• Study Resource
• Explore

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Statistics wikipedia, lookup

History of statistics wikipedia, lookup

Transcript
Introduction to Inference
Estimating with Confidence
IPS Chapter 6.1
© 2009 W.H. Freeman and Company
Objectives (IPS Chapter 6.1)
Estimating with confidence

Statistical confidence

Confidence intervals

Confidence interval for a population mean

How confidence intervals behave

Choosing the sample size
Overview of Inference

Methods for drawing conclusions about a population from sample
data are called statistical inference

Methods


Confidence Intervals - estimating a value of a population parameter

Tests of significance - assess evidence for a claim about a population
Inference is appropriate when data are produced by either

a random sample or

a randomized experiment
Statistical confidence
Although the sample mean, x, is a unique number for any particular
sample, if you pick a different sample you will probably get a different
sample mean.
In fact, you could get many different values for the sample mean, and
virtually none of them would actually equal the true population mean, .
But the sample distribution is narrower than the population distribution,
by a factor of √n.
n
Sample means,
n subjects
Thus, the estimates
x
x

gained from our samples
are always relatively
n
Population, x
individual subjects
close to
the population

parameter µ.

If the population is normally distributed N(µ,σ),

so will the sampling distribution N(µ,σ/√n),
95% of all sample means will

be within roughly 2 standard
n
deviations (2*/√n) of the
population parameter .

Distances are symmetrical
which implies that the
population parameter 
must be within roughly 2
standard deviations from
the sample average
x , in
95% of all samples.
Red dot: mean value
of individual sample
This reasoning is the essence of statistical inference.

The weight of single eggs of the brown variety is normally distributed N(65 g,5 g).
Think of a carton of 12 brown eggs as an SRS of size 12.
.

What is the distribution of the sample means x?
Normal (mean , standard deviation /√n) = N(65 g,1.44 g).

Find the middle 95% of the sample
 means distribution.
Roughly ± 2 standard deviations from the mean, or 65g ± 2.88g.
population
sample
You buy a carton of 12 white eggs instead. The box weighs 770 g.
The average egg weight from that SRS is thus x = 64.2 g.
Knowing that the standard deviation of egg weight is 5 g, what
can you infer about the mean µ of the white egg population?


There is a 95% chance that the population mean µ is roughly within
± 2/√n of x, or 64.2 g ± 2.88 g.
Confidence intervals
The confidence interval is a range of values with an associated
probability or confidence level C. The probability quantifies the chance
that the interval contains the true population parameter.
x ± 4.2 is a 95% confidence interval for the population parameter .
This equation says that in 95% of the cases, the actual value of  will be
within 4.2 units of the value of x.
Implications
We don’t need to take a lot of
random samples to “rebuild” the
sampling distribution and find 
at its center.
n
All we need is one SRS of
Sample
size n and rely on the
n
Population
properties of the sample
means distribution to infer
the population mean .

Reworded

With 95% confidence, we can say
that µ should be within roughly 2
standard deviations (2*/√n) from
our sample mean x .

In 95% of all possible samples of

this size n, µ will indeed fall in our
confidence interval.

In only 5% of samples would
farther from µ.

x be

n
A confidence interval can be expressed as:

Mean ± m
m is called the margin of error
 within x ± m
Example: 120 ± 6

A confidence level C (in %)

Two endpoints of an interval
 within ( x − m) to ( x+ m)
ex. 114 to 126


indicates the probability that the
µ falls within the interval.
It represents the area under the
normal curve within ± m of the
center of the curve.
m
m
Review: standardizing the normal curve using z
x 
z
 n
N(64.5, 2.5)
N(µ, σ/√n)
N(0,1)

x
z
Standardized height (no units)
Here,
we work with the sampling distribution,
and /√n is its standard deviation (spread).
Remember that  is the standard deviation of the original population.
Varying confidence levels
Confidence intervals contain the population mean  in C% of samples.
Different areas under the curve give different confidence levels C.
Practical use of z: z*
z* is related to the chosen
confidence level C.

C
C is the area under the standard
normal curve between −z* and z*.

