Chapter 8. Inferences on a Population Meanmathsci.kaist.ac.kr/~nipl/cc511/lectures/Chapter8.pdf ·...

Chapter 8. Inferences

on a Population Mean

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8.1 Confidence Intervals

8.2 Hypothesis Testing

8.3 Summary

8.1 Confidence Intervals8.1.1 Confidence Interval Construction(1/8)

• Confidence Intervals

– A confidence interval for an unknown parameter θ is an interval that

contains a set of plausible values of the parameter.

– It is associated with a confidence level 1-aaaa, which measures the

��

probability that the confidence interval actually contains the unknown

parameter value.

– Confidence levels of 90%, 95%, and 99% are typically used.

8.1.1 Confidence Interval Construction(2/8)

• Inferences on a Population Mean

– Inference methods on a population mean based upon the t-

procedure are appropriate for large sample sizes n ≥ 30 and

also for small sample sizes as long as the data can reasonably

��

also for small sample sizes as long as the data can reasonably

be taken to be approximately normally distributed.

– Nonparametric techniques (Chapter 15) can be employed for

small sample sizes with data that are clearly not normally

distributed.


• Two-Sided t-Interval

– A confidence interval with confidence level 1-a for a population mean µ

based upon a sample of n continuous data observations with a sample

mean and a sample standard deviation s isx

/ 2, 1 / 2 , 1n nt s t sα αµ − −

∈ − +

��

– The interval is known as a two-sided t-interval or variance unknown

confidence interval.

/ 2, 1 / 2 , 1,n nt s t s

x xn n

α αµ − −

∈ − +


• A two-sided t-interval

/ 2, 1nt sxn

α −− / 2, 1nt s

xn

α −+

/ 2, 1n

st

nα − ×

��

x

µ̂

ˆcritical point . .( )s e µ×


• The length of two-sided t-interval is

• As the standard error of decreases, so that becomes a more

/ 2, 12 ˆ2 critical point . .( )nt s

L s en

α µ−= = × ×

µ̂ ˆ xµ =

��

“accurate” estimate of .

• The length of a confidence interval also depends upon the confidence

level. As the confidence level increases, the length of the confidence

interval also increase.

µ


• We know that

and so the definition of the critical point of the t-distribution ensures

that

( )1n

n Xt

S

µ−

−�

��

that

( )/ 2, 1 / 2, 1 1n n

n XP t t

Sα αµ

α− − − − ≤ ≤ = −


• And

/ 2, 1 / 2, 1 1n nt S t S

P X Xn n

α αµ α− −

− ≤ ≤ + = −

��

• This probability statement should be interpreted as saying that there

is a probability of 1-aaaa that the random confidence interval limits

take values that “straddle” the fixed value µ.


• Technically speaking, has a t-distribution only when the

random variables are normally distributed.

• Nevertheless, the central limit theorem ensures that the

distribution of is approximately normal for reasonably large

( )n X Sµ−iX

��

distribution of is approximately normal for reasonably large

sample sizes, and in such cases it is sensible to construct t-intervals

regardless of the actual distribution of the data observations.

X

Example 14 : Metal Cylinder Production (p.365)

• Data : 60 metal cylinder diameters (page 290, Figure 6.5).

• Summary statistics:

n = 60 Median = 50.01 Max. = 50.36

= 49.999 Upper quartile = 50.07 Min. = 49.74x

��

s = 0.134 Lower quartile = 49.91

• Critical points:

Sample size n = 60

Confidence level 90%: t0.05,59 = 1.671



Example 14 : Metal Cylinder Production(2/4)

• Confidence interval with confidence level 90%:


1.671 0.134 1.671 0.13449.999 , 49.999 (49.970,50.028)

60 60× ×

− + =

��



2.001 0.134 2.001 0.13449.999 ,49.999 (49.964,50.033)

60 60× ×

− + =

2.662 0.134 2.662 0.13449.999 , 49.999 (49.953,50.045)

60 60× ×

− + =


49.970 50.028=49.999

90%

x

• Confidence intervals for mean metal cylinder diameter

��

49.964 50.033=49.999

95%

x

49.953 50.045=49.999

99%

x


• Conclusion with confidence interval:

With over 99% certainty, the average cylinder diameter lies

within 0.05 mm of 50.00mm, that is, within the interval (49.95,

50.05).

