96 terms

AP Stat Ch. 9

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significance test
a formal procedure for comparing observed data with a claim or hypothesis whose truth we want to asses
claim is a
statement about a parameter, like the population proportion p or population mean u
say the probability of the event that he makes a minimal amount GIVEN THAT he really makes a higher amount on average
Conditional Probability --> if the estimation of this probability is low, it's likely that the claim was false, and that the amount he makes on average is actually lower than he says it is
Two Things can occur when given evidence:
1) the claim is correct and the sample was simply a bad sample
2) the population proportion/mean is actually lower or higher than is stated
explanation 1 COULD be correct, but if it's
unlikely that that occurred after calculating the conditional probability, it's likely that it's false
**JUDGE USING THE 5% RULE**
***an outcome that would rarely happen if a claim were true is
good evidence that the claim is not true***
Null Hypothesis (Hnot)
the claim tested by a statistical test; the statement of "no difference"; the one we want to prove against
Alternative Hypothesis (Ha)
the claim about the population we're trying to find evidence to support; proves the Null Ho wrong
The hypothesis MUST be made before looking at the data, not
looking at the data first and then fitting a hypothesis to it
an Alternative Ha is ONE-SIDED if
it states that a parameter is LARGER/SMALLER than the null hypothesis value
it is TWO-SIDED if it states that
the parameter is DIFFERENT from the null hypothesis (either larger or smaller)
Null Hypothesis has the form:
Ho: parameter = value
Alternative Hypothesis has three forms:
Ha: parameter > value
Ha: parameter < value
Ha: parameter is NOT= to value
** ALL the hypothesis refer to the
POPULATION and NOT the sample, so always state Ho and Ha in terms of population parameters, NOT sample statistics like phat or xbar
P-value
the probability, assuming that Ho is true, that the statistic (such as phat or xbar) would take a value as extreme as or more extreme than the one actually observed.
the SMALLER the P-value, the
stronger the evidence against Ho is --> the observed result is unlikely to occur when Ho is true
larger P-values fail to give good evidence b/c they say that
the observed result is likely to occur by chance when Ho is true
the alternative hypothesis sets the direction that counts as evidence against Ho -->
if one-sided - only low/high counts as evidence against
if two-sided - both count as evidence against
if we can't prove a hypothesis wrong, that doesn't mean it's true, it simply means that
the data are consistent with Ho
never write "accept Ho" because
just because there's not enough evidence to prove its guilt, doesn't mean it's innocent. --> always say "reject" or "fail to reject"
If the P-value is smaller than alpha, we say that the data are STATISTICALLY SIGNIFICANT AT LEVEL "ALPHA" -->
reject the null hypothesis and conclude that there is convincing evidence in favor of the alternative hypothesis Ha
"Significant" means that it's
NOT LIKELY TO OCCUR BY CHANCE ALONE, not that it's important
the actual P-value is more informative than a statement of significance because
it allows us to asses significance at any level we choose (a result of P=0.03 is significant at the alpha = 0.05 leve but not at the alpha = 0.01 level)
When we use a fixed significance level to draw a conclusion in a statistical test,
P-value < alpha --> reject Ho, can conclude Ha
P-value > alpha --> fail to reject, cannot conclude Ha
most commonly used significance:
alpha = 0.5
if going to draw a conclusion based on statistical significance, the
significance level alpha must be stated BEFORE the data are produced (otherwise someone could set the alpha level after data have been analyzed in the attempt to manipulate the conclusion)
the purpose of a significance test is to
give a clear statement of the strength of evidence provided by the data against the null hypothesis. the P-value does this.
How small a P-value is convincing evidence against the null hypothesis?
1) How plausible is Ho --> if common misconception, need strong evidence (really small P-value!) to convince them otherwise
2) What are consequences of rejecting Ho? --> If rejecting Ho, it'll mean making an expensive change --> need strong evidence that it'll be beneficial
giving the P-value allows each of us to decide
individually if the evidence is strong enough
There is NO practical distinction between P-values 0.049 and 0.051 without the
decided alpha level 0.05 --> former causes us to reject Ho, latter causes us to fail to reject Ho
Type I error
reject when Ho is true
Type II error
fail to reject when Ho isn't true
deciding which error is more serious depends on
the context of the question
we can asses the performance of a significance test by looking at the probabilities of
the two types of error
***The significance level alpha of any fixed level test is the probability of a Type I error, meaning,
alpha is the probability that the test will reject the null hypothesis Ho when Ho is in fact true. CONSIDER THE CONSEQUENCES OF A TYPE I ERROR BEFORE CHOOSING THE SIGNIFICANCE LEVEL!
significance test makes a Type II error when it fails to reject
a null hypothesis that is really false
A high probability of a Type II error for a particular alternative means that the test is not
sensitive enough to ususally detect taht alternative
**In the significance test setting, it is more comon to report the probability that a test DOES reject Ho
when an alternative is true --> "power" of the test against that specific alternative --> the higher the probability, the more sensitive the test
the POWER of a test against a specific alternative is
the probability that the test will reject Ho at a chosen significance level alpha when the specified alternative value of the parameter is true
Type II error and Power are
closely linked
the power of a test gives
the probability of detecting a specific alternative value of the parameter --> the choice of that alternative value is made by someone with a vested interest in the situation
power of a test is a number between
0 and 1
power close to 0 -->
the test has almost no chance of detecting Ho as false
Significance Level of a Test:
the probability of reaching the wrong conclusion when the null hypothesis is true
the power of a test to detect a specific alternative is the
probability of reaching the right conclusion when that alternative is true --> THE PROBABILITY OF MAKING A TYPE II ERROR
the power of a test against any alternative is 1 minus the
probability of a Type II error for that alternative --> power = 1-B
How large a sample/how many observations do we need to make to carry out the Significance Test?
1) Significance Level: how much protection we want against a Type I error (getting a significant result from our sample when Ho is true)?