The confidence interval is thus:
x  z *
−z*
n
z*
Example: For an 80% confidence
level C, 80% of the normal curve’s
area is contained in the interval.
How do we find specific z* values?
We can use a table of z/t values (Table D). For a particular confidence
level, C, the appropriate z* value is just above it.
Example: For a 98% confidence level, z*=2.326
We can use software. In Excel:
=NORMINV(probability,mean,standard_dev)
gives z for a given cumulative probability.
Since we want the middle C probability, the probability we require is (1 - C)/2
Example: For a 98% confidence level, =NORMINV(.01,0,1) = −2.32635 (= neg. z*)
Link between confidence level and margin of error
The confidence level C determines the value of z* (in table C).
The margin of error also depends on z*.
m  z *
n
Higher confidence C implies a larger
margin of error m (thus less precision
in our estimates).

C
A lower confidence level C produces a
smaller margin of error m (thus better
precision in our estimates).
m
−z*
m
z*
Different confidence intervals for the same
set of measurements
Density of bacteria in solution:
Measurement equipment has standard deviation
 = 1 * 106 bacteria/ml fluid.
Three measurements: 24, 29, and 31 * 106 bacteria/ml fluid
Mean: x = 28 * 106 bacteria/ml. Find the 96% and 70% CI.

96% confidence interval for the
 true density, z* = 2.054, and write
x  z*

= 28 ± 2.054(1/√3)
n
= 28 ± 1.19 x 106
bacteria/ml
70% confidence interval for the
true density, z* = 1.036, and write


x  z*
= 28 ± 1.036(1/√3)
n
= 28 ± 0.60 x 106
bacteria/ml
Impact of sample size
The spread in the sampling distribution of the mean is a function of the
number of individuals per sample.
 The larger the sample size, the smaller
the standard deviation (spread) of the
sample mean distribution.
Standard error  ⁄ √n
 But the spread only decreases at a rate
equal to √n.
Sample size n
Sample size and experimental design
You may need a certain margin of error (e.g., drug trial, manufacturing
specs). In many cases, the population variability () is fixed, but we can
choose the number of measurements (n).
So plan ahead what sample size to use to achieve that margin of error.
m  z*

n
z *  
n  

 m 
2

Remember, though, that sample size is not always stretchable at will. There are
typically
 costs and constraints associated with large samples. The best
approach is to use the smallest sample size that can give you useful results.
What sample size for a given margin of error?
Density of bacteria in solution:
Measurement equipment has standard deviation
σ = 1 * 106 bacteria/ml fluid.
How many measurements should you make to obtain a margin of error
of at most 0.5 * 106 bacteria/ml with a confidence level of 90%?
For a 90% confidence interval, z* = 1.645.
 z * 
 1.645 *1 
2
n
  n
  3.29  10.8241
 m 
 0.5 
2
2
Using only 10 measurements will not be enough to ensure that m is no
more than 0.5 * 106. Therefore, we need at least 11 measurements.
Introduction to Inference
Tests of Significance
IPS Chapter 6.2
© 2009 W.H. Freeman and Company
Objectives (IPS Chapter 6.2)
Tests of significance

The reasoning of significance tests

Stating hypotheses

The P-value

Statistical significance

Tests for a population mean

Confidence intervals to test hypotheses
Reasoning of Significance Tests
We have seen that the properties of the sampling distribution of x help us
estimate a range of likely values for population mean .

We can also rely on the properties of the sample distribution
to test hypotheses.
Example: You are in charge of quality control in your food company. You sample
randomly four packs of cherry tomatoes, each labeled 1/2 lb. (227 g).
The average weight from your four boxes is 222 g. Obviously, we cannot expect
boxes filled with whole tomatoes to all weigh exactly half a pound. Thus,

Is the somewhat smaller weight simply due to chance variation?

