Hypothesis ((P-value) T Tests (Statistics II Review of 8.2+) page 1 of 7 BETA FORM (still developing developing))
Tip: You need to read sections 8.1 and 8.2 in your text. You can also reference http://www.rmower.com/statistics/stat_lecture/130_brie http://www.rmower.com/sta tistics/stat_lecture/130_brief_hypothe f_hypothesis_testing.pdf sis_testing.pdf
ef: Stat Essen for Dummies and Cartoon Guide to Stats 1993 R ef: Hypothesis testing is a statistician's way of trying to confirm or deny a claim about a population using data from a sample. For example, you might read on the Internet that the average price of a home in your city is $150,000 and wonder if that number is true for the whole city. Or you hear that 65% of all Americans are in favor of a smoking ban in public places. Is this a credible result? Rare Event Rule: If, under a given assumption, the probability of a particularobserved event event is exceptionally small, we conclude that the assumption is probably
not correct. Small or “unusual” is less than 5% = .05. In terms of standard deviations, small is more than 2 standard deviations from the mean.
Doing a a H Hypothesis T Test
(single s sample h hypothesis ttest) A hypothesis test is a statistical procedure that's designed to test a claim. Typically, the claim is being made about a population population parameter parameter (A (A “ pp” is Step 1. Formulate the NULL and a number that measures a characteristic of the entire population). Because ALTERNATE Hypotheses: Hypothese s: parameters tend to be unknown quantities, claims are made about what their values may be. For example, the claim that, “25% (or p = 0.25) of all H0: The null hypothesis is always “=” such See the table below as H 0 0: p = p 0 . women in the population eat chocolate once a week”, is a claim about the (population) proportion of women who eat chocolate once a week. (In this ≠ ”, ”, H1: The alternate hypothesis is always “ ≠ case, p is the population population parameter parameter and p0 = .25 is the hypothesized or “<” or “>” such as H 1: p < p 0 . assumed “ pp”.) “ pp http://www.rmower.com/statistics/stat_lecture/130_Rare_Event_Rule.pdf Note: H 1 determines the tail type. p0 is the H 1: p < p0 is “left 1-tail”. up tthe N Null a and A Allter nate H Hypothesis Step 1 1. Set u hypothesized H : p > p is “right 1-tail”. 1 0 The claim in this case, p = 0.25, is the null hypothesis, H 0. (Note that the value H : p p ≠ 1 0 is “both 2-tail”. claim is not always H 0.) If we're out to test this claim, we're questioning the 0 or H 1. claim and have a hypothesis of our own that is called the research Note: The “CLAIM can be either H 0 hypothesis, or alternative hypothesis, H 1. We could hypothesize, for example, that due to the sugar in chocolate causing weight gain, the proportion is less than than 0.25. 0.25. In other circumstances, circumstances, our alternate alternate hypothesis could be greater than 0.25. In our case, we're simply questioning whether the actual proportion is 0.25, so our alternative hypothesis is, " p ≠ 0.25."
T ai l C r l T T y y p es, C ri i t t i ic c a l Z Z S S i ig g n s for Hypothesis Tests
The tail type is determined by H 1 Equal o or at least
H 0
H 1
Type
Critical Value Sign
=
<
1-tail Left
♦
negative (-)
Equal
♦
> 1-tail Right positive (+) o or at = In addition to testing hypotheses about proportions, we can also test most hypotheses about means or variances, such as the average commuting Equal = 2-tail Both pos & neg (±) ≠ to time for people working in Los Angeles or their average household income. Confidence e Intervals are 2- ail Note: All Confidenc In these cases, the parameter of interest is the population mean (denoted ). When testing testing the variance, the parameter would be the µ ). Would you reject this assumption for H 0? variance or standard deviation (denoted σ 2 or σ ). ). Again, the null hypothesis is that this parameter is equal to a certain value, µ 0 or U N L UC UC K K Y C H A AP P, σ 0, versus some alternative. W A AT T S S O O N N… ♦
Setting up the Hypotheses and Notation Every hypothesis test contains two hypotheses. The first hypothesis is called the null hypothesis, denoted H 0. The null hypothesis always states that the population parameter is equal to the hypothesized (or assumed) value. For example, if the claim is that the average time to make Jiffy Muffin Mix is less than five minutes, the statistical shorthand notation for the null hypothesis would be as follows: H 0: µ = 5. Since the claim is < 5 minutes and H 0 is always “=”, the alternate hypothesis and the claim would be H 1: µ < 5 (claim).
