basics of hypothesis testing
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Basics of Hypothesis Testing. Review of Inferential Basics Hypothesis Testing Procedure One-Sample z Test ( σ known) One-sample t test ( σ estimated with s ). Review of Some Basics. Population all possible values Sample a portion of the population - PowerPoint PPT PresentationTRANSCRIPT
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HS 167 Basics of Hypothesis Testing 1
Basics of Hypothesis Testing(a) Review of Inferential Basics
(b) Hypothesis Testing Procedure
(c) One-Sample z Test (σ known)
(d) One-sample t test (σ estimated with s)
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HS 167 Basics of Hypothesis Testing 2
Review of Some Basics
Population all possible valuesSample a portion of the population Statistical inference generalizing from a sample to a population with calculated degree of certainty Two forms of statistical inference
Estimation Hypothesis testing
Parameter a numerical characteristic of a population, e.g., population mean µ, population standard deviation σStatistic a numerical characteristic calculated in the sample, e.g., sample mean x-bar, sample standard deviation s
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Recall the distinction between parameters and statistics
Parameters Statistics
Source Population Sample
Notation Greek ()
Roman (xbar, s)
Vary No Yes
Calculated No Yes
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Recall how sample means (xbars) vary
The sampling distribution of the mean (SDM) tends to be Normal with mean µ and a standard deviation called the SE
),(~ SENxn
SE
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Hypothesis Testing Procedure
A. HypothesesB. Test StatisticC. P-valueD. Significance level (optional)
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Step A. Hypotheses
Take the research question and convert it to statistical hypotheses State statistical hypotheses in null and alternative forms H0 Null hypothesis “no difference in the
population” Ha Alternative hypothesis “difference in
the population”
Seek evidence against H0 as a way of bolstering H1
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Step B. Test StatisticCalculate the appropriate test statistic There are many types of test statistics, depending on the conditions of the dataThis Chapter introduces this particular one-sample z statistic:
nSE
SE
x
and
trueis hypothesis null theassumingmean population the where
z
0
0stat
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Step C. P-valueConvert zstat to P-value using table or softwareThe P-value is the AUC in the tail of the SDM beyond the test statistics.The P-value answer the question: What is the probability of the observed test statistics or one that is more extreme assuming Hassuming H00 is true is true?Small P-value good evidence against H0 Although it unwise to draw too-firm Although it unwise to draw too-firm cutoffs, here are guidelines for the cutoffs, here are guidelines for the beginner:beginner:
P > 0.10 evidence against H0 not significant
0.05 < P 0.10 evidence against H0 marginally significant
0.01 < P 0.05 evidence against H0 significant
P 0.01 evidence against H0 highly significant
Smaller and smaller P-values provide stronger and stronger evidence against
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Step D. Significance level (optional)
Declare acceptable false rejection (Type I) error rate α
α can be set at any level (e.g., 1 in 20, 1 in 50, 1 in 100)
When P < α reject H0 at α level of significance
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Illustrative Example: Lake Wobegon, Study Design
Research question: Does a particular population of children have higher than average intelligence scores?Study design
Weschler intelligence scores are Normal with µ = 100 and = 15
Take a SRS Measure WIS {116, 128, 125, 119, 89, 99, 105,
116, 118} Calculate sample mean: x-bar = 112.8 Ask: Does this provide sufficient evidence that
population mean (μ) is greater than expected (100)?
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Illustrative Example: Lake Wobegon, Step A
Under the null hypothesis of “no difference” H0: µ = 100
Under the alternative hypothesis of “difference” Ha: µ > 100
Note: (a) Hypotheses address the parameter (not the statistic) (b) The value of the parameter under H0 is based on the research question (NOT the data)
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Illustrative Example: Lake Wobegon, Step B
The test statistics for this problem is the one-sample z statisticThis statistic assumes H0 is correctIt then standardizes the sample mean (i.e., turns it into a z-score):
56.25
1008.112
59
15
0stat
SEM
xz
nSEM
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Illustrative Example: Lake Wobegon, Step C
The test statistic is converted to a P-valueAssuming H0 true, where does zstat fall on the curve?P value area under curve beyond zstat
Use Z table (or software) to find Pr(Z ≥ zstat) = Pr(Z ≥ 2.56) = 0.0052The P-value of 0.0052 provides strong (“highly significant”) evidence against H0
)1,0(~
)(~
under that Recall
stat
,0
0
Nz
SENx
H
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Illustrative Example: Lake Wobegon Step D (Optional)
P = 0.0052 is highly significant evidence against H0
The smaller the P, the more significant the evidence.