��

50.05).

• Comment:

It is important to remember that this confidence interval is for the

mean cylinder diameter, and not for the actual diameter of a

randomly selected cylinder.

8.1.2 Effect of the Sample Sizeon Confidence Intervals(1/4)

• Recall:

• For a fixed critical point, a confidence interval length L is inversely

/ 2, 12 nt sLn

α −=

��

• For a fixed critical point, a confidence interval length L is inversely

proportional to the square root of the sample size n.

• Notice that this dependence of the critical point on the sample size

also serves to produce smaller confidence intervals with large

sample sizes.


• If a confidence interval with a length no large than L0 is required, then

sample size

2/ 2, 1

0

4 nt s

nL

α − ≥ ×

��

must be used. This inequality can be used to find a suitable sample

size n if approximate values or upper bounds are used for ta/2, n-1 and s.

0L


• Example (p. 364): For the mean thickness of plastic sheets, an

experimenter wishes to construct a 95% confidence interval with a

length no larger than L0 = 2.0 mm.

– Known fact: the standard deviation (s) ≤ 4.0 mm

– Assumption: t0.025, n-1 ≤ 2.1, for a large enough sample size

��

– Assumption: t0.025, n-1 ≤ 2.1, for a large enough sample size

– Then a sample size is/ 2, 1

0

2

2.1 4.02 2

2.1 4.04 70.56

2.0

nt s Ln n

n

α − × ≤ ≤

⇓

× ≥ × =

� �


• Additional sampling:

– For a sample size n1 and a sample standard deviation s, the length of the

confidence interval is 1/ 2, 1

1

2 nt sLn

α −=

��

– To reduce the confidence interval length to L0 < L, the size of the

additional sample required is

1

2/ 2,

11

0

, 4 nt s

where nL

n n α −

≥ ×

−

Example 14 : Metal Cylinder Production (p. 365)

• n = 60, 99% confidence interval (49.953, 50.045) and confidence length

= 0.092 mm

• Question: How much additional sampling is required to provide the

increased precision of a confidence interval with a length of 0.08 mm at

��

the same confidence level?

• Answer: A total sample size required is

– Therefore, an additional sample of at least 80-60 = 20 cylinders is

needed.

2 20.005,59

0

2.662 0.1344 4 79.53

0.08

t sn

L × ≥ × = × =

8.1.4 Simulation Experiment(1/2)

• Figure 8.8(page 369) shows the 500 simulation results (a mean of µ = 10, a

sample size of n = 30) and 95% confidence intervals.

• Notice that, in simulations 24 and 37, the confidence intervals do not

include the correct value µ = 10.

��

• Each simulation provides a 95% confidence interval which has a probability

of 0.05 of not containing the value µ = 10.

8.1.4 Simulation Experiment(2/2)

• Since the simulations are independent of each other, the number of

simulations out of 500 for which the confidence interval does not

contain µ = 10 has a binomial distribution with n = 500 and p = 0.05.

• In practice, an experimenter observes just one data set, and it has a

��

• In practice, an experimenter observes just one data set, and it has a

probability of 0.95 of providing a 95% confidence interval that does

indeed straddle the true value µ.

8.1.5 One-Sided Confidence Intervals(1/4)

• One-Sided t-Interval: One-sided confidence intervals with confidence

levels 1-a for a population mean µ based on a sample of n

continuous data observations with a sample mean and a

sample standard deviation s arex

t s

��

which provides an upper bound on the population mean µ, and

which provides a lower bound on the population mean µ.

, 1, nt s

xn

αµ −

∈ −∞ +

, 1 ,nt s

xn

αµ −

∈ − ∞


• c.f.