Is it evidence that the calibrating machine that sorts
cherry tomatoes into packs needs revision?
Stating hypotheses
A test of statistical significance tests a specific hypothesis using
sample data to decide on the validity of the hypothesis.
In statistics, a hypothesis is an assumption or a theory about the
characteristics of one or more variables in one or more populations.
What you want to know: Does the calibrating machine that sorts cherry
tomatoes into packs need revision?
The same question reframed statistically: Is the population mean µ for the
distribution of weights of cherry tomato packages equal to 227 g (i.e., half
a pound)?
The null hypothesis is a very specific statement about a parameter of
the population(s). It is labeled H0.
The alternative hypothesis is a more general statement about a
parameter of the population(s) that is exclusive of the null hypothesis. It
is labeled Ha.
Weight of cherry tomato packs:
H0 : µ = 227 g (µ is the average weight of the population of packs)
Ha : µ ≠ 227 g (µ is either larger or smaller)
One-sided and two-sided tests
A two-tail or two-sided test of the population mean has these null
and alternative hypotheses:

H0 : µ = [a specific number] Ha : µ  [a specific number]
A one-tail or one-sided test of a population mean has these null and
alternative hypotheses:

H0 : µ = [a specific number] Ha : µ < [a specific number]
OR
H0 : µ = [a specific number] Ha : µ > [a specific number]
The FDA tests whether a generic drug has an absorption extent similar to
the known absorption extent of the brand-name drug it is copying. Higher or
lower absorption would both be problematic, thus we test:
H0 : µgeneric = µbrand
Ha : µgeneric  µbrand
two-sided
How to choose?
What determines the choice of a one-sided versus a two-sided test is
what we know about the problem before we perform a test of statistical
significance.
A health advocacy group tests whether the mean nicotine content of a
brand of cigarettes is greater than the advertised value of 1.4 mg.
Here, the health advocacy group suspects that cigarette manufacturers sell
cigarettes with a nicotine content higher than what they advertise in order
to better addict consumers to their products and maintain revenues.
Thus, this is a one-sided test:
H0 : µ = 1.4 mg
Ha : µ > 1.4 mg
It is important to make that choice before performing the test or else
you could make a choice of “convenience” or fall into circular logic.
The P-value
The packaging process has a known standard deviation  = 5 g.
H0 : µ = 227 g versus Ha : µ ≠ 227 g
The average weight from your four random boxes is 222 g.
What is the probability of drawing a random sample such as yours if H0 is true?
Tests of statistical significance quantify the chance of obtaining a
particular random sample result if the null hypothesis were true. This
quantity is the P-value.
This is a way of assessing the “believability” of the null hypothesis, given
the evidence provided by a random sample.
Interpreting a P-value
Could random variation alone account for the difference between
the null hypothesis and observations from a random sample?

A small P-value implies that random variation due to the sampling
process alone is not likely to account for the observed difference.

With a small p-value we reject H0. The true property of the
population is significantly different from what was stated in H0.
Thus, small P-values are strong evidence AGAINST H0.
But how small is small…?
P = 0.2758
P = 0.1711
P = 0.0892
P = 0.0735
Significant
P-value
???
P = 0.05
P = 0.01
When the shaded area becomes very small, the probability of drawing such a
sample at random gets very slim. Oftentimes, a P-value of 0.05 or less is
considered significant: The phenomenon observed is unlikely to be entirely
due to chance event from the random sampling.
Tests for a population mean
To test the hypothesis H0 : µ = µ0 based on an SRS of size n from a
Normal population with unknown mean µ and known standard deviation
σ, we rely on the properties of the sampling distribution N(µ, σ/√n).
The P-value is the area under the sampling distribution for values at
least as extreme, in the direction of Ha, as that of our random sample.
Sampling
distribution
Again, we first calculate a z-value
and then use Table A.
x 
z
 n
σ/√n
x
µ
defined by H0
P-value in one-sided and two-sided tests
One-sided
(one-tailed) test
Two-sided
(two-tailed) test
To calculate the P-value for a two-sided test, use the symmetry of the
normal curve. Find the P-value for a one-sided test and double it.
Does the packaging machine need revision?


x  222g
H0 : µ = 227 g versus Ha : µ ≠ 227 g
What is the probability of drawing a random sample such
as yours if H0 is true?
  5g
x   222  227
z

 2
 n
5 4
n4
From table A, the area under the standard
normal curve to the left of z is 0.0228.
Sampling
distribution
Thus, P-value = 2*0.0228 = 4.56%.
σ/√n = 2.5 g
2.28%
The probability of getting a random
2.28%
sample average so different from
µ is so low that we reject H0.
217
The machine does need recalibration.
222
227
232
x,
µ (H0)weight (n=4)
Average
package
z  2
237
The significance level a
The significance level, α, is the largest P-value tolerated for rejecting a
true null hypothesis (how much evidence against H0 we require). This
value is decided arbitrarily before conducting the test.

