Hypothesis Tests Regarding a Parameter – Single Mean & Single Proportion

Hypothesis TestsRegarding a Parameter –

Single Mean & Single Proportion

Overview• This is the other part of inferential statistics, hypothesis

testing• Hypothesis testing and estimation are two different

approaches to two similar problems– Estimation is the process of using sample data to

estimate the value of a population parameter– Hypothesis testing is the process of using sample

data to test a claim about the value of a population parameter

The Language ofHypothesis Testing

Determine the null and alternative hypotheses from a claim

Hypothesis Testing

• The environment of our problem is that we want to test whether a particular claim is believable, or not

• The process that we use is called hypothesis testing

• This is one of the most common goals of statistics

Hypothesis Testing

• Hypothesis testing involves two steps– Step 1 – to state what we think is true– Step 2 – to quantify how confident we are in

our claim• The first step is relatively easy• The second step is why we need statistics

Hypothesis Testing

• We are usually told what the claim is, what the goal of the test is

• Now similar to estimation in the previous unit discussed, we will again use the material regarding the sampling distribution of the sample mean to quantify how confident we are in our claim

Example• An example of what we want to quantify

– A car manufacturer claims that a certain model of car achieves 29 miles per gallon

– To test for the claim, we then test some number of cars

– We calculate the sample mean … it is 27– Is 27 miles per gallon consistent with the

manufacturer’s claim? How confident are we that the manufacturer has significantly overstated the miles per gallon achievable?

Example

• How confident are we that the gas economy is definitely less than 29 miles per gallon?

• We would like to make either a statement“We’re pretty sure that

the mileage is less than 29 mpg”or

“It’s believable that themileage is equal to 29 mpg”

Level of Significance

• A hypothesis test for an unknown parameter is a test of a specific claim– Compare this to a confidence interval which

gives an interval of numbers, not a “believe it” or “don’t believe it” answer

• The level of significance reflects the confidence we have in our conclusion

Null Hypothesis

• How do we state our claim?• Our claim

– Is the statement to be tested– Is called the null hypothesis– Is written as H0 (and is read as “H-naught”)

Alternative Hypothesis

• How do we state our counter-claim?• Our counter-claim

– Is the opposite of the statement to be tested– Is called the alternative hypothesis– Is written as H1 (and is read as “H-one”)

Two-tailed Test• There are different types of null hypothesis / alternative

hypothesis pairs, depending on the claim and the counter-claim

• One type of H0 / H1 pair, called a two-tailed (or two-sided) test, tests whether the parameter is either equal to, versus not equal to, some value– H0: parameter = some value

– H1: parameter ≠ some value

Example• An example of a two-tailed test• A bolt manufacturer claims that the diameter of the bolts

average 10 mm– H0: Diameter = 10– H1: Diameter ≠ 10

• An alternative hypothesis of “≠ 10” is appropriate since– A sample diameter that is too high may be a problem– A sample diameter that is too low may also be a problemThat is, we may reject the claim under the H0 , if the sample value

is either too high or too low• Thus this is a two-tailed test

Left-tailed Test

• Another type of pair, called a left-tailed test, tests whether the parameter is either equal to, versus less than, some value– H0: parameter = some value (This actually means

Parameter some value)

– H1: parameter < some value

Note: Equality sign appears only in the null hypothesis.

Example• An example of a left-tailed test• A car manufacturer claims that the mpg of a certain model car is at least 29.0

– H0: MPG = 29.0 (In fact, this does not mean MPG is only 29.0. it means MPG 29.0 )– H1: MPG < 29.0

• An alternative hypothesis of “< 29” is appropriate since– A mpg that is too low is a problem– A mpg that is too high is not a problemThat is, we reject the claim under the H0, if the sample mpg observed is too low,

much lower than 29.• Thus this is a left-tailed test. (The side of the tail depends on the direction under H1

which tends to support a lower value of MPG. And a lower value is located on the left of a higher value on a number line.

Note: By convention, we always only put the equality sign for the claim in H0 , even though it should be MPG 29.0. This is because we can tell the actual direction of the inequality sign under H0 by just looking at the sign in H1 (H0 is the opposite of H1. Since MPG is less than 29 in H1, MPG will be no less than 29 under H0.)