Since

( )1n

n Xt

S

µ−

−�

��

the definition of the critical point ta,n-1 implies that

S

( ), 1 1n

n XP t

Sαµ

α− − − ≤ = −


This may be rewritten

so that

, 1 1nt S

P Xn

αµ α−

≤ + = −

��

so that

is a one-sided confidence interval for µ with a confidence level of 1-

a.

, 1, nt s

xn

αµ −

∈ −∞ +


• Figure 8.10: Comparison of two-sided and one-sided confidence

intervals

, 1nt sxn

α −+xOne-sided(upper bound)

��

/ 2, 1nt sxn

α −− / 2, 1nt s

xn

α −+xTwo-sided

, 1nt sxn

α −− xOne-sided(lower bound)

Example 45 : Hospital Worker Radiation Exposures

• n = 28, sample mean = 5.145,

• sample standard deviation s = 0.7524,

• critical point t0.01, 27 = 2.473.

• 99% one-sided confidence interval for µ is

x

( ), 1 2.473 0.7524nt sαµ −

× ∈ −∞ + = −∞ + = −∞

��

• Consequently, with a confidence level of 0.99 the experimenter can

conclude that the average radiation level at a 50cm distance from a

patient is no more than about 5.5.

( ), 1 2.473 0.7524, ,5,145 ,5.49628

nt sxn

αµ − ×

∈ −∞ + = −∞ + = −∞

8.1.6 z-Intervals(1/2)

• Two-Sided z-Interval

If an experimenter wishes to construct a confidence interval for a

population mean µ based on a sample of size n with a sample mean

and using an assumed known value for the population standard x

��

deviation s, then the appropriate confidence interval is

which is known as a two-sided z-interval or variance known

confidence interval.

/ 2 / 2,z z

x xn n

α ασ σµ ∈ − +

��

8.1.6 z-Intervals(2/2)

• One-Sided z-Interval

One-sided 1-a level confidence intervals for a population mean µ

based on a sample of n observations with a sample mean and

using a known value of the population standard deviation s are

x

��

using a known value of the population standard deviation s are

These confidence intervals are known as one-sided z-intervals.,

zx

nασµ ∈ −∞ +

,

zx

nασµ ∈ − ∞

and

8.2 Hypothesis Testing8.2.1 Hypotheses(1/2)

• Hypothesis Tests of a Population Mean

– A null hypothesis H0 for a population mean µ is a statement that

designates possible values for the population mean.

– It is associated with an alternative hypothesis HA, which is the

��

– It is associated with an alternative hypothesis HA, which is the

“opposite” of the null hypothesis.

– A two-sided set of hypotheses is

H0 : µ = µ0 versus HA : µ ≠ µ0

for specified value of µ.

8.2.1 Hypotheses(2/2)

– A one-sided set of hypotheses is either

H0 : µ ≤ µ0 versus HA : µ > µ0

or

H0 : µ ≥ µ0 versus HA : µ < µ0

��

0 0 A 0

Example 14 : Metal Cylinder Production

• The machine that produces metal cylinders is set to make

cylinders with a diameter of 50 mm.

• The two-sided hypotheses of interest are

H0 : µ = 50 versus HA : µ ≠ 50

��

H0 : µ = 50 versus HA : µ ≠ 50

where the null hypothesis states that the machine is calibrated

correctly.

Example 47 : Car Fuel Efficiency

• A manufacturer claim : its cars achieve an average of at least 35

miles per gallon in highway driving.

• The one-sided hypotheses of interest are

H0 : µ ≥ 35 versus HA : µ < 35

��

H0 : µ ≥ 35 versus HA : µ < 35

• The null hypothesis states that the manufacturer’s claim regarding

the fuel efficiency of its cars is correct.

8.2.2 Interpretation of p-values(1/4)

• Types of error

– Type I error: An error committed by rejecting the null hypothesis

when it is true.

– Type II error: An error committed by accepting the null hypothesis

��

when it is false.