If the P-value is equal to or less than α (P ≤ α), then we reject H0.

If the P-value is greater than α (P > α), then we fail to reject H0.
Does the packaging machine need revision?
Two-sided test. The P-value is 4.56%.
* If α had been set to 5%, then the P-value would be significant.
* If α had been set to 1%, then the P-value would not be significant.
When the z score falls within the
the tail-side), the p-value is
smaller than α and you have
shown statistical significance.
z = -1.645
One-sided
test, α = 5%
Two-sided
test, α = 1%
Z
Rejection region for a two-tail test of µ with α = 0.05 (5%)
A two-sided test means that α is spread
between both tails of the curve, thus:
-A middle area C of 1 − α = 95%, and
-An upper tail area of α /2 = 0.025.
0.025
0.025
Table C
upper tail probability p
0.25
0.20
0.15
0.10
0.05
0.025
0.02
0.01
0.674
50%
0.841
60%
1.036
70%
1.282
80%
1.645
90%
1.960
95%
2.054
96%
2.326
98%
0.005 0.0025
0.001 0.0005
(…)
z*
Confidence interval C
2.576
99%
2.807 3.091 3.291
99.5% 99.8% 99.9%
Confidence intervals to test hypotheses
Because a two-sided test is symmetrical, you can also use a
confidence interval to test a two-sided hypothesis.
In a two-sided test,
C = 1 – α.
C confidence level
α significance level
α /2
α /2
Packs of cherry tomatoes (σ = 5 g): H0 : µ = 227 g versus Ha : µ ≠ 227 g
Sample average 222 g. 95% CI for µ = 222 ± 1.96*5/√4 = 222 g ± 4.9 g
227 g does not belong to the 95% CI (217.1 to 226.9 g). Thus, we reject H0.
Logic of confidence interval test
Ex: Your sample gives a 99% confidence interval of x  m  0.84  0.0101.
With 99% confidence, could samples be from populations with µ = 0.86? µ = 0.85?
Cannot
 reject
H0:  = 0.85
Reject H0 :  = 0.86
99% C.I.
x

A confidence interval gives a black and white answer: Reject or don't reject H0.
But it also estimates a range of likely values for the true population mean µ.
A P-value quantifies how strong the evidence is against the H0. But if you reject
H0, it doesn’t provide any information about the true population mean µ.
Introduction to Inference
Use and Abuse of Tests; Power and Decision
IPS Chapters 6.3 and 6.4
© 2009 W.H. Freeman and Company
Objectives (IPS Chapters 6.3 and 6.4)
Use and abuse of tests
Power and inference as a decision


Power of a test

Type I and II errors

Error probabilities
Choosing the significance level α
Factors often considered:


What are the consequences of rejecting the null hypothesis
(e.g., global warming, convicting a person for life with DNA evidence)?
Are you conducting a preliminary study? If so, you may want a larger α so
that you will be less likely to miss an interesting result.
Some conventions:

We typically use the standards of our field of work.

There are no “sharp” cutoffs: e.g., 4.9% versus 5.1 %.

It is the order of magnitude of the P-value that matters: “somewhat
significant,” “significant,” or “very significant.”
Practical significance
Statistical significance only says whether the effect observed is
likely to be due to chance alone because of random sampling.
Statistical significance may not be practically important. That’s because
statistical significance doesn’t tell you about the magnitude of the
effect, only that there is one.
An effect could be too small to be relevant. And with a large enough
sample size, significance can be reached even for the tiniest effect.

A drug to lower temperature is found to reproducibly lower patient
temperature by 0.4°Celsius (P-value < 0.01). But clinical benefits of
temperature reduction only appear for a 1° decrease or larger.
Don’t ignore lack of significance

Consider this provocative title from the British Medical Journal: “Absence
of evidence is not evidence of absence”.