Right-tailed Test• Another third type of pair, called a right-tailed test,

tests whether the parameter is either equal to, versus greater than, some value– H0: parameter = some value (This actually means

parameter Some value.)– H1: parameter > some value

Note: Equality sign appears only under the null hypothesis H0

Example• An example of a right-tailed test• A bolt manufacturer claims that the defective rate

of their product is at most 1 part in 1,000– H0: Defect Rate = 0.001– H1: Defect Rate > 0.001

• An alternative hypothesis of “> 0.001” is appropriate since– A defect rate that is too low is not a problem– A defect rate that is too high is a problemThat is, higher defective rate observed tends to be in

favor of H1, , but against H0. • Thus this is a right-tailed test

One-tailed and Two-tailed Tests

• A comparison of the three types of tests• The null hypothesis

– We believe that this is true• The alternative hypothesis

Type of test Sample value that is too low

Sample value that is too high

Two-tailed test A problem A problemLeft-tailed test A problem Not a problemRight-tailed test Not a problem A problem

Example 1

• A manufacturer claims that there are at least two scoops of cranberries in each box of cereal

• What would be a problem?– The parameter to be tested is the number of scoops

of cranberries in each box of cereal– If the sample mean is too low, that is a problem– If the sample mean is too high, that is not a problem

• This is a left-tailed test– The “bad case” is when there are too few

Example 2

• A manufacturer claims that there are exactly 500 mg of a medication in each tablet

• What would be a problem?– The parameter to be tested is the amount of a

medication in each tablet– If the sample mean is too low, that is a problem– If the sample mean is too high, that is a problem too

• This is a two-tailed test– A “bad case” is when there are too few

Example 3

• A manufacturer claims that there are at most 8 grams of fat per serving

• What would be a problem?– The parameter to be tested is the number of grams of

fat in each serving– If the sample mean is too low, that is not a problem– If the sample mean is too high, that is a problem

• This is a right-tailed test– The “bad case” is when there are too many

Reject or Not to reject H0

• There are two possible results for a hypothesis test

• If we believe that the null hypothesis could be true, this is called not rejecting the null hypothesis– Note that this is only “we believe … could be”

• If we are pretty sure that the null hypothesis is not true, so that the alternative hypothesis is true, this is called rejecting the null hypothesis– Note that this is “we are pretty sure that … is”

Understand Type I and Type II errors

Decision Errors• In comparing our conclusion (not reject or reject the null

hypothesis) with reality, we could either be right or we could be wrong– When we reject (and state that the null hypothesis is

false) but the null hypothesis is actually true– When we not reject (and state that the null hypothesis

could be true) but the null hypothesis is actually false• These would be undesirable errors

Type I and II Errors• A summary of the errors is

• We see that there are four possibilities … in two of which we are correct and in two of which we are incorrect

Type I and II Errors• When we reject the null hypothesis (and state

that the null hypothesis is false) but the null hypothesis is actually true … this is called a Type I error

• When we do not reject the null hypothesis (and state that the null hypothesis could be true) but the null hypothesis is actually false … this called a Type II error

• In general, Type I errors are considered the more serious of the two

Example

• A very good analogy for Type I and Type II errors is in comparing it to a criminal trial

• In the US judicial system, the defendant “is innocent until proven guilty”– Thus the defendant is presumed to be

innocent– The null hypothesis is that the defendant is

innocent– H0: the defendant is innocent

Example (continued)

• If the defendant is not innocent, then– The defendant is guilty– The alternative hypothesis is that the

defendant is guilty– H1: the defendant is guilty

• The summary of the set-up– H0: the defendant is innocent

– H1: the defendant is guilty

Example (continued)• Our possible conclusions• Reject the null hypothesis

– Go with the alternative hypothesis – H1: the defendant is guilty– We vote “guilty”

• Do not reject the null hypothesis– Go with the null hypothesis– H0: the defendant is innocent– We vote “not guilty” (which is not the same as voting

innocent! Voting “not guilty” does not prove the defendant is innocent, we just do not have enough evidence to against the defendant.)

