Confidence Interval of the Mean , Independent-, and Paired-Samples T-tests

CONFIDENCE INTERVAL OF THE MEAN , INDEPENDENT-, AND PAIRED-SAMPLES T-TESTS

CONFIDENCE INTERVAL OF THE MEAN

xtailed stcritx 2

We often use limited observations (samples) to talk about or estimate the population values from which they come.

For example: My driver friend wants to know if he should take the F train; approximately how frequently does this train arrive?

If I tell him approximately 8 minutes, how good is this estimate? How justified am I in using my SAMPLE mean here?

5 min

12 min

7 min

2 min

6 min

16 min

10.5

8

s

x

Confidence Interval of the Mean (95% or 99%)


xtailed stcritx 2

5 min

12 min

7 min

2 min

6 min

16 min

10.5

8

s

x 95 % Confidence Interval of the Mean

8


xtailed stcritx 2

5 min

12 min

7 min

2 min

6 min

16 min

10.5

8

s

x 95 % Confidence Interval of the Mean

8 + 2.57(5.10/√6) = 13.358 - 2.57(5.10/√6) = 2.65

There is a 95% probability that the TRUE population mean is between 2.65 and 13.35 minutes.


xtailed stcritx 2

5 min

12 min

7 min

2 min

6 min

16 min

10.5

8

s

x 99% Confidence Interval of the Mean


xtailed stcritx 2

5 min

12 min

7 min

2 min

6 min

16 min

10.5

8

s

x 99% Confidence Interval of the Mean

8 + 4.03(5.10/√6) = 16.398 - 4.03(5.10/√6) = -.39

There is a 99% probability that the TRUE population mean is between -.39 and 16.39 minutes.

xtailed stcritx 2

Confidence Interval of the Mean

sample mean

t critical value(look up in table)If 95% CI, use a=.05If 99% CI, use a=.01

Remember, df = N-1

Standard error

N

ssx

Paired-Samples T-test

Experimental design: One group, experiencing both treatments.

D

obt s

Dt

Seven people are recruited to test Proactiv acne treatment. Each person’s face is examined by a dermatologist who reports the number of pimples on each person’s face. Individuals are then instructed to use the Proactiv system of products for 3 months, after which they return to have their face pimples counted again. Test the hypothesis that Proactiv produces a difference in pimple number, using an alpha level of .05.

Before After

5 7

6 6

7 9

5 5

6 6

7 9

5 5

H0: Proactiv does not produce a difference in pimple number.H1: Proactiv produces a difference in pimple number.

Two-tailed, alpha .05, df = 6

tcrit = -2.45 and +2.45

Step 1: State the null and alternative hypotheses:

Step 2: Find the critical value.


Before After

5 7

6 6

7 9

5 5

6 6

7 9

5 5

D

-2

0

-2

0

0

-2

0

Step 3: Calculate the obtained statistic:

D

obt s

Dt =

-.88____

)1(

NN

SSs DD

)17(7

86.6

Ds

40.Ds


Before After

5 7

6 6

7 9

5 5

6 6

7 9

5 5

D

-2

0

-2

0

0

-2

0

Step 3: Calculate the obtained statistic:

D

obt s

Dt =

-.88____.40

= - 2.15

Step 4: Make a decision.

-2.45I

2.45I

Retain the null hypothesis.

Independent-Samples T-test

21

21

xxobt s

xxt

Experimental design: Two separate groups, each experiencing a

different treatment.

A food writer would like to review the pricing of cocktails in big cities. She is looking specifically to compare the price of cocktails in Boston and New York to examine whether or not the average cocktail price is different. She goes to 7 bars in Boston and 7 bars in New York, recording the price of each bar’s Cosmopolitan. Below is the data. Test the hypothesis that Boston and New York charge significantly different prices for cocktails using an alpha level of .05.

Boston NY

5 7

6 6

7 9

5 5

6 6

7 9

5 5

H0: Boston and NY do not charge different prices for cocktails.H1: Boston and NY do charge different prices for cocktails.

For an independent-groups t-test, we use df = N-2

tcrit = -2.18 and +2.18Alpha = .05, 2-tailed, df = 12

Step 1: State the null and alternative hypotheses:

Step 2: Find the critical value.


Boston NY

5 7

6 6

7 9

5 5

6 6

7 9

5 5

Step 3: Calculate the obtained statistic

21

21

xxobt s

xxt

x 5.86 6.71SS 4.86 17.43

= 5.86 – 6.71 __________

2121

2121

11

2 nnnn

SSSSs xx

7

1

7

1

277

43.1786.421 xxs

7

2

12

29.2221 xxs

29.86.121 xxs

54.21 xxs

73.21 xxs


Boston NY

5 7

6 6

7 9

5 5

6 6

7 9

5 5

Step 3: Calculate the obtained statistic

Step 4: Make a decision

21

21

xxobt s

xxt

xSS

5.86 6.714.86 17.43

= 5.86 – 6.71 __________

-2.18I

2.18I

.73= -1.16

Retain the null hypothesis.

So far we have learned how to do five types of hypothesis tests:

Test Statistic df Used when comparing: Note:

Sign test Outcome (given in question)

N/A Comparing outcome to binomial distribution

Coin flip;Number of pluses/minuses

z-test z N/Ato µ when σ is known

Comparing sample mean to pop mean when pop sd

known

One sample t-test

t N-1 to µ when σ is unknown

Comparing sample mean to pop mean when pop sd is not

known

Paired-Samples t-

test

t N-1to

Comparing two sample means when they are from the same group of people

Independent-samples

t-test

t N-2to

Comparing two sample means when they come from

different groups of people

x

x

1x 2x

1x 2x

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Confidence Interval of the Mean , Independent-, and Paired-Samples T-tests