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Chapter 7 Hypothesis Testing

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7-1 Basics of Hypothesis Testing

7-2 Testing a Claim about a Mean: Large Samples

7-3 Testing a Claim about a Mean: Small Samples

7-4 Testing a Claim about a Proportion

7- 5 Testing a Claim about a Standard Deviation (will cover with chap 8)

7-1

Basics of

Hypothesis Testing

Hypothesis

in statistics, is a statement regarding a characteristic of one or more populations

Statement is made about the population

Evidence in collected to test the statement

Data is analyzed to assess the plausibility of the statement

Components of aFormal Hypothesis Test

Form Hypothesis

Calculate Test Statistic

Choose Significance Level

Find Critical Value(s)

Conclusion

A hypothesis set up to be nullified or refuted in order to support an alternate hypothesis. When used, the null hypothesis is presumed true until statistical evidence in the form of a hypothesis test indicates otherwise.

Statement about value of population parameter like m, p or s

Must contain condition of equality

=, , or

Test the Null Hypothesis directly

RejectH0 or fail to rejectH0

Must be true if H0 is false

, <, >

‘opposite’ of Null

sometimes used instead of

H1

Ha

If you are conducting a study and want to use a hypothesis test to support your claim, the claim must be worded so that it becomes the alternative hypothesis.

The null hypothesis must contain the condition of equality

Set up the null and alternative hypothesis

The packaging on a lightbulb states that the bulb will last 500 hours. A consumer advocate would like to know if the mean lifetime of a bulb is different than 500 hours.

A drug to lower blood pressure advertises that it drops blood pressure by 20%. A doctor that prescribes this medication believes that it is less. Set up the null and alternative hypothesis. (see hw # 1)

a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis

Testing claims about the population proportion

x - µ

σ

Z*=

n

Critical Region - Set of all values of the test statistic that would cause a rejection of the null hypothesis

Critical Value - Value or values that separate the critical region from the values of the test statistics that do not lead to a rejection of the null hypothesis

One Tailed Test

Critical

Region

Critical Value

( z score )

One Tailed Test

Critical

Region

Critical Value

( z score )

Two Tailed Test

Critical

Regions

Critical Value

( z score )

Critical Value

( z score )

Denoted by

The probability that the test statistic will fall in the critical region when the null hypothesis is actually true.

Common choices are 0.05, 0.01, and 0.10

Two-tailed,Right-tailed,Left-tailed Tests

The tails in a distribution are the extreme regions bounded

by critical values.

H0: µ = 100

H1: µ 100

is divided equally between

the two tails of the critical

region

Means less than or greater than

Reject H0

Fail to reject H0

Reject H0

100

Values that differ significantly from 100

H0: µ 100

H1: µ > 100

Fail to reject H0

Reject H0

Points Right

Values that

differ significantly

from 100

100

H0: µ 100

H1: µ < 100

Points Left

Reject H0

Fail to reject H0

Values that

differ significantly

from 100

100

Traditional Method

Reject H0if the test statistic falls in the critical region

Fail to reject H0if the test statistic does not fall in the critical region

P-Value Method

Reject H0if the P-value is less than or equal

Fail to reject H0if the P-value is greater than the

Finds the probability (P-value) of getting a result and rejects the null hypothesis if that probability is very low

Uses test statistic to find the probability.

Method used by most computer programs and calculators.

Will prefer that you use the traditional method on HW and Tests

Two tailed test

p(z>a) + p(z<-a)

One tailed test (right)

p(z>a)

One tailed test (left)

p(z<-a)

Where “a” is the value of the calculated test statistic

Used for HW # 3 – 5 – see example on next two slides

Determine P-value

Sample data: x = 105

or

z* = 2.66

Reject

H0: µ = 100

Fail to Reject

H0: µ = 100

*

µ = 73.4

or z = 0

z = 1.96

z* = 2.66

Just find p(z > 2.66)

Determine P-value

Sample data: x = 105

or

z* = 2.66

Reject

H0: µ = 100

Reject

H0: µ = 100

Fail to Reject

H0: µ = 100

*

z = - 1.96

µ = 73.4

or z = 0

z = 1.96

z* = 2.66

Just find p(z > 2.66) + p(z < -2.66)

Always test the null hypothesis

Choose one of two possible conclusions

1. Reject the H0

2. Fail to reject the H0

Never “accept the null hypothesis, we will fail to reject it.

