statistics and quantitative analysis u4320
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Statistics and Quantitative Analysis U4320. Segment 7 : Hypothesis Testing Prof. Sharyn O’Halloran. Hypothesis Testing. I. Introduction A. Review of Confidence Intervals. Introduction (cont.). B. Hypothesis Testing: Basic Definitions - PowerPoint PPT PresentationTRANSCRIPT
Statistics and Quantitative Analysis U4320
Segment 7 :Hypothesis Testing
Prof. Sharyn O’Halloran
Hypothesis Testing I. Introduction
A. Review of Confidence Intervals
SE
-1.96*SE 1.96*SE
-1.96*SE 1.96*SEX
Introduction (cont.)
B. Hypothesis Testing: Basic Definitions 1. A Hypotheses is a statement about the
population 2. Null Hypothesis
The Null Hypothesis (Ho)- the statement about our data that we want to test.
It is always stated as an equality. For instance; Ho: = 82, where is the average test score Or, H0: = 0, where is the difference
between men's and women' salaries is zero.
Introduction (cont.)
3. Alternative Hypothesis Every Null Hypothesis has an associated
Alternative Hypothesis, denoted Ha. This is always stated as an inequality either ,
>, or <. For instances, the alternative hypothesis
to the test scores having a mean of 82 might be Ha: 82.
The alternative hypothesis to men's and women's' salaries being equal might be Ha: > 0.
Introduction (cont.)
4. One Tail vs. Two Tail Tests If the alternative hypothesis is in terms of a
sign, it is called a two-tailed test. If the alternative hypothesis is in terms of a <
or > sign, it is called a one-tailed test.
Introduction (cont.)
C. Three Methods for Testing Hypothesis
1. Method I: Testing hypotheses using confidence intervals.
2. Method II: Testing hypotheses using p-values.
3. Method III: Testing hypotheses using critical values.
Hypothesis Testing Using Confidence Intervals II. Method I: Hypothesis Testing
Using Confidence Intervals Note: This method works only for two-tail tests
H0: = 0
Ha: 0
Hypothesis Testing Using Confidence Intervals (cont.)
A. Example: Differences in Means In a large university, 10 male professors
and 5 female professors were randomly sampled. Their salaries were:
Men (X1) Women (X2)
13 20 9 11 14 12 19 17 8 15 14 10 22 15 16
X1 = 16 X2 = 11
Hypothesis Testing Using Confidence Intervals (cont.)
1. Step 1: Define Hypothesis We are interested in the difference between
the means of men's and women's salaries. Call this difference = (1-2),
The males state that = 0, The females say that = 7,
Do the data support both of these hypotheses, one of them, or neither?
We will test these hypotheses at the 5 % -level.
H0: = 0, Ha: 0.
H0: = 7, Ha: 7.
Hypothesis Testing Using Confidence Intervals (cont.)
2. Step 2: Calculate a Confidence Interval Form a 95% confidence interval:
Notice that our data are two samples, one of men and other of women, from the same larger population of university professors. So we can pool our sample variances.
= (X1-X2) t.025 * sp * 11
12nn
X1 = 16
X2 = 11
n1 = 10
n2 = 5
sX X X X
n np2 1 1
22 2
2
1 21 1
( ) ( )
( ) ( ) = 146 / [(10-1) + (5-1)]= 11.23
Hypothesis Testing Using Confidence Intervals (cont.)
(cont.)
So the 95% confidence interval is from 1 to 9 thousand dollars.
sp = 146/13 = 3.35
SE = 3.35*11015 = 3.35 * .548= 1.84
d.f. = 13
t.025 = 2.16 (look in the t-tables)
= 5 2.16 * 1.84
= 5 4.0 1 < (1-2) < 9
Hypothesis Testing Using Confidence Intervals (cont.)
3. Step 3: Accept or Reject the Hypothesis According to these data, is the claim that = 0
plausible?
We must reject the hypothesis that = 0 because it falls outside the 95% confidence interval
What about the hypothesis that = 7?
SE=1.84
1
9
-t.025
*SE t.025
*SE
5
reject
reject
We cannot reject the null hypothesis H0: = 7 at the 5% level.
Hypothesis Testing Using Confidence Intervals (cont.)
4. Summary: Step by Step Procedure
1. Step 1: Define Hypothesis Pick a significance level; the usual one is 5%.
2. Step 2: Construct confidence interval
Formula depends on type of data, (matched or pooled variance) and how confident you want to be.
3. Step 3: Accept or Reject If falls within this interval, then we fail to
reject the null, otherwise we reject it.
Define the null hypothesis H0: = 0.
