size 10 okay
-- if it's skewed, 20 or greater
Example
Population: mean = 30.5 SD = 3.3
Sample size: n = 105
Sample mean: y-bar = 29.8
Sampling distribution: N(30.5, 3.3/sqrt{105})
[Valid if (a) sample random, (b) sampled elements independent of each other, ( c) sample < 10% of population (d) sample large enough (> 20, or even smaller if population distribution unimodal and symmetric]
z-score for y-bar = (29.8 - 30.5)/0.32 = -2.1875
y-bar is 2.18 standard deviations below the mean. This is quite small but perhaps not unusually so.
(There is a 1.4% of being 2.18 SDs below the mean.)
hypothesis test
null hypothesis H0 : p = 0.5
alternative hypothesis HA:
p > 0.5 one sided upper tail (ONE PROP Z-TEST)
p not equal 0.5 two sided (TWO PROP Z- TEST)
p < 0.5 one sided lower tail.
Given sample of size n with sampled value equal to phat.
Can apply normal model under following assumptions
-- independence assumption
-- randomization conditio
-- 10% condition
-- Success failure condition.
Compute SD(p^) = sqrt{p q/n}
compute
zhat = phat - p / SD (p^)
from that compute p- value
Conditions needed:
random sample,
independence of samples
< 10% of population
np > 10, nq > 10,
p value
Probability of getting these results | null hypothesis is true.
= Probability (observed statistic value | H0 = true)
HA = p > 0.5
p value = Pr (of having z > zhat)
HA = p < 0.5
p value = Pr (of having z < zhat)
HA = p not equal 0.5
p value = Pr (of having |z | > |zhat|)
p-value large (e.g. > 0.2)
There is insufficient evidence to reject the null hypothesis.
The sample is not inconsistent with the null hypothesis,
and could be accounted for by sampling variability.
Big p value means that what we've observed is not surprising given the null hypothesis.
"don't reject the null hypothesis"
p-value small (e.g. < 0.05)
If the null hypothesis were true, the probability of observing a sample with this value of statistic (or further from mean) is extremely unlikely, so I reject the null hypothesis.
There is evidence that the alternative hypothesis is correct.
"reject the null hypothesis"
alpha level or significance level
significance level
the level of p-value below which we reject null hypothesis.
common value is 0.05 (5 %)
If < 0.05, say we "reject ht null hypothesis at the 5% level of significance.
confidence interval
used to estimate what proportion of a population has a certain characteristic.
TAke a sample of size n
observe fraction p^ with characteristic.
compute
SE(phat) = sqrt{ phat q-hat/ n)
Assume sampling model is Normal
For a 95% confidence interval
need z^* = 1.96
(Probability that a normal(0.1) random variable within -1.96 standard deviations and 1.96 standard deviations is 0.95)
Margin of Error
ME = z^* SE(p^)
p-hat plus/minus Margin of Error
E.g. if have a 95% confidence interval which is
(0.55 - 0.64), then you are 95% confidence that between 0.55 and 0.64 fraction of people have this characteristic.
sometimes use confidence interval to decide whether to accept or reject hypothesis
If the null hypothesis lies within a 95% confidence interval for sampled data, then don't reject the null hypothesis.
type I error
null hypothesis true, but we mistakenly reject it.
type II error
null hypothesis false, but we fail to reject it.
H0 -- null hypothesis
HA - alternative hypothesis
P value small
Conclusion:
Conclusion: I reject the null hypothesis.
There is evidence to support the alternative hypothesis.
H0 -- null hypothesis
HA - alternative hypothesis
P value big > 20%
What is the conclusion?
Conclusion: I fail to reject the null hypothesis.
There is insufficient evidence to support the alternative hypothesis.
What does the P value mean?
If the [null hypothesis was true], then we would expect t see [this data] about P percent of the time.
H0 -- null hypothesis
HA - alternative hypothesis
P value small
What does the P value mean?
What does the P value mean?
If the null hypothesis was true, it would be very unlikely to see this data.
H0 -- null hypothesis
HA - alternative hypothesis
P value big > 20%
What does the P value mean?
What does the P value mean?
If the [null hypothesis was true], then we would expect t see [this data] about P percent of the time.
Usually
Null hypothesis: status quo
Alternative hypothesis: result of investing money in something.
Type 1 error: status quo true, but reject it.
Waste money on something
Type 2 error: investment of money worthwhile, miss the opportunity to improve sales.
If Null hypothesis is in confidence interval
trials of new thing (e.g. advertising) doesn't provide convincing evidence that the new marketing campaign works.
advantage and disadvantage of larger alpha
if null is false, more likely to reject it, and take advantage of new marketing campaign
if null is true, more likely to reject it and waste money on new marketing campaign.
advantage and disadvantage of smaller alpha
if null is true, more likely to accept it and not waste money on useless marketing campaign
if null is false, more likely to accept it
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