Having no proof of who committed a murder does not imply that the
murder was not committed.
Indeed, failing to find statistical significance in results is not
rejecting the null hypothesis. This is very different from actually
accepting it. The sample size, for instance, could be too small to
overcome large variability in the population.
When comparing two populations, lack of significance does not imply
that the two samples come from the same population. They could
represent two very distinct populations with similar mathematical
properties.
Interpreting effect size: It’s all about context
There is no consensus on how big an effect has to be in order to be
considered meaningful. In some cases, effects that may appear to be
trivial can be very important.

Example: Improving the format of a computerized test reduces the average
response time by about 2 seconds. Although this effect is small, it is
important since this is done millions of times a year. The cumulative time
savings of using the better format is gigantic.
Always think about the context. Try to plot your results, and compare
them with a baseline or results from similar studies.
The power of a test
The power of a test of hypothesis with fixed significance level α is the
probability that the test will reject the null hypothesis when the
alternative is true.
In other words, power is the probability that the data gathered in an
experiment will be sufficient to reject a wrong null hypothesis.
Knowing the power of your test is important:

When designing your experiment: select a sample size large enough to
detect an effect of a magnitude you think is meaningful.

When a test found no significance: Check that your test would have had
enough power to detect an effect of a magnitude you think is meaningful.
Test of hypothesis at significance level α 5%:
H0: µ = 0 versus Ha: µ > 0
Can an exercise program increase bone density? From previous studies, we
assume that σ = 2 for the percent change in bone density and would consider a
percent increase of 1 medically important.
Is 25 subjects a large enough sample for this project?
A significance level of 5% implies a lower tail of 95% and z = 1.645. Thus:
z  ( x   ) (
x    z * (
n)
n)
x  0  1.645 * (2 / 25 )
x  0.658
All sample averages larger than 0.658 will result in rejecting the null hypothesis.
What if the null hypothesis is wrong and the true population mean is 1?
The power against the alternative
 P( x  0.658 when   1)
µ = 1% is the probability that H0 will
 x   0.658  1 

 P



n
2
25


 P( z  0.855)  0.80
be rejected when in fact µ = 1%.
We expect that a
sample size of 25
would yield a
power of 80%.
A test power of 80% or more is considered good statistical practice.
Factors affecting power: Size of the effect
The size of the effect is an important factor in determining power.
Larger effects are easier to detect.
More conservative significance levels (lower α) yield lower power.
Thus, using an α of .01 will result in less power than using an α of .05.
Increasing the sample size decreases the spread of the sampling
distribution and therefore increases power. But there is a tradeoff
between gain in power and the time and cost of testing a larger sample.
A larger variance σ2 implies a larger spread of the sampling distribution,
σ/sqrt(N). Thus, the larger the variance, the lower the power. The
variance is in part a property of the population, but it is possible to
reduce it to some extent by carefully designing your study.
Ho: µ = 0
σ = 10
n = 30
α = 5%
red area
is b
1. Real µ is 3 => power = .5
2. Real µ is 5.4 => power = .905
3. Real µ is 13.5 => power = 1
 larger differences
are easier to detect
http://wise.cgu.edu/power/power_applet.html
Ho: µ = 0
σ = 10
Real µ = 5.4
α = 5%
1. n = 10 => power = .525
2. n = 30 => power = .905
3. n = 80 => power = .999
 larger sample sizes
yield greater power
Ho: µ = 0
Real µ = 5.4
n = 30
α = 5%
1. σ is 15 => power = .628
2. σ is 10 => power = .905
3. σ is 5 => power = 1
 smaller variability
yields greater power
Type I and II errors

A Type I error is made when we reject the null hypothesis and the
null hypothesis is actually true (incorrectly reject a true H0).
The probability of making a Type I error is the significance level a

A Type II error is made when we fail to reject the null hypothesis
and the null hypothesis is false (incorrectly keep a false H0).
The probability of making a Type II error is labeled b.
The power of a test is 1 − b.
Running a test of significance is a balancing act between the chance α
of making a Type I error and the chance b of making a Type II error.
Reducing α reduces the power of a test and thus increases b.
It might be tempting to emphasize greater power (the more the better).

However, with “too much power” trivial effects become highly significant.

A type II error is not definitive since a failure to reject the null hypothesis
does not imply that the null hypothesis is wrong.
Related documents
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}