H 0 : A Accident ccident CLAIM
H 1 : N o accident
H 0 : A P P A R E N T T L L Y H E M E T W I T H A AN N AC A C C I D E N T …
H 0 : A Accident ccident H 1 : N o accident
Rare event rule: If, under a given assumpt assumption ion (that his death was an accident), the probability of a particular observed Looks like we should apply the, “ Rare Event Rule”. event (axe, gun, sword, arrows, needle, pills) is exceptionally small, we conclude that the assumption is probably not correct. In other words, there is sufficient evidence to warrant rejection of the claim that it was an accident. http://www.rmower.com/statistics/stat_lecture/130_Rare_Event_Rule.pdf
Hypothesis T Tests Continued (Statistics II Review) page 2 of 7 When the claim is NOT equality (Not H 0 as in the muffin example), the claim will be
H 1: µ < 5 Claim. If we collect evidence and conclude that the null hypothesis is true (we fail-to-reject H 0), we will then say that, “There is not sufficient sample evidence to support the claim that H 1 < 5”. (See the, “Determining the Final Conclusion”, flowchart.)
Again, when the claim is NOT equality, the claim will be H 1: µ < 5 Claim. But, if we collect evidence and conclude that the null hypothesis in not true (we reject H 0), we will then say that, “The sample data support the claim that H 1 < 5”.
Start The claim IS H 0 Yes
Hypothesis Tests: Deter mining tthe F Final C Conclusion
Is the claim H 0?
Did you reject H 0?
When the claim IS equality (H 0: µ = 5 Claim) and we fail-to-reject H 0, We will then say that, “There is not sufficient evidence to warrant rejection of the claim that H 0: µ = 5”.
The claim is NOT H 0
H 0 IS Yes rejected The claim IS H 0 and
No
No
H 0 is NOT rejected H 0 IS Yes rejected
However, when the claim is equality (H 0: µ = 5) and we reject H 0, we will then say that, “ There is
Did you reject H 0?
sufficient evidence to warrant rejection of the claim that H 0: µ = 5”.
No
The claim is NOT H 0 and
H 0 is NOT rejected
There is sufficient evidence to warrant rejection of the claim that…(original claim) There is not sufficient evidence to warrant rejection of the claim that…(original claim) The sample data support the claim that…(original claim) There is not sufficient sample evidence to support the claim that…(original claim)
Choice for H 0: ♦ The
population parameter is always equal to the hypothesized value (H 0: µ = 5).
Choices for H 1: ♦ The
population parameter is not equal to the hypothesized value (H 1: µ ≠ 5). ♦ The population parameter is greater than the hypothesized value (H 1: µ > 5). ♦ The population parameter is less than the hypothesized value (H 1: µ < 5). Which alternative hypothesis we choose will depend on the stated claim. If equal or not equal is claimed, then H 1: µ ≠ 5. If greater than is claimed, then H 1: µ > 5. If less than is claimed, then H 1: µ < 5. Remember that H 0: µ = 5 is always the case. Also, always LABEL THE CLAIM. Special cases: If greater than or equal is claimed, then H 1: µ < 5 . If less than or equal is claimed, then H 1: µ > 5. In the special cases, the claim will be H 0 and H 0 will still be H 0: µ = 5 (claim). ”YOU MADE A TYPE I ERROR !