• Significance level

– is specified as the upper bound of the probability of type I error.


• p-value of a test (or observed level of significance)

– Definition: The p-value of a test is the probability of obtaining a

given data set or worse when the null hypothesis is true.

– A data set can be used to measure the plausibility of null

��

hypothesis H0 through the construction of a p-value.

– The smaller the p-value, the less plausible is the null hypothesis.

(why?)


• Rejection of the Null Hypothesis

– If a p-value is smaller than the significance level, then the

hypothesis H0 is rejected in favor of the alternative hypothesis

HA.

��

• Acceptance of the Null Hypothesis

– A p-value larger than 0.10 is generally taken to indicate that the null

hypothesis H0 is a plausible statement. The null hypothesis H0 is

therefore accepted.

– However, this does not mean that the null hypothesis H0 has been

proven to be true.


• Intermediate p-values

– A p-value in the range 1% ~ 10% is generally taken to indicate

that the data analysis is inconclusive. There is some evidence

that the null hypothesis is not plausible, but the evidence is not

overwhelming.

��


• Q : whether the machine can be shown to be calibrated

incorrectly.

• H0 : µ = 50, HA : µ ≠ 50

• With a small p-value,

– The null hypothesis is rejected and the machine is demonstrated

��

to be miscalibrated.

• With a large p-value,

– The null hypothesis is accepted and the experimenter concludes

that there is no evidence that the machine is calibrated

incorrectly.

8.2.3 Calculation of p-values

• Two-sided t-test

– Consider testing

–

0 0 0: :AH vs Hµ µ µ µ= ≠� ��

2 ( | |)p value P X t

where

− = × ≥

��

0

/

where

xX

s n

µ− =

• One-sided t-test


– Then0 0 0: :AH vs Hµ µ µ µ≤ >� ��

( )p value P X t− = ≥

��


– Then 0 0 0: :AH vs Hµ µ µ µ≥


• The data set of metal cylinder diameters: n = 60, = 49.99856,

s = 0.1334

• µ0 = 50.0 è

• p-value = 2 x P(X≥0.0836), X ~ t-distribution with n-1=59 d.f.

x

( )49.99856 50.00.0836

0.1334 60t

−= = −

H0: µ = µ0 versus HA: µ ≠ µ0

��

• p-value = 2 x P(X≥0.0836), X ~ t-distribution with n-1=59 d.f.

• p-value = 2 x 0.467 = 0.934

• With such a large p-value,

the null hypothesis is accepted.

t59 distribution

0 ltl=0.0836

0.467

Example 47 : Car Fuel Efficiency (1/2)

• n=20,

• the fuel efficiency with a sample mean =34.271 miles/gallon

• sample standard deviation s=2.915 miles/gallon.

• µ0 = 35.0

x

0 : 35 : 35AH versus Hµ µ≥ <

��

• µ0 = 35.0

• The alternative hypothesis is HA: µ < 35, so that

where X has a t-distribution with n-1=19 d.f.

( )20 34.271 35.01.119

2.915t

−= = −

( )1.119p value P X− = ≤ −


• The value can be shown to be

p-value = 0.1386.

• This p-value is larger than 0.10

and so the null hypothesis

t19 distribution

��

and so the null hypothesis

should be accepted.

0-1.119p-value

8.2.4 Significance Levels(1/14)

• Significance Level of a Hypothesis Test

– A hypothesis test with a significance level or size a

rejects the null hypothesis H0 if a p-value smaller than a is obtained and

accepts the null hypothesis H0 if a p-value larger than a is obtained.

��

0

• P-values are more informative than knowing whether a size a test accepts

or rejects the null hypothesis. (Why?)


• Two-Sided Problems

– Two-Sided Hypothesis Test for a Population Mean

• A size a test for the two-sided hypotheses

rejects the null hypothesis H if the test statistic ltl falls in


��

rejects the null hypothesis H0 if the test statistic ltl falls in

the rejection region

and accepts the null hypothesis H0 if the test statistic ltl falls in the

acceptance region

• What is the role of here?