Define the alternative hypothesis Ha: 0.
0
Hypothesis Testing Using Confidence Intervals (cont.)
B. Another Example: Matched Data A firm producing plate glass has developed a
less expensive tempering process to allow glass for fireplaces to rise to a higher temperature without breaking. To test it, five different plates of glass were drawn randomly from a production run, then cut in half, with one half tempered by the new process and one half tempered by the old. The two halves were then heated until they broke. The results of the experiment look like this: (next slide)
Hypothesis Testing Using Confidence Intervals (cont.)
Matched Data (cont.)
We want to test the hypothesis that the two processes are equal at the 95% confidence level or at the = .05 significance level.
New Old D 485 475 10 438 436 2 493 495 -2 486 483 3 433 426 7
DD
ni
i
n
1
4
Hypothesis Testing Using Confidence Intervals (cont.)
1. Step 1: Define Hypothesis H0: = 0;
Ha: 0;
Significance level = 5%. 2. Step 2: Calculate a 95% Confidence
interval. (unknown) D = 4 s2D= (D-D)2/(n-1) = (10-4)2 + (2-4)2 + (-2-4)2 + (3-4)2 +(7-4)2 / (5-1)
s2D= 21.5 sD = 4.64
d.f. = 4 t.025 = 2.78
Hypothesis Testing Using Confidence Intervals (cont.)
Step 2 (cont.)
SE = SnD464
5208
..
= D t.025 * SE 4 2.78*2.08
= 4 5.76 -1.76 to 9.77 -1.76 < (1-2) < 9.768
Hypothesis Testing Using Confidence Intervals (cont.)
3. Step 3: Accept or reject null hypothesis?
So we do not reject the hypothesis that H0: = 0 because 0 falls within that range. The two processes are seen as indistinguishable.
-1.76 9.77D=4
p-Values III. Method II: p-Values
P-values are essentially the significance level.
In essence, we are calculating the probability that the hypothesis is true. It summarizes the credibility of the null hypothesis.
p-Values A. known
1. Step 1: State the Hypothesis A manufacturing process produces TV. tubes
with an average life=1200 hours and = 300 hours. A new process is thought to give tubes a higher average life. And out of a sample of 100 tubes we find that they have an average life = 1265 hours. Is the new process really any better than the old?
X
p-Values Step 1 (cont.)
H0: = 1200
Ha: > 1200
= .05 or 5% significance-level
This is a one-tailed test because we have put all the area in one-tail of the distribution. We are interested in those values that are greater than the mean.
Reject Region
p-Values 2. Step 2: Calculate p-value
We know and n is large so we can use the normal distribution.
0 = 1200, and = 300 and n= 100 Standard error = /n = 300/ 100 = 30. The observed value = 1265.
a. Standardize We then standardize (get the z-value )
X
Z = (X- 0) / (/n) Z = 1265-1200 / 30 = 2.17
p-Values b. Find z-score (probability of the event
occurring) Pr (X 1265) = Pr(Z 2.17) = .015 (from the z-table)
Reject Region
X=1265
area=1.5%
area=5%
p-Values 3. Step 3: Accept or Reject the
Hypothesis This suggests that if the null hypothesis was
true that there would be only a 1.5% probability of observing as larger as 1265.
Since 1.5% lies to the right of our initial 5% significance level, we can reject the null hypothesis.
X
p-Values 4. Two-Tailed Test
H0: = 1200
Ha: 1200
= .05 or 5% significance-level
Reject Region
X=1265
area=1.5%
area=2.5%
Reject Regionarea=2.5%
p-Values Accept or Reject
Since the area to the right of 1265 is only 1.5%, we can again reject H0.
p-Values B. unknown
Usually is unknown and has to be estimated with the sample standard deviation s. The test statistic is then t instead of Z.
tXs n
Xestimated SE
/.
t = estimate - null hypothesis estimated SE
p-Values 1. Step 1: State Hypothesis (e.g.,
difference in men's and women's salaries)
We know from the above example, ( - ) = 5 Standard Error = 1.84
Is this a one or a two tailed test?
H0: = 0; Ha: > 0 ; at = 5%.
X 1 X 2
Reject Regionarea=5%
p-Values 2. Step 2: Calculate p-value
a. Standardize
t = estimate-null = 5.0 - 0 = 2.72 SE 1.84
p-Values b. Find probability of event from t-
table Degrees of freedom = (n-1) = 13 So the probability of observing a t-value of
2.72 lies beyond This means that the tail probability is
smaller than .01. That is, p-value < .01.
t.01=2.65.