Will discuss this in more detail in a moment

We are not proving the null hypothesis

Sample evidence is not strong enough to warrant rejection (such as not enough evidence to convict a suspect – guilty vs. not guilty)

Need to formulate correct wording of finalconclusion

Wording of final conclusion

1. Reject the H0

Conclusion: There is sufficient evidence to conclude………………………(what ever H1 says)

2. Fail to reject the H0

Conclusion: There is not sufficient evidence to conclude ……………………(what ever H1 says)

State a conclusion

The proportion of college graduates how smoke is less than 27%. Reject Ho:

The mean weights of men at FLC is different from 180 lbs. Fail to Reject Ho:

Used for #6 on HW

The mistake of rejecting the null hypothesis when it is true.

(alpha) is used to represent the probability of a type I error

Example: Rejecting a claim that the mean body temperature is 98.6 degrees when the mean really does equal 98.6 (test question)

the mistake of failing to reject the null hypothesis when it is false.

ß (beta) is used to represent the probability of a type II error

Example:Failing to reject the claim that the mean body temperature is 98.6 degrees when the mean is really different from 98.6 (test question)

True State of Nature

H0 True

H0 False

Reject H0

Correct

decision

Type I error

Decision

Fail to Reject H0

Type II error

Correct

decision

In this class we will focus on controlling a Type I error. However, you will have one question on the exam asking you to differentiate between the two.

a = p(rejecting a true null hypothesis)

b = p(failing to reject a false null hypothesis)

n, a and b are all related

Identify the type I and type II error.

The mean IQ of statistics teachers is greater than 120.

Type I: We reject the mean IQ of statistics teachers is 120 when it really is 120.

Type II: We fail to reject the mean IQ of statistics teachers is 120 when it really isn’t 120.

For any fixed sample size n, as decreases, increases and conversely.

To decrease both and , increase the sample size.

Power of a Hypothesis Test

is the probability (1 - ) of rejecting a false null hypothesis.

Note: No exam questions on this. Usually covered in a more advanced class in statistics.

7-2

Testing a claim about the mean

(large samples)

Goal

Identify a sample result that is significantly different from the claimed value

By

Comparing the test statistic to the critical value

Determine H0 and H1. (and if necessary)

Determine the correct test statistic and calculate.

Determine the critical values, the critical region and sketch a graph.

Determine Reject H0 or Fail to reject H0

State your conclusion in simple non technical terms.

Traditional (or Classical) Method of Testing Hypotheses (MAKE SURE THIS IS IN YOUR NOTES)

Can Use

Traditional method

Or

P-value method

1) Traditional method

2) P-value method

3) Confidence intervals

for testing claims about population means

1) The sample is a random sample.

2) The sample is large (n > 30).

a) Central limit theorem applies

b) Can use normal distribution

3) If is unknown, we can use sample standard deviation s as estimate for .

x - µx

Z*=

n

Reject the null hypothesis if the test statistic is in the critical region

Fail to reject the null hypothesis if the test statistic is not in the critical region

Claim: = 69.5 years

H0 : = 69.5

H1 : 69.5

Example:A newspaper article noted that the mean life span for 35 male symphony conductors was 73.4 years, in contrast to the mean of 69.5 years for males in the general population. Test the claim that there is a difference. Assume a standard deviation of 8.7 years. Choose your own significance level.

Step 1: Set up Claim, H0, H1

Select if necessary level:

= 0.05

Step 2: Identify the test statistic and calculate

x - µ 73.4 – 69.5

z*=== 2.65

8.7

n

35

Step 3: Determine critical region(s) and critical value(s) & Sketch

= 0.05

/2= 0.025 (two tailed test)

0.4750

0.4750

0.025

0.025

z = - 1.96 1.96

Critical Values - Calculator

Step 4: Determine reject or fail to reject H0:

Sample data: x = 73.4

or

z = 2.66

Reject

H0: µ = 69.5

Reject

H0: µ = 69.5

Fail to Reject

H0: µ = 69.5

*

z = - 1.96

µ = 73.4

or z = 0

z = 1.96

z = 2.66

P-value = P(z > 2.66) x 2 = .0078

REJECT H0

Claim: = 69.5 years

H0 : = 69.5

H1 : 69.5

There is sufficient evident to conclude that the mean life span of symphony conductors is different from the general population.