T ai l C r l T y y pe p es, C ri i t t i ic c a l Z S i ig g n s for Hypothesis Tests
The tail type is determined by H 1 Equal o or at least
H 0
H 1
Type
Critical Value Sign
=
<
1-tail Left
negative (-)
=
>
1-tail Right
positive (+)
=
≠
2-tail Both
pos & neg (±)
♦
Equal o or at most
♦
Equal to
♦
Note: All Confidence Intervals are 2- ail
Jury Trials In a sense, hypothesis tests are similar to jury trials. In a jury trial, H 0 is similar to the not-guilty verdict, and H 1's the guilty verdict. We assume in a jury trial that the defendant isn't guilty unless the prosecution can show beyond a reasonable doubt that he or she is guilty. If the jury says the evidence is beyond a reasonable doubt, they reject H 0, not guilty, in favor of H 1, guilty. Type I Error: Reject H 0: “Not Guilty”, when it is, in fact, true.
Step 4 4. Finding sample statistics
After we select our sample, the appropriate number crunching takes place. Our null hypothesis makes a statement about what the population parameter is (for example, the proportion of all women who eat chocolate or the average miles per gallon of a U.S.-built light truck). We need a measure of how much our results can be expected to change if we took a different sample. In statistical jargon, the data we collect measure that variable of interest, and the statistics that we calculate will include the sample statistic that most closely estimates the population parameter. If we're testing a cl aim about the proportion of women who eat chocolate , we need to calculate the proportion of women in our sample who eat chocolate. If we're testing a claim about the average miles per gallon of a U.S.-built light truck, our statistic should be the average miles per gallon of the light trucks in our sample.
Hypothesis T Tests Continued (Statistics II Review)
Standard error - def. the standard deviation of the sampling
distribution associated with the estimation method. Standard score - def. a standard score indicates how many standard deviations an observation or datum is above or below the mean or other parameter. Also called the Test Statistic . For means a and p pr opor tions: Sample d data → s sample s statistic a and s standar d d deviation → s standar d s scor e f f or mula ((test s stat a as a a z or t )
page 3 of 7
Standar dizing tthe e evidence: tthe ttest s statistic ˆ or After we have our sample statistic (such as x , p s), we may think we're done with the analysis part and are ready to make our conclusions - but we're not. The problem is we have no way to put our results into any kind of perspective just by looking at them in their regular units. The number of standard errors that a statistic lays above or below the mean is called a standard score (Ch 8/9 will be a z, t or F -score.). To interpret our statistic, we need to convert it from original units to a standard score.
Note: In a p-value test (as opposed to a 6-step traditional test on p7), we do not do steps 2 (find the critical value) and 3 (draw a picture). Also, we only do steps 4 and 5 if an α is given.
When finding a standard score for a sample mean or proportion, we take our statistic, subtract the mean, and divide the result by the standard error. This is the test stat formula in your formula reference sheet. In the case of hypothesis tests, we use the value in H 0 as the mean. (That's because the assumption is that H 0 is true, unless we have enough evidence against it.) This standardized version of our statistic is called a test statistic , and it's the main component of a hypothesis test.
Step 4. Compute the Test Stat and Determine the p-value: Use the formula for the test statistic that will assess the evidence against the null hypothesis. Then use the test statistic and appropriate table or software to determine the p-value. Note: A p-value is a probability statement that answers the question: If the Null Hypothesis were true, what is the probability of a test statistic at least as extreme as the one we just computed. The smaller the p-value, the stronger the evidence against H 0 . (Recall that, “at least”, means, “greater than or equa to and , “at most”, means, “less than or equa to”.)
Note: Most software programs supply the p-value.
FORMULAS: http://www.rmower.com/statistics/hyp_test_etc_ref_package/formulas_mower.pdf
The g gener al p pr ocedur e f f or conver ting a a s statistic tto a ttest s statistic ((standar d s scor e): 1. Look up the test stat formula. 2. Compute the test stat.