/ 2, 1nt tα −>

/ 2, 1nt tα −≤α


• The data set of metal cylinder diameters gives a test statistic of ltl =

0.0836

• a=0.10 à t0.05,59=1.671

• a=0.05 à t0.025,59=2.001


��

• a=0.05 à t0.025,59=2.001

• a=0.01 à t0.005,59=2.662

• The test statistic is smaller than each of these critical points, and so

the hypothesis tests all accept the null hypothesis.

• The p-value is therefore known to be larger than 0.10, and in fact the

previous analysis found the p-value to be 0.934.


• Relationship Between Confidence Intervals and Hypothesis Tests

– The value µ0 is contained within a 1-a level two-sided confidence

interval

/ 2, 1 / 2, 1,n nt s t s

x xn n

α α− − − +

��

if the p-value for the two-sided hypothesis test


is larger than a.

,x xn n

− +


• If µ0 is contained within the 1-a level confidence interval,

the hypothesis test with size a accepts the null hypothesis,

and

• if µ is not contained within the 1-a level confidence interval, the

��

• if µ0 is not contained within the 1-a level confidence interval, the

hypothesis test with size a rejects the null hypothesis.


0 0 0: :AH versus Hµ µ µ µ= ≠

• (FIGURE 8.36, Page 399)

��

/ 2, 1nt sxn

α −− / 2, 1nt s

xn

α −+x

1- level two-sided confidence intervalα

p value α− < p value α− ≥ p value α−

Example 14 : Metal Cylinder Production (1/2)

• A 90% two-sided t-interval for the mean cylinder diameter was found

to be (49.970, 50.028).

• This contains the value µ0=50.0

and so is consistent with

��

and so is consistent with

• the hypothesis testing problem

H0: µ = 50.0 versus HA: µ ≠ 50.0

having a p-value of 0.934, so that the null hypothesis is accepted at size

aaaa=0.10.

Example 14 : Metal Cylinder Production (2/2)

• The 90% confidence interval implies that the hypothesis testing

problem


has a p-value larger than 0.10 for 49.970≤ µ0≤50.028 and a p-value

��

has a p-value larger than 0.10 for 49.970≤ µ0≤50.028 and a p-value

smaller than 0.10 otherwise. (Why ?)


• One-Sided Inferences on a Population Mean(H0: µ ≤ µ0) (p.401)

– A size a test for the one-sided hypothesis

H0: µ ≤ µ0 versus HA: µ > µ0

rejects the null hypothesis when

��


t > t a,n-1

and accepts the null hypothesis when

t ≤ t a,n-1


• One-sided Inferences on a Population Mean(H0: µ ≤ µ0)

– The 1-a level one-sided upper confidence interval

, 1 ,nt s

x αµ −

∈ − ∞

��

consists of the values µ0 for which this hypothesis testing

problem has a p-value larger than a.

,xn

µ ∈ − ∞


• Relationship between hypothesis testing and confidence intervals for

one-sided problems (FIGURE 8.40, Page 403)

0 0 0: :AH versus Hµ µ µ µ≤ >

��

, 1nt sxn

α −− x

1- level one-sided confidence intervalα

p value α− < p value α− ≥


• One-sided Inferences on a Population Mean (H0: µ ≥ µ0)

– A size a test for the one-sided hypothesis

H0: µ ≥ µ0 versus HA: µ < µ0


��


t < -t a,n-1


t ≥ -t a,n-1


• One-sided Inferences on a Population Mean(H0: µ ≥ µ0)

– The 1-a level one-sided lower confidence interval

, 1, nt s

x αµ −

∈ −∞ +

��

consists of the values µ0 for which this hypothesis testing problem

has a p-value larger than a.

, xn

µ ∈ −∞ +


• Relationship between hypothesis testing and confidence intervals for

one-sided problems (FIGURE 8.39, Page 402)

0 0 0: :AH versus Hµ µ µ µ≥ <

��

, 1nt sxn

α −+x

1- level one-sided confidence intervalα

p value α−


• The one-sided hypotheses of interest here are

– H0: µ ≥ 35.0 versus HA: µ < 35.0

• t = -1.119 > –t0.10,19 = -1.328.