Example (continued)

• A Type I error– Reject the null hypothesis– The null hypothesis was actually true– We voted “guilty” for an innocent defendant

• A Type II error– Do not reject the null hypothesis– The alternative hypothesis was actually true– We voted “not guilty” for a guilty defendant

Example (continued)• Which error do we try to control?• Type I error (sending an innocent person to jail)

– The evidence was “beyond a reasonable doubt”– We must be pretty sure– Very bad! We want to minimize this type of error

• A Type II error (letting a guilty person go)– The evidence wasn’t “beyond a reasonable doubt”– We weren’t sure enough– If this happens … well … it’s not as bad as a Type I

error (according to the US system)

State Conclusion to Hypothesis Tests

Reject or Not to reject H0

• “Innocent” versus “Not Guilty”• This is an important concept• Innocent is not the same as not guilty

– Innocent – the person did not commit the crime– Not guilty – there is not enough evidence to convict …

that the reality is unclear• To not reject the null hypothesis – doesn’t mean

that the null hypothesis is true – just that there isn’t enough evidence to reject

Summary

• A hypothesis test tests whether a claim is believable or not, compared to the alternative

• We test the null hypothesis H0 versus the alternative hypothesis H1

• If there is sufficient evidence to conclude that H0 is false, we reject the null hypothesis

• If there is insufficient evidence to conclude that H0 is false, we do not reject the null hypothesis

Hypothesis Tests for a Population MeanAssuming the Population Standard

Deviation is Known

Learning Objectives

• Understand the logic of hypothesis testing

• Test hypotheses about a population mean with σ known using the classical approach

• Test hypotheses about a population mean with σ known using P-values approach

• Test hypotheses about a population mean with σ known using confidence intervals approach

Understand the logic of hypothesis testing

Decision Rule

• Hypothesis test is to set up a decision rule for the sample data to reject or not to reject the null hypothesis – How do we quantify “unlikely” the null

hypothesis is true?– What is the exact procedure to get to a “do

not reject” or “reject” conclusion?

Methods of Hypothesis Testing

• There are three equivalent ways to perform a hypothesis test

• They will reach the same conclusion• The methods

– The classical approach– The P-value approach– The confidence interval approach

Methods of Hypothesis Testing• The classical approach

– If the sample value observed is too many standard deviations away from the true value claimed under H0, then it must be too unlikely H0 is true

• The P-value approach– If the probability of the sample value being that far

away is small, then it must be too unlikely H0 is true• The confidence interval approach

– If we are not sufficiently confident that the parameter is likely enough, then it must be too unlikely

• Don’t worry … we’ll be explaining more

Basic Steps to Test the Hypothesis

Step 1: We set up the null hypothesis that the actual mean μ is equal to a value μ0 and the alternative hypothesis

Step 2: We set up a criterion (to reject H0)– A criterion that quantifies “unlikely” the null

hypothesis that the actual mean μ being equal to a specified value of μ0 is true. That is, the actual mean is unlikely to be equal to μ0

Collect Sample Data• The three methods all need information

– We run an experiment– We collect the data– We calculate the sample mean

• The three methods all make the same assumptions to be able to make the statistical calculations– That the sample is a simple random sample– That the sample mean has a normal distribution

Choose a Test Statistic• We first assume that the population standard deviation σ

is known• We use a sample estimate, for instance, a sample mean to test for the population parameter - the population

mean μ• We can apply our techniques if either

– The population has a normal distribution– Our sample size n is large (n ≥ 30)

• In those cases, the distribution of the sample mean is normal with mean μ and standard deviation σ / √ n

x

x

Check the criterion for Unlikely• The three methods all compare the observed results with

the criterion that quantifies “unlikely”: – Classical – how many standard deviations– P-value – the size of the probability– Confidence interval – inside or outside the interval

• If the results are unlikely based on these criterion, we reject the claim under the null hypothesis.

