section 10.2 hypothesis testing for means (small samples) hawkes learning systems math courseware...

Section 10.2

Hypothesis Testing for Means (Small Samples)

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Copyright © 2008 by Hawkes Learning

Systems/Quant Systems, Inc.

All rights reserved.

What this lesson is about

• Learn to perform a hypothesis test• The previous lesson was only about how to set

up a hypothesis test.– Reading and understanding the real-life scenario.– Getting the right letter, μ or p.– Getting the right relational operators in the right

places: = and ≠, ≤ and >, ≥ and <.– Getting the right value of μ or p (and setting aside

the “noise” numbers in the problem statement.)

(Added content by D.R.S.)

Choice: Do a t Test or a z Test?

Small Samples: t Test• “Small” means “sample size

is n < 30.• There’s an assumption that

the population is normally distributed.

• If the population is not normally distributed, this method we use is NOT valid.

• Easy for today: everything we do is a t Test.

Large Samples: z Test• “Large” means “sample size

is n ≥ 30.• To be discussed in a later

lesson.• The Bluman book has

slightly different rules from the way this Hawkes book does it. Just be aware of that.

(Added content by D.R.S.)

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Test Statistic for Small Samples, n < 30:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

with d.f. = n – 1

To determine if the test statistic calculated from the sample is statistically significant we will need to look at the critical value.

The critical values for n < 30 are found from the t-distribution.

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Find the critical value:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

Find the critical t-score for a right-tailed test that has 14 degrees of freedom at the 0.025 level of significance.

Solution:

d.f. = 14 and a = 0.025

t0.025 = 2.145

(Added info)• It’s in Table C, Critical Values

of t

Inputs: • Column for α (alpha)• Choose the right column for

one- or two-tailed• Row for d.f., degrees of

freedom (= sample size n, minus 1)

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Rejection Regions:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

Determined by two things:1. The type of hypothesis test.

2. The level of significance, a.

Finding a Rejection Region:

1. Look up the critical value, tc, to determine the cutoff for the rejection region.

2. If the test statistic you calculate from the sample data falls in the a area, then reject H0.

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Types of Hypothesis Tests:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

Alternative Hypothesis

< Value> Value≠ Value

Type of Test

Left-tailed testRight-tailed testTwo-tailed test

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Rejection Regions for Left-Tailed Tests, Ha contains <:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

Reject if t ≤ –t

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Rejection Regions for Right-Tailed Tests, Ha contains >:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

Reject if t ≥ t

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Rejection Regions for Two-Tailed Tests, Ha contains ≠:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

Reject if | t | ≥ t/2

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Steps for Hypothesis Testing:

1. State the null and alternative hypotheses.2. Set up the hypothesis test by choosing the

test statistic [that is, make a decision on whether it’s a t or z problem] and determining the values of the test statistic that would lead to rejecting the null hypothesis [the critical value(s)].

3. Gather data and calculate the necessary sample statistics [t = or z = ].

4. Draw a conclusion [Stating it two ways: reject/fail to reject, and also in plain English].

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

IMPORTANT !!!!

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Draw a conclusion:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

27 college students are asked how many parking tickets they get a semester to test the claim that the average student gets more than 9 tickets a semester. The sample mean for the number of tickets was found to be 9.8 with a standard deviation of 1.5. Use a = 0.10.

Solution:

n = 27, = 9, = 9.8, s = 1.5, d.f. = 26, a = 0.10

t0.10 =

Since t is greater than t a , we will reject the null hypothesis.

1.315

2.771

H0: μ ≤ 9 tickets Ha: μ > 9 tickets.

This is the CRITICAL VALUE. Either use table or invT(0.10,26). Draw a PICTURE, too. Mark 1.315and highlight the critical region.

This is the TEST STATISTIC.Mark 2.771 on your picture.

Remarks about the parking ticket example

• There was a choice made to do a t Test because the sample size was < 30.

• There was an implicit assumption that the distribution of the count of parking tickets fits a normal distribution.

• It was a RIGHT-TAILED TEST because of the “>” in the alternative hypothesis.

(Added content by D.R.S.)

Remarks about the parking ticket example, continued

• Hypothesis tests are really essay questions. • The outline for the essay is the four-step

procedure described in the earlier slide.• Each of the four steps needs to be explained

plainly with a lot of words: Complete thoughts and complete sentences.

• The final statement is in plain English, suitable for the general public to understand.

(Added content by D.R.S.)

The Parking Ticket problem done as an essay question

1. State the hypotheses• We investigate the claim

that the average student receives more than nine parking tickets in a semester. Our hypotheses are:

• Null hypothesis, H0: μ ≤ 9• Alternative hypothesis:

Ha: μ > 9, more than nine tickets per semester.

2. Find the critical value• This is a t Test, right tailed.• The sample size is n = 27.• The degrees of freedom is

d.f. = n – 1 = 26.• The level of significance

chosen is α = 0.10• The critical value is

tα=0.10,d.f.=26 = 1.315

(Added content by D.R.S.)

The Parking Ticket problem done as an essay question

3. Compute the test statistic• (As shown on the earlier

slide – formula & details)

4. Conclusions• Since the test value 2.771 is

greater than the critical value 1.315, we reject the null hypothesis.

• “There is sufficient evidence to support the claim that the average student gets more than 9 parking tickets per semester.”

(Added content by D.R.S.)