2) Practical Importance: how large a difference between hypothesized parameter value and the actual parameter value is important in the practice?

3) Power: how confident do we want to be that our study will detect a difference of the size we think is important?
***Decreasing the Type I error probability alpha
INCREASES the Type II error probability B
(OPPOSITE IS ALSO TRUE)
the smaller the significance level, the
larger the sample size needed (a smaller significance level requires stronger evidence to reject the null hypothesis)
the higher the power, the
larger the sample needed (higher power gives a better chance of detecting a difference when it is really there)
at any significance level and desired power,
detecting a small difference requires a larger sample than detecting a large difference
to maximize the power of a test,
choose as high an alpha level (Type I error probability) as you are willing to risk AND as large a sample as can afford
9.2
STARTS HERE!
3 conditions must be met before conducting a significance test:
- Random
- Normal
- Independent
test statistic
says how far the sample result is from the null parameter value, and in what direction, on a standardized scale (Normal)
test statistic =
statistic-parameter
----------------
S.D. of statistic
when the conditions are met (RNI) the sampling distribution of ^p is
approximately Normal with
mean u^p = p and S.D. o^p = srt of (p(1-p)/n)
just like last time, to get the STANDARD ERROR, sub in ^p fro p in the
standard deviation formula
z =
^p -Po
--------------
srt of (Po(1-Po)/n)
***this z-statistic has approx. Normal distribution when
Ho is true
One-Sample "z" Test for a Proportion:
Choose an SRS of size n from a large population that contains an unknown proportion p of successes. To test the hypothesis Ho: p = Po, compute the z statistic:
^p -Po
--------------
srt of (Po(1-Po)/n)
see yellow box on page
553
*if the Normal condition and CLT don't apply,
can't do this.
if the evidence from the sample proportion doesn't support the Ha,
there's no need to do a significance test
Steps for Significance Tests:
1) what hypothesis are you using? what significance level? what parameters?
2) choose the method and check the conditions
3) compute the test statistic
4) find the P-value
when conditions are met, sampling distribution of p^ is approx. Normal with
mean up^ = p
and
standard dev. op^ = srt. of p(1-p)/n
for confidence intervals, sub in p^ for p
in the sd formula to get the standard error
BUT in a significance test,
null hypothesis specifies a value for p, which we will call pnot
test statistic:
z = p^ - pnot
----------
srt. of pnot (1-pnot)/n
z statistic is approx. Normal when the standard Normal distribution is true -->
P-values come from the standard Normal distribution
One-Sample Z Test for a Proportion:
choose an SRS of size n from a large population that contains an unknown proportion p of successes --> test the hypothesis Hnot: p = pnot
--> compute the z-statistic
--> find the P-value by getting the probability of getting a z stat this large/larger in the direction specified by the Ha hypothesis
Ha: p > pnot
shade right corner
Ha: p < pnot
shade left corner
Ha: p not = to pnot
shade both corners
*****USE THIS TEST ONLY WHEN:
# of successes/failures npnot and n(1-pnot) are AT least both 10 and 10% condition is met
if you get a sample that is greater than 0.08, it's clear that
you need to do a SIFNIFICANCE TEST --> find the P-value
if the P-value is greater than 5%, we can't
rule out sampling variability as a cause, and must reject Hnot
IF THE PROPORTION IS NOT greater than 0.08,
we don't need to do a significance test b/c we ALREADY fail to reject Ha
the probability of making a Type I error is equal to
alpha, and SO, if Type I error is the one you want to make the least, you should opt for a smaller significance level
A greater risk of a Type I error means a smaller risk of making a
Type II error and a HIGHER POWER to detect a specific alternative value of the parameter.
If they're equally bad, use the standard significance level,
0.05
*****When you reverse the success and failure, you are
changing the sign of the test statistic z. The p-value remains the same, and OUR CONCLUSION DOES NOT DEPEND ON OUR INITIAL CHOICES OF SUCCESS AND/OR FAILURE
confidence interval:
^p +/- z X srt. of (^p (1-^p) / n)
the confidence interval gives an
approximate range of Po's that would NOT be rejected by a two-sided test at the 0.05 significance level (therefore a range of plausible values for the true population parameter p
100(1-significance level)%
___% confidence interval
standard deviation is still
srt of (^p (1-^p) / n)
a two-sided test at Ho:p = po at significance level alpha gives
roughly the same conclusion as 100(1-alpha)% confidence interval
P-value is LESS THAN alpha
we reject Ho
P-value is GREATER THAN alpha
we fail to reject Ho
9.3 BEGINS HERE!!!!!!!!!!!!!!!!!!
9.3 BEGINS HERE!!!!!!!!!!!!
if the sample size is small, its still possible to use it, it just means that
we have to look at the data in order to be able to tell if we can use it (HISTOGRAM)
test Ho: u = uo, statistic is sample mean
xbar
it's stand. dev. is
ox = o/ srt. of n
ideal world test statistic:
z = xbar - uo
--------------
o/ srt. of n
BUT because the actual standard deviation is
unknown, we have to put the SAMPLE s.d. in it's place
question on page
568