P-Values Left tail: Area to the left of the test statistic Right tail: Area to the right of the test statistic Two tail: Test Stat to left of center: Twice the area to the
left of the test statistic. Test Stat to right of center: Twice the area to the right of the test statistic. (To find the areas, use the “Strategies to Find Areas” included in the Hypothesis package page 2 at the right. )
Our test statistic represents the distance between our actual sample results F a t o o r r e j e f α < p - v a r e e j e f α ≥ p F a i l i l t j e c t c t H o i i f v a l u l u e . O r , r j e c t c t H o i i f and the assumed population value in terms of number of standard errors v a v a l u l u e . (number of z, t or F -scores). If we see that the distance between the assumption 0 and the test statistic is small in terms of standard errors, our sample isn't far from 0 and our data are telling us to stick with H 0. If that distance is large, however, our data are showing less and less support for H 0. The next question is, how large of a distance is large enough to reject H 0?
Weighing the evidence and making decisions: p-values (Assumption is always “H 0: parameter = hypothesized value”. Recall that the CLAIM can be either H 0 or H 1.) To test whether the assumption H 0 is true, we're looking at our test statistic taken from our sample, and seeing whether it supports H 0. And how do we determine that? By looking at where our test statistic ends up on its corresponding sampling distribution - see Chapter 6, “Central Limit Theorem”. In the case of means or proportions (if certain conditions are met) we look at α α where our test statistic ends up on the standard normal z or t 2 2 distribution. The z -distribution has a mean of 0 and a standard deviation of 1. If our test statistic is close to 0, or at least within that range where most of the results should fall, then we fail to reject H 0. If our test statistic is out in a tail of the standard normal distribution, far from 0, it means the results of
Hypothesis T Tests Continued (Statistics II Review) page 4 of 7 this sample do not verify the assumption. Hence we reject H 0. But how far is "too far from 0"? If the null hypothesis is true, most (about 95%) of the samples will result in test statistics that lie roughly within 2 standard errors (2 z ’s or t ’s) of 0. Any test statistic outside this range will result in H 0 being rejected (see the Figure – for a 2-tail test). If your test statistic is close to 0, you can't reject H 0. However, this does not mean you accept the clai m as truth either. Because H 0 is on trial, and the test statistic is the evidence, either there is enough evidence to reject “H 0: Not Guilty” or there isn't. In a real trial, the jury's conclusion is either guilty or not guilty. They never conclude "innocent." Similarly, in a hypothesis test we either say "reject H 0" or "fail to reject H 0" - we never say "accept H 0"
α
α
2
2
Finding tthe p p-value You can be more specific about your conclusion by noting exactly how far out on the standard normal distribution the test statistic fall s, so everyone knows where the result stands and what that means in terms of how strong the evidence is against the claim. In the case of means or proportions (if certain conditions are met), you do this by looking up the test statistic on the standard normal distribution (z -distribution, Table A-2 or other) and finding the probability of being at that value or beyond it (in the same direction). This p-value measures how likely it was that you would have gotten your sample results if the null hypothesis were true. The farther out your test statistic is on the tails of the standard normal distribution, the smaller the p-value will be, and the more evidence you have against the null hypothesis being true. P-Values Left tail: Area to the left of
To find the p-value for your test statistic: 1. Look up the location of your test statistic on the standard normal distribution (see Table A-2).
the test statistic Right tail: Area to the right of the test statistic Two tail:
2. Find the percentage chance of being at or beyond that value in the same direction: a. If H 1 contains a less-than alternative (left tail), find the probability from Table A-2 that corresponds to your test statistic. b. If H 1 contains a greater-than alternative (right tail), find the probability from Table A-2 that corresponds to your test statistic, and then take 1 minus that. (You want the percentage to the right of your test statistic in this case, and table gives you the percentage to the left.) 3. Double this probability if (and only if) H 1 is the not-equal-to alternative. Doubling adjusts for t he split alpha in the two tails.