See guidelines in earlier slide.
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The one-sided alternative
The prior test made a supposition about the direction of the difference The test had a “one-sided H1”
We looked only at one side of the
SDM
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The two-sided alternative
Allows for unanticipated findings either up or down from expectedThe requires a two-two-sided test sided test The two-sided test looks at both tails Just double the one-sided PLake Wobegon example two-sided P = 2 × 0.0052 = 0.0104
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Illustrative example: “Anemia”
Research question: A public health official suspects a particular population is anemic Hemoglobin is a protein in red blood cells that
carries oxygen People with less than 12 g/dl hemoglobin are
anemic We know that hemoglobin levels in children are
Normal with standard deviation σ = 1.6 g/dl
Data: The researcher measures hemoglobin in 50 children and finds x-bar = 11.3
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Step A: Anemia example
We seek evidence against µ = 12. Thus, H0: µ = 12
The one-sided alternative is H1: µ < 12The two-sided alternative is H1: µ 12Let’s do a two-sided test (much more common in practice)
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Step B: Test statistic
10.3226.0
123.11
226.050
6.1
0stat
SEM
xz
nSEM
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Step C. P-value
Pr(Z < -3.10) = 0.0010Double this b/c the alternative is two-sided: P = 2 × 0.0010 = 0.0020Comment: Under H0, xbar~N(12, 0.226), I’ll draw this on the board so you see that an x-bar of 11.3 is in the far left tail
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Step D: Conclusion
P = 0.0020
evidence against H0 is highly significant
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Conditions necessary for z test
Simple random sample (SRS)Normal population or large sample known (not calculated)
What do you do when σ is not known? Use a t procedure!
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Diabetic weight illustrative example
Claim: “diabetics are over-weight”Data are “% of ideal body weight” n = 18 Sample mean (x-bar) = 112.778 Sample standard deviation (s) =
14.424
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Step A: Hypotheses (diabetic weight)
Research claim is “diabetics are over-weight”Convert claim to a null hypothesis “Diabetics are not overweight” Not overweight = 100 of ideal body weight Therefore, H0: µ = 100
Alternative hypothesis can be H1: µ 100 (two-sided) H1: µ > 100 (one-sided to right) H1: µ < 100 (one-sided to left)
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Step B: Test statistic
ndf n
ssem
sem
xt
1
,hypothesis null under themean population where
0
0stat
tstat tells you how many standard errors the sample mean falls from hypothesized population mean
171181
76.3400.3
100778.112
400.318
424.14
0stat
ndfsem
xt
n
ssem
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Step C: P value (Diabetic weight)
t table: wedge tstat between t landmarksExample: tstat = 3.76 on t with 17 df falls between 3.646 (right tail 0.001) and 3.965 (right tail 0.0005) Two-tailed P is twice the one-tailed values: P less than 0.002 and more than 0.001P = 0.0016 (via software)
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Step D: Conclusion
P = 0.0016 provides highly significant evidence against H0
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Consequences of test decisions
TRUTHDECISION
H0 true H0 false
Retain H0 Correct retention
type II error
Reject H0type I error
Correct rejection
Probability (type I error) =
Probability (type II error) =
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Type II errors (β)
We have considered only type I () errors In the next chapter we will take up the issue of type II () errors Pr(rejecting a false H0) The complement of is 1 –
”Power”
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Fallacies of hypothesis testing
Failure to reject H0 = acceptance of H0 (WRONG!)
P value = probability H0 is incorrect (WRONG!)Statistical significance implies biological or social importance (WRONG!)
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Beware!
Hypothesis testing addresses random error only. It does not account for many problems encountered in practice, such as measurement error and sampling biases