Post on 06-Jan-2016
description
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%)
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 95 % Confidence Interval of the Mean
8
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 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.
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
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.
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
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
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
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.
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
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
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
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