Test Stat to left of center: Twice the area to the left of the test statistic. Test Stat to right of center: Twice the area to the right of the test statistic. (To find the areas, use the
“Strategies to Find Areas” included in the Hypothesis package page 2 at the right. ) F a i l tt o r r e j e c t H o ii f α < p - va l u e . O r , r r e e j e c t H o ii f α ≥ p - va l u e .
Step 5 5. Inter pr eting a a p p-value ne x t t pa pag e T h i r d Roya l F lu s h?
H e ’s che at in g ! (C l ai m )
H 0: (Assume) He is not guilty of cheating H 1: He is guilty of cheating (Claim) Apply the rare event rule: If, under a given assumption (that he’s not guilty of cheating), the probability of a particular observed event (he got 3 royal flushes) is exceptionally small (one out of millions, maybe billions), we conclude that the assumption is probably not correct. Hence, we Reject H 0. The evidence supports the claim that, “He’s guilty of cheating”.
Rev iew) p page 5 5 o of 7 Hypothesis T Tests Continued ((Statistics III R See: http://www.rmower.com/statistics/prog_calc_stat/conf_int_hyp_handouts/p_value_concept.PDF >General p-value See: http://www.rmower.com/statistics/prog_calc_stat/conf_int_hyp_handouts/calculator_pvalue.PDF >p-values on the TI-83/4
See: http://www.rmower.com/statistics/stat_lecture/132_hypothesis_testing_and_p-values.pdf
>p-value examples
a p p-value (links aabove) Step 5 5. Inter pr eting a To make a proper decision about whether or not to reject H 0, you determine your cutoff probability for your p-value before doing a hypothesis test; this cutoff is called an alpha level α . . Typical values for α are 0.05 or 0.01. Here's how to interpret your results for any given alpha level: (Recall that the p-value is a probability (or area under the curve) – see the, “P-Values”, box below right.)
α
α
2
critical # HSS
SS MS
(Recall that α is the probability (or area under the curve) of a Type I Error – the probability of rejecting H 0 when it is, in fact, true. A Type II Error – β is the probability of failing to reject H 0 when it is, in fact, false. An α is chosen by the person in charge of the test where a β error must be computed.)
H 0 mean z = 0
NS Location of test stat
2
critical # MS
SS
HSS
p-Value and Significance (when significant you Reject) HSS p-val <.01 highly statistically significant SS .01 < p-val <.05 statistically significant MS p-val close .05 marginally significant NS p-val > .05 non-significant
Just try to remember the bold one and you can figure the rest from that.
Step 5. Reject or Fail-to-Reject H 0, or State the p-value.
Here's how you interpret your results objectively for any alpha:
Compare the p-value to a fixed significance level α α to determine whether to Reject or Fail-to-Reject H 0 .
♦If
the p-value reject H 0.)
≥
α , fail-to-reject H 0. (Or, If α α < p -value, fail-to-
The test stat will be in the fail-to-reject H 0 zone. ♦If
the p-value < α , reject H 0. (Or, If the α ≥ p-value, reject H 0.)
The test stat will be in the reject H 0 zone. ♦ p-values
on the borderline (very close to α ) are treated as marginal
results. Here's how you interpret your results subjectively if you use an alpha level of 0.05: ♦If
the p-value is less than 0.01 (very small), the results are considered highly statistically significant - reject H 0.
♦If
the p-value is between 0.05 and 0.01 (but not close to 0.05), the results are considered statistically significant -reject H 0.
♦If
the p-value is close to 0.05, the results are considered marginally significant - decision could go either way.
α is given, state, “Fail to reject H 0 for If no α any alpha less than (give the p-value)”.