Statistical Significance

• The three methods all conclude similarly– We do not reject the null hypothesis, or– We reject the null hypothesis

• When we reject the null hypothesis, we say that the result is statistically significant

Perform Hypothesis Testing

• We now will cover how each of the– Classical– P-value, and– Confidence interval

approaches will show us how to conclude whether the result is statistically significant or not

Test hypotheses about a population mean with σ known using the classical approach

The Classical Approach

• We compare the sample mean to the hypothesized population mean μ0

– Measure the difference in units of standard deviations, which is called the test statistic:

– A lot of standard deviations is far … few standard deviations is not far

– Just like using a general normal distribution

x

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α Level of Significance • How far is too far?• For example, we can set α = 0.05 as the size of “unlikely”,

so-called “the level of significance”• “Unlikely” means that this difference occurs with

probability α = 0.05 of the time, or less under the null hypothesis

• This concept applies to two-tailed tests, left-tailed tests, and right-tailed tests

Note: α is often determined subjectively before the experiment. It sets up a rule to reject the null hypothesis. So, it is also the size of the risk for committing a type I error of rejecting the null hypothesis by mistake.

5% Level of Significance • For two-tailed tests

– The least likely 5% is the lowest 2.5% and highest 2.5% (below –1.96 and above +1.96 standard deviations) … –1.96 and +1.96 are the critical values (There are two critical values for 2-tailed test)

– The region outside this is the rejection region (or critical region) which covers the range of “ unlikely” values for the test statistic to reject H0

5% Level of Significance• For left-tailed tests

– The least likely 5% is the lowest 5% (below –1.645 standard deviations) … –1.645 is the critical value (only one critical value for one-tailed test.)

– The region less than this is the rejection region

5% Level of Significance• For right-tailed tests

– The least likely 5% is the highest 5% (above 1.645 standard deviations) … +1.645 is the critical value

– The region greater than this is the rejection region

Example 1• An example of a two-tailed test• A bolt manufacturer claims that the diameter of

the bolts average 10.0 mm– H0: Diameter = 10.0– H1: Diameter ≠ 10.0

• We take a sample of size 40– (Somehow) We know that the standard deviation of

the population is 0.3 mm– The sample mean is 10.12 mm– We’ll use a level of significance α = 0.05

Example 1 (continued)• Do we reject the null hypothesis?

– 10.12 is 0.12 higher than 10.0– The standard error is (0.3 / √ 40) = 0.047– The test statistic is 2.53– One of the two critical normal values is z/2, for α =

0.05, is 1.96– 2.53 is more than 1.96, which is in the rejection

region.• Our conclusion

– We reject the null hypothesis– We have sufficient evidence that the population mean

diameter is not 10.0

Example 2• An example of a left-tailed test• A car manufacturer claims that the mpg of a

certain model car is at least 29.0– H0: MPG = 29.0– H1: MPG < 29.0


the population is 0.5– The sample mean mpg is 28.89– We’ll use a level of significance α = 0.05


– 28.89 is 0.11 lower than 29.0– The standard error is (0.5 / √ 40) = 0.079– The test statistic is -1.39– -1.39 is greater than -1.645, which is the left-tailed

critical value -z, with α = 0.05. -1.39 is not in the rejection region.

• Our conclusion– We do not reject the null hypothesis– We have insufficient evidence that the population

mean mpg is less than 29.0

Example 3• An example of a right-tailed test• A bolt manufacturer claims that the defective

rate of their product is at most 1.70 per 1,000– H0: Defect Rate = 1.70– H1: Defect Rate > 1.70


the population is .06– The sample defect rate is 1.78– We’ll use a level of significance α = 0.05

Example 3 (continued)

• Do we reject the null hypothesis?– 1.78 is 0.08 higher than 1.70– The standard error is (0.06 / √ 40) = 0.009– The test statistic is 8.43– 8.43 is more than 1.645 which is the right-tailed

critical value z, with α = 0.05

• Our conclusion– We reject the null hypothesis– We have sufficient evidence that the population mean

rate is more than 1.70

Critical Value(s) and Rejection Region

• Two-tailed test – The critical values are zα/2 and -zα/2

– The rejection region includes {less than -zα/2} and {greater than z1-α/2}

• Left-tailed test– The critical value is -zα

– The rejection region is {less than -zα}• Right-tailed test

– The critical value is zα

– The rejection region is {greater than zα}

Summary

• The general picture for a level of significance α

Decision Rule for Classical Approach

• Calculate a test

• For a significance level α provided, we locate the critical value(s) and corresponding rejection region.

• Reject the null hypothesis, if a calculated test statistic z0 is in the rejection region; Do not reject, otherwise.