2.771

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Draw a conclusion:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

A hometown department store has chosen its marketing strategies for many years under the assumption that the average shopper spends no more than $100 in their store. A newly hired store manager claims that the current average is higher, and wants to change their marketing scheme accordingly. A group of 24 shoppers is chosen at random and found to have spent on average $104.93 with a standard deviation of $9.07. Test the store manager’s claim at the 0.010 level of significance.

Solution:

First state the hypotheses:H0:Ha:

Next, set up the hypothesis test and determine the critical value: d.f. = 23, a = 0.010t0.010 =Reject if t ≥ t , or if t > 2.500.

m ≤ 100m > 100

2.500

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Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

Solution (continued):

Gather the data and calculate the necessary sample statistics:n = 24, = 100, = 104.93, s = 9.07,

Finally, draw a conclusion:Since t is greater than t a , we will reject the null hypothesis.

2.663

Added content

• Repeating several of the slides with extra comments about TI-84

• Also an important reminder: using this method for small sample sizes requires that the population being studied is NORMALLY DISTRIBUTED. Not uniform, not skewed, but a bell curve distribution is assumed. (This book somewhat glosses over this point.

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Find the critical value:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

Find the critical t-score for a right-tailed test that has 14 degrees of freedom at the 0.025 level of significance.

Solution:d.f. = 14 and a = 0.025t0.025 = 2.145

The critical values for n < 30 are found from the t-distribution.

invT(area to left, d.f.) = t valuePlus or Minus Sign? Either by symmetry or by adjusting the area value for a right-tailed test.You still have to understand whether it’s left-tailed, right-tailed, or two-tailed. The calculator won’t do that for you !

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Draw a conclusion:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

27 college students are asked how many parking tickets they get a semester to test the claim that the average student gets more than 9 tickets a semester. The sample mean for the number of tickets was found to be 9.8 with a standard deviation of 1.5. Use a = 0.10.

Solution:

n = 27, = 9, = 9.8, s = 1.5, d.f. = 26, a = 0.10t0.10 = 1.315

Again, fix up the sign by knowing that it’s a right-tailed test, therefore positive critical value. The calculator will not do this thinking for you.

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Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

(continued from previous slide)

Solution:

n = 27, = 9, = 9.8, s = 1.5, d.f. = 26, a = 0.10

t0.10 =

Since t is greater than t a , we will reject the null hypothesis.

1.315

2.771

EXTRA ( ) around complicated numerators and denominators !!!

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Draw a conclusion:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

A hometown department store has chosen its marketing strategies for many years under the assumption that the average shopper spends no more than $100 in their store. A newly hired store manager claims that the current average is higher, and wants to change their marketing scheme accordingly. A group of 24 shoppers is chosen at random and found to have spent on average $104.93 with a standard deviation of $9.07. Test the store manager’s claim at the 0.010 level of significance.

Solution:

First state the hypotheses:H0:Ha:

Next, set up the hypothesis test and determine the critical value: d.f. = 23, a = 0.010t0.010 =Reject if t ≥ t , or if t > 2.500.

m ≤ 100m > 100

2.500

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Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

Solution (continued):

Gather the data and calculate the necessary sample statistics:n = 24, = 100, = 104.93, s = 9.07,

Finally, draw a conclusion:Since t is greater than t a , we will reject the null hypothesis.

2.663

TI-84 T-Test

• The TI-84 has a built in Hypothesis Testing tool• STAT menu, TESTS submenu, 2:T-Test• You must understand how to do hypothesis

testing with charts and formulas, however. The calculator is not a substitute for that. Mere button smashing will lead you to failure.

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Example:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

27 college students are asked how many parking tickets they get a semester to test the claim that the average student gets more than 9 tickets a semester. The sample mean for the number of tickets was found to be 9.8 with a standard deviation of 1.5. Use a = 0.10.

Solution: Choose “Data” if the 27 data values were in TI-84 Lists,Stats if we have summary statistics already calculated

Null hypothesis’s mean

Sample’sMean, Standard deviation, and Size Direction of the

Alternative Hypothesis

Highlight “Calculate” and press ENTER

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Example, continued:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples). . . . Use a = 0.10.

Verify that it did theTest you wanted and that it has the correct Alternative Hypothesis. Verify that the sample data is correct.

The t= is the Test Statistic. It comes from the same formula as the one we’ve been using.

The p = is the p-value. It is the area to the right of that t value (in the case of this right-tailed test.) It is the probability of getting a t value as extreme as the t value we got.

When using the calculator’s T-Test, we use the “p-value method”. You don’t need a t critical value. Instead, you compare your p-value to the α (alpha) level of significance. If your p < α(alpha), thenthe decision is “Reject H0”.

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The other example, done with TI-84 T-Test and the p-value method:

Hypothesis Testing

10.2 Hypothesis Testing for Means

(Small Samples)

A hometown department store has chosen its marketing strategies for many years under the assumption that the average shopper spends no more than $100 in their store. A newly hired store manager claims that the current average is higher, and wants to change their marketing scheme accordingly. A group of 24 shoppers is chosen at random and found to have spent on average $104.93 with a standard deviation of $9.07. Test the store manager’s claim at the 0.010 level of significance.

H0:Ha:

m ≤ 100

m > 100

Compare your p-value p=.0069501788 to alpha: α=0.010and make the decision: Should we reject H0?