In general: If α < p-value, Fail-to-Reject H 0 . (Or, if the p-value ≤ α , Reject H 0 ). Note: α acts as a cut-off point below which we agree that an effect is statistically significant. That is, if p-value ≤ α , (or if α > p-value) then Reject H 0 and agree that the alternate hypothesis is what’s going on. Note: TI-83/4 programs to use: see http://rmower.com/s_calculators_general/applied_ statistics_calc_detail.html
Note: Many software programs supply this step when an alpha is provided.
♦If
the p-value is greater than (but not close to) 0.05, the results are considered non-significant – fail to reject H 0.
When you hear about a result that has been found to be statistically significant, ask for the p-value and make your own decision. Alpha levels and resulting decisions will vary from researcher to researcher.
P-Values Left tail: Area to the left of the test statistic Right tail: Area to the right of the test statistic Two tail: Test Stat to left of center: Twice the area to the
left of the test statistic. Test Stat to right of center: Twice the area to the right of the test statistic. (To find the areas, use the “Strategies to Find Areas” included in the Hypothesis package page 2 at the right. ) F a t o o r r e e j e f α < p - v a r e j e f α ≥ p F a i l i l t j e c t c t H o i i f v a l u l u e . O r , r , r j e c t c t H o i i f v a v a l u l u e .
Hypothesis T Tests Continued (Statistics II Review) page 6 of 7 Step 6. Write a final conclusion using the flowchart. Flowchart: http://www.rmower.com/statistics/hyp_test_etc_ref_package/snd_tdist_use_t_or_z.pdf
Hypothesis Tests: Deter mining tthe F Final C Conclusion
Start
Is the claim H 0?
Yes
Did you reject H 0?
Yes
No
No
There is sufficient evidence to warrant rejection of the claim that…(original claim) There is not sufficient evidence to warrant rejection of the claim that…(original claim)
Yes Did you reject H 0? No
The sample data support the claim that…(original claim) There is not sufficient sample evidence to support the claim that…(original claim)
Gener al s steps f f or a h hypothesis ttest ((also ssee tthe ne x t t pa pag e) Here's a boiled-down summary of the steps involved in doing a hypothesis test. (Particular formulas needed to find test statistics for any of the most common hypothesis tests are provided on your formula sheet.) 1. Formulate the NULL and ALTERNATE Hypotheses: H 0 and H 1. In real life, before going on to the next step, collect a random sample of individuals from the population and calculate the sample statistics (means and standard deviations). 4. Compute the Test Stat and Determine the p-value: 5. Reject or Fail-to-Reject H0, or State the p-value. 6. State the Final Conclusion
Hypothesis T Tests Continued (Statistics II Review) page 7 of 7 Summary for Reference: p-value test.
Step 1. Formulate the NULL and ALTERNATE Hypotheses: H0: The null hypothesis is always “= ” such as H 0: p = p 0 .
See the table below
H1: The alternate hypothesis is always “ ≠ ”, “<” or “>” such as H 1: p < p 0 . Note: H 1 determines the tail type. p0 is the H 1: p < p 0 is “left 1-tail”. hypothesized H 1: p > p 0 is “right 1-tail”. value H 1: p ≠ p0 is “both 2-tail”. Note: The “CLAIM can be either H 0 or H 1.
No Step 2 or 3 for p-value test. Step 4. Compute the Test Stat and Determine the p-value: Use the formula for the test statistic that will assess the evidence against the null hypothesis. Then use the test statistic and appropriate table or software to determine the p-value.
Note: When doing a traditional 6step hypothesis test, we (step 2) determine a critical z , t or F similar
to the way we did for confidence intervals , and then (step 3) use the critical number(s) to build an interval (similar to a confidence interval) around the population parameter whose z or t is 0. Then (step 4), after collecting our sample evidence and computing the test mean or parameter, etc., we convert it to a “Test Statistic” Standard score by using our z , t or F formulas, ex.
z =
x − s /
6 S Step Test Tr aditional Hypothesis Test 6 6 Steps 1) *Set up H 0 & H 1
. Then
H 0 : Use the “=” or indicate equality, independence, etc.
n
H 1 :
Use the “≠”, “>”, “<” or indicate inequality, dependence, etc. Label
(step 5), if the test stat is outside the critical number(s) that make up the interval, we Reject H 0. If not, we Fail-to-reject H 0 (step 5).
the Claim t h he 2) D e t e t e r m i n e t c r i c c r i t i t i c a l n u m b e r ( s ( s ) Use the Z -table for all proportions. For means: When σ is known, use the Z table. When σ is NOT known (and s is given), use the ttable. For others (Chi-square, regression, F-dist, ANOVA, etc.) refer to the proper table or method.
