# Lecture Note 17 ```ST 361 Ch8.2 Testing Hypotheses about Means
(Part I: Testing for a Population Mean  )
Topics: Hypothesis testing with population means
► One-sample problem: Testing for a population mean 
1. Assume population SD is known: use a z test statistic
2. Assume population SD is not known: use a t test statistic
► A Special Case: the Paired t test
► Two-sample problem: : Testing for 2 population means  1 and 2 
-------------------------------------------------------------------------------------------------------------------► Steps for Testing for a Population mean  -----
Need x ~Normal !!!
Step 1. Specify H 0 and H a
H 0 :    0 vs. H a :   0 (this is referred to as _______________________)
H a :   0 (this is referred to as _______________________)
H a :    0 (this is referred to as _______________________)
Step 2. Determine the test level  (also called significance level)
Step 3. Compute the test statistic
A test statistic is defined as_________________________________________________
When the population SD  is known, a test statistic is ________________________
When the population SD  is NOT known, a test statistic is _____________________
Step 4. Calculate the p-value
Step 5. Draw conclusions
If p-value &lt;  __________________ and draw conclusion based on _______
Otherwise we _________________________and draw conclusion based on ______
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► One-sample problem: Testing for a population mean  assume the population SD is known
A Working Example: (adapted from 8.14 p.355 of the textbook) Light bulbs of a certain type are
advertised as having an average lifetime of 750 hours. The price of these bulbs is very favorable,
so a potential customer has decided to go ahead with a purchase arrangement unless the true
average lifetime is smaller than what is advertised. A random sample of 50 bulbs was selected.
The sample data and result are presented below: (Assume the population SD of the bulbs lifetime
is 38.2.) What conclusion would be appropriate for a significance level of 0.05?
Variable
n
50
Z
p-value
Mean X
SE of
Mean  X
738.44
5.4
-2.14
0.016
Sample
Step 1: parameter of interest =
H0 :
Ha :
Step 2: significance level  =
Step 3: test statistic =
Step 4: p-value =
Step 5: Conclusion:
► The p-value
 Rationale:
The p-value quantifies how the null hypothesis ( H 0 ) is supported by the observed data,
i.e.,
 The smaller the p-value is, the more contradictory is the data to H 0 .
 To get p-values, one begins with assuming H 0 is true (e.g., in the working example, it
means ______________________________________________________). We then
calculate that when H o is true, how likely is it to observe a sample with X = 738.44?
 i.e., want to calculate P( X  738.444 | H 0 is true (i.e., μ  750) ) = ___________
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 To resolve this problem, one will instead calculate
the probability of observing X = 738.44 or the
more extreme cases under H 0 . Note that the
“more extreme cases” are determined by the
_____________hypothesis. Here H a :  ___ 750 ,
so we calculate
Such probability is called “p-value”
 Calculation of p-value (If the population SD is known….)
p-value = P( X  738.444 | μ  750 )
► In general, the calculation of p-value can be simplified in the following steps:
Let X * =
First calculate the test statistic z 
*
X *  0
X

.
Then, the p-value = ______________
 Interpretation of p-value:
(1) If the p-value, P( X  738.444 | μ  750 ), is very small (i.e., _______________), it
implies that ________________________________________________________
_______________________________________. In other words, the sample data
______________ support H0.
 Reject / Do not reject (pick one) H0.
(2) If the p-value, P( X  738.444 | μ  750 ), is not very small (i.e., _____________), it implies
that the chance of observing sample with Y =738.44 or smaller when  = 750 is
___________________ In other words, the sample data and H0
___________________________________________
 Reject / Do not reject (pick one) H0.
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 In summary,
1. The p-value describes the probability of seeing your data or more extreme IF the null hypothesis is
true. Recall that the more extreme cases are determined by_______.
If H a :    0 , p-value =
=________________
If H a :    0 , p-value =
=________________
If H a :    0 , p-value =
= ________________
2. To get p-value, all we need is to calculate the test statistic z 
*
X *  0
X

X *  0

n
(where X * is the observed sample mean) and then find the corresponding p-value by
P( Z  z* ), or P( Z  z* ), or 2  P( Z   z * ), depending on the alternative hypothesis.
3. To draw conclusion, compare p-value with the test level  :
Reject H 0 if p-value___________, and conclude based on H a .
Otherwise we do not reject H 0 , and conclude the test based on H 0 .
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Ex1. Let X = the scores on the Verbal SAT exam this year. The score X varies according to a normal
distribution with mean  and variance 80. A sample of 64 students was collected, and x  580. Records
showed that the mean SAT score of two years ago is 570. Based on the data, has the average SAT score
increased over the two years? Perform a 0.05-level of test.
Step 1: parameter of interest =
H0 :
Ha
Step 2: significance level  =
Step 3: test statistic =
Step 4: p-value =
Step 5: Conclusion:
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Ex2. Consider the true mean stopping distances at 50 mph for cars equipped with the braking system of
brand A. It is known that the average stopping distances for braking system B is 120 inches. Result
based on 36 cars have mean 115. Assume the population SD of the stopping distance is 20. Do the
stopping distances of the two systems differ? Perform a 0.01-level of test.
Step 1: parameter of interest =
H0 :
Ha
Step 2: significance level  =
Step 3: test statistic =
Step 4: p-value =
Step 5: Conclusion:
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► One-sample problem: Testing for a Population mean  when  is unknown
If the population SD  is unknown, the testing procedure is the same as what we do when the
population SD  known, except that
1. the __________________ is used
2. (as a result of 1,) the test statistic is a ___________________ , which is a _________instead of a z
statistic

Need to know how to use Table VI to find p-values
EX. (From textbook Question 8.17)
1. Upper-tailed test, df=8, t=2.0
2. Lower-tailed test, df=11, t= -2.4
3. Two-tailed test, df=15, t= -1.6
4. Two tailed test, df=40, t=4.8
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Ex1. Life of electric bulb: Industrial standard for the bulb life is 6000 hours. A company claims that their
bulbs are better than the industrial standard. To test their claim, a sample of 16 light bulbs was
collected and has mean 6.5 (unit=1000 hours) and SD 1 (unit = 1000 hours). (a) Perform a test at
5% level. (b) What assumption do we need to conduct a hypothesis?
(a) Step 1: parameter of interest =
H0 :
Ha
Step 2: significance level  =
Step 3: test statistic:
Step 4: p-value:
Step 5: Conclusion:
(b) Assumption needed:
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Ex2. A certain pen has been designed so that the true average writing lifetime is 10 hours. A random
sample of 18 pens is selected and the writing lifetime of each is determined: the mean lifetime of
the 18 pens is 10.5 hours with SD=1.2 hours. Perform a 0.01 level of test to examine if the design
specification has been satisfied.
(a) Step 1: parameter of interest =
H0 :
Ha
Step 2: significance level  =
Step 3: test statistic:
Step 4: p-value:
Step 5: Conclusion:
(b) Assumption needed: (select any that apply)
_______ The sample mean lifetime follows a normal distribution
_______ The lifetime follows a normal distribution
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