• So

��

• So

a size a = 0.10 hypothesis test accepts the null hypothesis.

• This conclusion is consistent with the previous analysis where the p-

value was found to be 0.1386, which is larger than a =0.10.


• Furthermore, the one-sided 90% lower t-interval

contains the value µ = 35.0, as expected.

( ), 1 1.328 2.915, ,34.271 ,35.1420

nt sxn

αµ − ×

∈ −∞ + = −∞ + = −∞

��

contains the value µ0 = 35.0, as expected.

• In fact, this confidence interval indicates that the hypothesis testing

problem

– H0: µ ≥ µ0 versus HA: µ < µ0

has a p-value larger than 0.10 for any value of µ0 ≤ 35.14.


• Power of a hypothesis Test

– The power of a hypothesis test is defined to be

• power = 1 – (probability of Type II error | )

which is the probability that the null hypothesis is rejected when it

AH

��

is false.

• Larger power levels and shorter confidence intervals are both

indications of an increase in the “precision” of an experiment ( why? ).


Hypothesis Testing Sample size n

FIGURE 8.42 The specification of two quantities determines the third quantity

��

Confidence Intervals

Significance level a

Power of hypothesis test

Sample size n

Confidence level 1-a

Length of confidence interval


• Relationship Between Power and Sample Size

– For a fixed significance level a, the power of a hypothesis test

increases as the sample size n increases. (Why? )

��

8.2.5 z-Tests (1/7)

• Two-Sided z-test

– The p-value for the two-sided hypothesis testing problem


based upon a data set of n observations with a sample mean

, n-1(t-test, s, t ) (z-test, , )α ασ⇔ z

��


and an assumed “known” population standard deviation s, is

where is the standard normal cumulative distribution

function.

x

( )2p value z− = ×Φ −( )xΦ

8.2.5 z-Tests (2/7)

• Two-Sided z-test

which is known as the z-statistic.

• A size test rejects the null hypothesis if the test statistic l l falls in the

( )0n xz µσ

−=

��

• A size a test rejects the null hypothesis H0 if the test statistic lzl falls in the

rejection region

and accepts the null hypothesis H0 if the test statistic lzl falls in the acceptance

region

/ 2z zα>

/ 2z zα≤

8.2.5 z-Tests (3/7)

– The 1- a level two-sided confidence interval

•

consists of the value µ0 for which this hypothesis testing problem

/ 2 / 2,z z

x xn n

α ασ σµ ∈ − +

��

0

has a p-value larger than a.

8.2.5 z-Tests (4/7)

• One-Sided z-Test (H0: µ ≤ µ0)

– The p-value for the one-sided hypothesis testing problem

H0: µ ≤ µ0 versus HA: µ > µ0


��


and an assumed known population standard deviation s, is

This testing procedure is called a one-sided z-test.

x

( )1p value z− = − Φ

8.2.5 z-Tests (5/7)

• One-Sided z-Test (H0: µ ≤ µ0)

– A size a test rejects the null hypothesis when


z zα>

��


– The 1- a level upper one-sided confidence interval

z zα≤

,z

xn

ασµ ∈ − ∞

8.2.5 z-Tests (6/7)

• One-Sided z-Test (H0: µ ≥ µ0)

– The p-value for the one-sided hypothesis testing problem

• H0: µ ≥ µ0 versus HA: µ < µ0


��


and an assumed known population standard deviation s, is

•

This testing procedure is called a one-sided z-test.

x

( )p value z− = Φ

8.2.5 z-Tests (7/7)

• One-Sided z-Test (H0: µ ≥µ0)

– A size a test rejects the null hypothesis when


z zα< −

��


– The 1- a level lower one-sided confidence interval

z zα≥ −

,z

xn

ασµ ∈ −∞ +

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