If we are doing a p-value test, we will simply convert the test stat to an area (the p-value, below left) and compare it to the alpha (if we have one).
c u 3 ) D r a w a w a c u r r v v e
Step 5. Reject or Fail-to-Reject H 0, or State the p-value.
Label the “Fail to Reject H 0 Zone” and, label and shade the “Reject H 0 Zone”
Compare the p-value to a fixed significance level α α to determine whether to Reject or Fail-to-Reject H 0 . If no α α is given, state, “Fail to reject H 0 fo any alpha less than (give the p-value)”. In general: If α < p-value, Fail-to-Reject H 0 . (Or, if the p-value ≤ α , Reject H 0 ).
t h he 4) * D e t e t e r m i n e t
T ai l C r l T y y p es, C ri i t t i ic c a l Z S i ig g n s
t e s t i t e s t s t s t a t a t i t i s t i c c - ss e e
for Hypothesis Tests
the applicable formula sheet.
The tail type is determined by H 1 Equal o or at least
and plot it.
H 0
H 1
Type
Critical Value Sign
=
<
1-tail Left
negative (-)
Equal o or at most
♦
=
>
1-tail Right
positive (+)
Equal to
=
≠
2-tail Both
pos & neg (±)
5) * R e R e j e j e c t t H 0 o r F a i l t o o R R e e j e c t H l t t H 0 .
♦
♦
Step 6. State the Final Conclusion (Use Only if α is given).
and plot the critical number(s).
Examples will follow.
Note: All Confidence Intervals are 2- ail
1 ) R R e j e c t ii f t s t a t ii s f t h e tt e s t s i n tt h e R R e j e c t Z Z o n e oo r 2 ) II f u Fa i l f u s i n g p - va l u e s . F t o r r e j e c t ii f α < p - va l u e , o o r R e j e c t ii f α ≥ p - va l u e .
t h h e f f i i n a l 6) * W W r r i i t e t e t c o c o n c l l u u s i o n -- ss e e tt h e f l f l o w c h a r t bb e l o w
*Steps for p-value tests are 1, 4, 5, 6
Use the flowchart. P-Values Left tail: Area to the left of the test statistic Right tail: Area to the right of the test statistic Two tail: Test Stat to left of center: Twice the area to the
left of the test statistic. Test Stat to right of center: Twice the area to the right of the test statistic.
Hypothesis Tests: Deter mining tthe F Final C Conclusion
Start
Is the claim H 0 ?
Yes
No
Did you reject H 0 ? No
(To find the areas, use the “Strategies to Find Areas” included in the Hypothesis package page 2 at the right. ) F a t o o r r e j e f α < p - v a r e e j e f α ≥ p F a i l i l t j e c t c t H o i i f v a l u l u e . O r , r j e c t c t H o i i f v a v a l u l u e .
Did you reject H 0?
There is sufficient evidence to warrant rejection of the claim that…(original claim) There is not sufficient evidence to warrant rejection of the claim that…(original claim)
Yes
No
Formulas: http://www.rmower.com/statistics/hyp_test_etc_ref_package/formulas_mower.pdf Flowchart: http://www.rmower.com/statistics/hyp_test_etc_ref_package/snd_tdist_use_t_or_z.pd
Yes
The sample data support the claim that…(original claim) There is not sufficient sample evidence to support the claim that…(original claim)