OR

There is sufficient evidence to conclude that mean life span of symphony conductors is different from 69.5 years.

Step 5: Restate in simple nontechnical terms

REJECT

Hypothesis Test using z (large sample)

- Press STAT
- Cursor to TESTS
- Choose ZTest
- Choose Input: STATS
- Enter σ and x and two tail, right tail or left tail
- Cursor to calculate or draw
*If your input is raw data, then input your raw data in L1 then use DATA

We reject a claim that the population parameter has a value that is not included in the confidence interval

Typically only used for two-tailed tests

For one-tailed test the degree of confidence would be 1 – 2a (don’t worry about this)

95% confidence interval of 35 conductors (that is, 95% of samples would contain true value µ )

70.5 < µ < 76.3

69.5 is not in this interval

Therefore it is very unlikely that µ = 69.5

Thus we reject claim µ = 69.5 (same conclusion as previously stated)

- Claim: mean age = 69.5 years,
- where n = 35, x = 73.4 ands= 8.7

7- 3

Testing a claim about the mean

(small samples)

for testing claims about population means (student t distribution)

1) The sample is a random sample.

2) The sample is small (n 30).

3) The value of the population standard deviation is unknown.

4) population is approximately normal.

Critical Values

Found in Table A-3

Degrees of freedom (df) = n -1

Critical t values to the left of the mean are negative

x -µx

t* =

s

n

Start

Use normal distribution with

x - µx

Is

n > 30

?

Yes

Z

/ n

(If is unknown use s instead.)

No

Is the

distribution of

the population essentially

normal ? (Use a

histogram.)

No

Use nonparametric methods, which don’t require a normal distribution.

Yes

Use normal distribution with

Is

known

?

x - µx

Z

/ n

No

(This case is rare.)

Use the Student t distribution

with

x - µx

t

s/ n

- Use z if
- known orn is large

- Use t if
- is unknown and n is small and population is approximately normal
MAKE SURE THIS IS IN YOUR NOTES

- is unknown and n is small and population is approximately normal

Table A-3 includes only selected values of

Specific P-values usually cannot be found from table

Use Table to identify limits that contain the P-value – very confusing

Some calculators and computer programs will find exact P-values

Hypothesis Test using t (small sample)

- Press STAT
- Cursor to TESTS
- Choose TTest
- Choose Input: STATS
- Enter s and x and two tail, right tail or left tail
- Cursor to calculate or draw
*If your input is raw data, then input your raw data in L1 then use DATA

Sample statistics of GPA include n=20, x=2.35 and s=.7

Test the claim that the GPA is greater than 2.0

Use traditional method

Use Calculator

Find exact p-value (see excel – TDIST function)

7-4

Testing a claim about a proportion

for testing claims about population proportions

1) The sample observations are a random sample.

2) The conditions for a binomial experiment are satisfied

If np 5 and nq 5 are satisfied we

Use normal distribution to approximate binomial with µ = np and = npq

p = population proportion (used in the null hypothesis)

q= 1 - p

n = number of trials

p = x/n(sample proportion)

p - p

z*=

pq

n

(determining the sample proportion of households with cable TV)

p sometimes is given directly

“10% of the observed sports cars are red”

is expressed as

p = 0.10

p sometimes must be calculated

“96 surveyed households have cable TV

and 54 do not” is calculated using

x

96

p = = = 0.64

n

(96+54)

When the calculation of p results in a decimal with many places, store the number on your calculator and use all the decimals when evaluating the z test statistic.

Large errors can result from rounding p too much.

Z* =

p - p

pq

n

x np

x - µ x - np n n p - p

z = = = =

pq

npq

npq

n

n

Hypothesis Test using z (proportions)

- Press STAT
- Cursor to TESTS
- Choose 1-PropZTest
- Enter x and n and two tail, right tail or left tail
- Cursor to calculate or draw