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Test hypotheses about a population mean with σ known

using P-values

The P-value Approach

• The P-value is the probability of observing a sample mean that is as or more extreme than the observed

• The probability is calculated assuming that the null hypothesis is true

• We use the P-value to quantify how unlikely the sample mean is

P-value• Just like in the classical approach, we calculate the test

statistic

• We then calculate the p-value, the probability that the sample mean would be this, or more extreme, if the null hypothesis was true.

• It measures how likely the observed z, i.e. z0 will occur under the null hypothesis.

• The two-tailed, left-tailed, and right-tailed calculations are slightly different

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P-value• For the two-tailed test, the “unlikely” region are values that are too

high and too low• Small P-values corresponds to situations where it is unlikely to be

this far away

P-value• For the left-tailed test, the “unlikely” region are values that are too

low• Small P-values corresponds to situations where it is unlikely to be

this low

P-value• For the right-tailed test, the “unlikely” region are values

that are too high• Small P-values corresponds to situations where it is

unlikely to be this high

Summary

• For all three models (two-tailed, left-tailed, right-tailed)– The larger P-values mean that the difference

is not relatively large … that it’s not an unlikely event

– The smaller P-values mean that the difference is relatively large … that it’s an unlikely event

Example• Larger P-values

– A P-value of 0.30, for example, means that this value, or more extreme, could happen 30% of the time

– 30% of the time is not unusual

• Smaller P-values– A P-value of 0.01, for example, means that this value,

or more extreme, could happen only 1% of the time– 1% of the time is unusual

Decision Rule for P-value Approach

• The decision rule is• For a significance level α provided

– Do not reject the null hypothesis if the P-value is greater than α

– Reject the null hypothesis if the P-value is less than α• For example, if α = 0.05

– A P-value of 0.30 is likely enough, compared to a criterion of 0.05 level of significance

– A P-value of 0.01 is unlikely, compared to a criterion of 0.05 level of significance

Example 1• An example of a two-tailed test• A bolt manufacturer claims that the diameter of the bolts average

10.0 mm– H0: Diameter = 10.0– H1: Diameter ≠ 10.0

• We take a sample of size 40– (Somehow) We know that the standard deviation of the

population is 0.3 mm– The sample mean is 10.12 mm– We’ll use a level of significance α = 0.05

Note: The claim is about the average is equal to 10.0 or not. It does not indicate it is going to be greater than or less than 10.0 if it is not equal to 10.0. So, we design a two-tailed hypotheses, since we do not know the direction.


– 10.12 is 0.12 higher than 10.0– The standard error is ( ) = 0.047– The test statistic z0 is 2.53: (10.12-0.12)/0.047) = 2.53– The 2-sided P-value of 2.53 is 0.0114 < 0.05 = α

P value = or 2 x normalcdf(2.53, E99) = 0.0114 or 2 x normalcdf(10.12, E99, 10, ) = 0.0114

• Our conclusion– We reject the null hypothesis– We have sufficient evidence that the population mean diameter is not

10.0

).x(P 12102

4030 /.

4030 /.

).z(P 5322

Example 2• An example of a left-tailed test• A car manufacturer claims that the mpg of a certain model car is at

least 29.0– H0: MPG = 29.0– H1: MPG < 29.0

• We take a sample of size 40– (Somehow) We know that the standard deviation of the

population is 0.5– The sample mean mpg is 28.89– We’ll use a level of significance α = 0.05

Note: Since the equality sign should always appear only in the null hypothesis H0, , the claim of at least (greater than or equal to) 29.0 is placed under the null hypothesis. Since the alternative hypothesis H1 is the opposite of the null hypothesis H0, so MPG < 29 under H1.


– 28.89 is 0.11 lower than 29.0– The standard error is ( ) = 0.079– The test statistic is -1.39– The 1-sided P-value of -1.39 is 0.0823 > 0.05 = α P value = or

normalcdf(-E99, -1.39) = 0.0823 or normalcdf(-E99, 28.89, 29, ) = 0.0823• Our conclusion

– We do not reject the null hypothesis– We have insufficient evidence that the population mean mpg is

less than 29.0

).Z(P 391).x(P 8928

4050 /.

4050 /.