Reject Regionarea=5%
2.721.77
p-Values 3. Step 3: Accept or Reject Hypothesis
Since the p-value is a measure of the credibility of H0, such a low value (below = 5%) leads us to conclude that H0 is implausible.
Therefore, we reject the null hypothesis.
p-Values C. Getting t-values from Computers
(Review of Homework) 1. Calculate t-values
How does the computer calculate the t-value?
The t-value = (X-)sn
0
p-Values 2. Calculate p-value
The 2-tail probability gives the area to the right of the t-value times two.
If this value is less than your significance level for a 2-tail test, then reject your null hypothesis.
p-Values 3. Example: Sample Homework
For example, the difference of means test between men and women's incomes, produced a t-value = 6.60 and an associated p-value of .00.
Therefore, I can reject the hypothesis that 1-2 = 0 because .00 is less than .025.
Reject Regionarea=2.5%
6.601.96-1.96
area= .000
p-Values D. Summary
1. Step 1: Define Hypothesis Choose H0, Ha and a significance level
(default is 5%). 2. Step 2: Calculate p-value
Calculate your p-value from the statistics if known
if is unknown
ZX
nXexact SE
/.
tXs n
Xestimated SE
/.
p-Values 3. Step 3: Accept or Reject hypothesis
Reject H0 if p-value For a One-Tailed Test
Reject H0 if the p-value is less than the significance level .
Accept H0 otherwise. For a Two-tailed Test
Reject H0 if the p-value is less than 1/2 the significance level. (i.e., 1/2 = .025)
Accept H0 otherwise.
Critical Values IV. Method III: Critical Values
Classical hypothesis testing is very similar to the p-value approach.
A. Example: Manufacturing of TV tubes
1. State the Hypothesis: H0: = 1200 n=100
Ha: > 1200 0=1200
= 5%. =300
Critical Values 2. Test Hypothesis: Find the Critical
Values A. In General
What z-value is associated with 5% of the area under the curve?
From the z-tables we see that the area of 5% is associated with a z-value of 1.64.
The question is what value on the x-axis corresponds to a z-value of 1.64?
Reject Regionarea=5%
z=1.64
Critical Values B. Critical Value
The critical value is the X-value that corresponds to a Z-value.
We obtain the critical value by arbitrarily setting = 5% and calculating:
C. Calculating the Critical Value for Manufacturing TV Tubes
We know that the 0=1200, and SE=300/100=30.
The Critical Value then is:
Xc = 0 + Z.05*SE
Xc = 1200 + 1.64*30 = 1249.
Critical Values 3. Step 3: Reject or Accept the Hypothesis
To accept or reject our hypothesis we collect data and see if our sample mean is greater then this critical value.
From the above example we observed a sample mean = 1265.
Therefore we reject H0: =1200 because 1265>1249.
So we once again conclude that the new process is better than the old.
X
Reject Regionarea=5%
z=1.64X = 1249
c
X=1265 (observed)
Critical Values B. Example of 2-tailed test
How do we construct a two-tailed test at the 5% significance value?
1. Step 1: State Hypothesis H0: = 1200Ha: 1200
= 5%.
Critical Values 2. Step 2: Calculate Critical Value
We use Z.025 instead of Z.05. In this case, we would get c = 0 Z.025*SE.
c = 1200 1.96*30 = 1141 and 1259.
XX
Critical Values 3 Step 3: Accept or reject null Hypothesis
We would reject H0 if the observed fell below 1141 or above 1259.
Again 1265 exceeds the critical value so we still reject H0.
Reject Regionarea=2.5%
z=1.96X = 1259
c
X=1265 (observed)z=-1.96
X = 1141c
Critical Values C. Summary:
1. Step 1: Define Hypothesis State H0; State Ha; and Choose a significance level .
Critical Values 2. Step 2: Calculate Critical Value
Draw a normal curve and find the critical values at the level of significance you arbitrarily set. Usually at the .05 significance-level.
For two-tailed test: known: c = 0 Z.025*SE. unknown: c = 0 + t.025*SE(estimated)
For one-tailed test: known: c = 0 + Z.05*SE. unknown: c = 0 + t.05*SE(estimated)
Critical Values 3. Step 3: Accept or Reject
Then collect sample data. If the sample mean exceeds the critical value,
then reject H0; otherwise accept H0.
Notes About the Exam V. Notes About the Exam
1. Hand in your homework at the beginning of class
2. The exam will cover the material through today's lecture.
3. Problems, no definitions. 4. You may bring a calculator and one 3 X 5
index card with whatever you want written on it.
5. Z-tables and t-tables will be supplied.
Review Session
Review Session: Saturday March 8 11 to 1 PM
Room 411 IAB