Example 3• An example of a right-tailed test• A bolt manufacturer claims that the defective rate of their product is

at most 1.70 per 1,000– H0: Defect Rate = 1.70 (This is the claim of at most 1.00 per 1,0000)

– H1: Defect Rate > 1.70• We take a sample of size 40

– (Somehow) We know that the standard deviation of the population is .06

– The sample defect rate is 1.78– We’ll use a level of significance α = 0.05


– 1.78 is 0.08 higher than 1.70– The standard error is ( ) = 0.009– The test statistic is 8.43– The 1-sided P-value of 8.43 is extremely small P-value = or normalcdf(1.74, E99, 1.70, ) = 1.75E-17 normalcdf(8.43, E99) = 1.75E-17

• Our conclusion– We reject the null hypothesis– We have sufficient evidence that the population mean rate is

more than 1.70

).x(P 781 ).z(P 43840060 /.

40060 /.

Classical and P-value Approaches

• Compare the rejection regions for the classical approach and the P-value approach

• They are the sameClassical P-Value

Note: The classical approach sets a criteria for “unlikely” in terms of a z value; the p-value approach sets a criteria in terms of a probability.

Test hypotheses about a population mean with σ known using confidence intervals

Level of Significance α and Level of Confidence (1 – α)

• The confidence interval approach yields the same result as the classical approach and as theP-value approach

• We compare– A hypothesis test with a level of significance α

to– A confidence interval with confidence (1 – α) •100%

• These are the same α’s

Decision Rule for Confidence Interval Approach

• The relationship is

• The hypothesis test calculation and the confidence interval calculation are very similar

Not rejecting thehypothesis

μ0 is inside theConfidence interval

Rejecting thehypothesis

μ0 is outside theConfidence interval

Example 1• An example of a two-tailed test• A bolt manufacturer claims that the diameter of

the bolts average 10.0 mm– H0: Diameter = 10.0– H1: Diameter ≠ 10.0


this measurement is 0.3 mm– The sample mean is 10.12 mm– We’ll use a level of significance α = 0.05


– 10.12 is 0.12 higher than 10.0– The standard error is (0.3 / √ 40) = 0.047– The confidence interval is 10.12 ± 1.96 • 0.047, or

10.03 to 10.21– 10.0 is outside (10.03, 10.21)


diameter is not 10.0

Note: 1.96 is z0.025 for a 95% confidence interval.

Example 2• An example of a left-tailed test• A car manufacturer claims that the mpg of a

certain model car is at least 29.0– H0: MPG = 29.0– H1: MPG < 29.0


the population is 0.5– The sample mean mpg is 28.89– We’ll use a level of significance α = 0.05


– 28.89 is 0.11 lower than 29.0– The standard error is (0.5 / √ 40) = 0.079– The upper confidence interval limit is 28.89 + 1.645 •

0.079, or 29.02– 29.0 is inside (-∞, 29.02)

• Our conclusion– We do not reject the null hypothesis– We have insufficient evidence that the population

mean mpg is less than 29.0Note: 1.645 is z0.05 for a 95% upper confidence interval

limit.

Example 3• An example of a right-tailed test• A bolt manufacturer claims that the defective

rate of their product is at most 1.70 per 1,000– H0: Defect Rate = 1.70– H1: Defect Rate > 1.70


the population is .06– The sample defect rate is 1.78– We’ll use a level of significance α = 0.05


– 1.78 is 0.08 higher than 1.70– The standard error is (0.06 / √ 40) = 0.009– The lower confidence interval limit is 1.78 – 1.645 • 0.009 = 1.76– 1.70 is outside (1.76, ∞)


rate is more than 1.70Note: 1.645 is z0.05 for a 95% lower confidence interval

limit.

Summary• A hypothesis test of means compares whether the true

mean is either– Equal to, or not equal to, μ0

– Equal to, or less than, μ0

– Equal to, or more than, μ0

• There are three equivalent methods of performing the hypothesis test– The classical approach– The P-value approach– The confidence interval approach

Hypothesis Tests for aPopulation Mean in Practice

Test hypotheses about a population mean with σ unknown

Test of Mean in Practice• So far, we assumed that the population standard

deviation, σ, was known• This is not a realistic assumption• There is a parallel between last unit and this unit

– solving the problems assuming that σ was known

– solving the problem assuming that σ was not known

• σ not being known is a much more practical assumption

Test of Mean in Practice• The parallel between Confidence Intervals and

Hypothesis Tests carries over here too• For Confidence Intervals

– We estimate the population standard deviation σ by the sample standard deviation s

– We use the Student’s t-distribution with n-1 degrees of freedom

• For Hypothesis Tests, we do the same– Use s for σ– Use the Student’s t for the normal

t-test Statistic

• Thus instead of the z-test statistic knowing σ

we calculate a t-test statistic using s

• This is the appropriate test statistic to use when σ is unknown

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n/sxt 0

Hypotheses

• We can perform our hypotheses for tests of a population proportion in the same way as when the sample standard deviation is known

Two-tailed Left-tailed Right-tailedH0: μ = μ0

H1: μ ≠ μ0

H0: μ = μ0

H1: μ < μ0

H0: μ = μ0

H1: μ > μ0

Test of Mean in Practice

• The process for a hypothesis test of a mean, when σ is unknown is not different from the test of a men, when σ is known– Set up the problem with a null and alternative

hypotheses– Collect the data and compute the sample

mean– Compute the test statistic


• Either the Classical and the P-value approach can be applied to determine the significance

P-value approachClassical approach

Test of Mean in Practice

• There are thus only differences between this process and the one using the normal distribution previously– We use the sample standard deviation s

instead of the population standard deviation σ– We use the Student’s t-distribution, with n-1

degrees of freedom, instead of the normal distribution

Example

• An example• A gasoline manufacturer wants to make sure

that the octane in their gasoline is at least 87.0– The testing organization takes a sample of size 40– The sample standard deviation is 0.5 ( i.e. s = 1.5)– The sample mean octane is 86.94

• Our null and alternative hypotheses– H0: Mean octane = 87

– HA: Mean octane < 87

Example (continued)• Do we reject the null hypothesis under 0.05 level of significance?

– 86.94 is 0.06 lower than 87.0– The standard error is (0.5 / √ 40) = 0.08– 0.06 is 0.75 standard error less

– The critical t value t0.05, with 39 degrees of freedom, is -1.685 [obtained from TI calculator invT(.95,39)=-1.685] – -1.685 < - 0.75, it is not unusual

• Our conclusion– We do not reject the null hypothesis– We have insufficient evidence that the true population mean

(mean octane) is less than 87.0

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..t

Classical Approach:

Example (continued)P-value approach:• Do we reject the null hypothesis under 0.05 level of significance?

– 86.94 is 0.06 lower than 87.0– The standard error is (0.5 / √ 40) = 0.08– 0.06 is 0.75 standard error less– The 1-sided P-value of -0.75 is 0.2289 > 0.05 = α P value = = 0.2289 or

tcdf(-E99, -0.75, 39) = 0.2289 • Our conclusion

– We do not reject the null hypothesis– We have insufficient evidence that the true population mean

(mean octane) is less than 87.0

).t(p 750

Compare t-test with z-test• Comparing using the classical approach

Compare t-test with z-test• Comparing using the P-value approach

Summary• A hypothesis test of means, with σ unknown, has the

same general structure as a hypothesis test of means with σ known

• Any one of our three methods can be used, with the following two changes to all the calculations– Use the sample standard deviation s in place of the

population standard deviation σ– Use the Student’s t-distribution in place of the normal

distribution

Hypothesis Tests for aPopulation Proportion

Test hypotheses about a population proportion using the normal model

Test of Population Proportion

• In a sample of size n, with x successes, the best estimate of the population proportion is

• Similar to tests for means, we have– Two-tailed tests– Left-tailed tests– Right-tailed tests

nxp̂

Standard Error of Sample Proportion

• Just as for confidence intervals, the standard error of the sample mean proportion is

n)p(p 1

Standard Error of the Sample Proportion under H0

• To test for the population proportion, we use the following standard error of the sample proportion:

and not

n)p(p 00 1

n)p̂(p̂ 1

(Yes, use this)

(No, don’t use this)

z-test Statistic for Testing the Proportion

• Because we assume that the null hypothesis H0: p = p0 is true, we should use

as the standard error

• The test statistic is thus

n)p(p 00 1

n)p(p

pp̂00

01

Z =

Hypothesis Test of Proportion

• We can perform our hypotheses for tests of a population proportion in the same way as the hypothesis tests of a population mean

Two-tailed Left-tailed Right-tailedH0: p = p0

H1: p ≠ p0

H0: p = p0

H1: p < p0

H0: p = p0

H1: p > p0

Hypothesis Test of Proportion• The process for a hypothesis test of a proportion is

– Set up the problem with a null and alternative hypotheses

– Collect the data and compute the sample proportion– Compute the test statistic

n)p(p

pp̂00

01

Z =


• Either the Classical and the P-value approach can be applied to determine the significance

Classical approach P-value approach

Example• An example• We believe that 60% of students prefer hamburgers

over hot dogs• A random sample of 200 students found that 102 of

them preferred hamburgers. • At α = 0.05, does the data support our belief?

– The sample size n = 200– The hypothesized proportion p0 = 0.60– The sample proportion 510200102 ./p

Example (continued)

• Our hypotheses– H0: p = 0.60

– H1: p ≠ 0.60

• The standard error is

• The test statistic is

03501 00 .n

)p(p

6021 00

0 .

n)p(p

pp̂

Example (continued)

• The critical values for α = 0.05 are ± z0.025 = ± 1.96

• The test statistic –2.60 is outside the critical values, so we reject the null hypothesis

• There is significant evidence that the proportion of students who prefer hamburgers is not 60%

Summary

• We can perform hypothesis tests of proportions in similar ways as hypothesis tests of means– Two-tailed, left-tailed, and right-tailed tests

• The normal distribution should be used to compute the critical value(s) for this test

Putting It All Together:Which Method Do I Use?

Determine the appropriate hypothesis test to perform

Which Test?

• Parallels between hypothesis tests and confidence intervals– Both use the concept of the variability of a

sample statistic– Both use critical values from the normal and

Student’s t-distributions– Both have means with known σ, means with

unknown σ, proportions, and standard deviations cases

Which Test?

• It should not be surprising that the decision process for which hypothesis test to use is very similar to the decision process for which confidence interval to use

• Start with– Is the parameter a mean?– Is the parameter a proportion?

Which Test?• In analyzing population means• Is the population variance known?

– If so, then we can use the normal distribution• If the population variance is not known

– If we have “enough” data (30 or more values), we still can use the normal distribution

– If we don’t have “enough” data (29 or fewer values), we should use the t-distribution

• We don’t have to ask this question in the analysis of proportions

Which Test?

• For the test of a population mean• If

The data is OK (reasonably normal)The variance is known

then we can use the normal distribution with a test statistic of

n

x0

Z=

Which Test?

• For the test of a population mean• If

The data is OK (reasonably normal)The variance is NOT known

then we can use the Student’s t-distribution with a test statistic of

ns

x 0t =

Which Test

• For the test of a population proportion• If the sample size is large enough,

we can use the proportions method with a test statistic of

n)p(p

pp̂00

01

Z=

Summary

• The main questions that determine the method• Is it a

– Population mean?– Population proportion?

• In the case of a population mean, we need to determine– Is the population variance known?– Does the data look reasonably normal?

Summary• The process of hypothesis testing is very similar across

the testing of different parameters• The major steps in hypothesis testing are

– Formulate the appropriate null and alternative hypotheses

– Calculate the test statistic– Determine the appropriate critical value or values– Reach the reject / do not reject conclusions

Summary

• Similarities in hypothesis test processesParameter Mean (Std

Dev known)Mean (Std

Dev unknown)Proportion

H0: μ = μ0 μ = μ0 p = p0

(2-tailed) H1: μ ≠ μ0 μ ≠ μ0 p ≠ p0

(L-tailed) H1: μ < μ0 μ < μ0 p < p0

(R-tailed) H1: μ > μ0 μ > μ0 p > p0

Test statistic Difference Difference DifferenceCritical value Normal Student t Normal

Summary• We can test whether sample data supports a hypothesis

claim about a population mean or a proportion• We can use any one of three methods

– The classical method– The P-Value method– The Confidence Interval method

• The commonality between the three methods is that they set a criterion for rejecting or not rejecting the test statistic. The classical approach sets a criteria in terms of a z value; the p-value approach sets a criteria in terms of a probability.

Hypothesis Tests Regarding a Parameter – Single Mean & Single Proportion

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