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Statistics 400 - Lecture 11
 Today: Finish 8.4; begin Chapter 9
 Assignment #4: 8.54, 8.103, 8.104, 9.20, 9.30
 Not to be handed in
 Next week…Case Studies and Review
 No Lab Friday and Next Tuesday
Example
 A company wishes to monitor customer service
 A senior manager claim that the average waiting time is 3 minutes
 The waiting times of 75 randomly selected calls to a customer service hotline were recorded
 The calls had a sample mean of 3.4 minutes and sample standard deviation
of 2.4 minutes
 Test this claim with a significance level of 0.05
Types of Errors
 What is the probability of rejecting H0 when it is indeed true?
 If H0 is true, how often do we make the right decision?
 Have only considered probability of rejecting when H0 is true
 Suppose H0 is not true. Ideally, what happens?
 Have 2 types of error:
 Type I error: Reject H0 when H0 is true
 Type II error: Fail to Reject H0 when H0 is false
 The probability of a type I error is:
 Probability of type II error is:
 Power of a Test is:
Small Sample Inference for Normal Populations
 Do not always have large samples
 When is it reasonable to use the Z -test statistic with an unknown
standard deviation?
X 
S/ n
 This statistic has a different sampling distribution called the
Student’s t-distribution
Student’s t-Distribution
 Have a random sample of size n from a normal population
 The distribution of the sample mean is:
 Distribution of Z is:
 When population standard deviation is unknown, it is estimated by
the sample standard deviation
 Will use
T
X 
as our test statistic
S/ n
Student’s t-Distribution
 If x1, x2, …, xn is a random sample from a normal population with
mean  , and standard deviation  , then
t
X 
S/ n
has a t-distribution with (n-1) degrees of freedom
 t-distributions are:
 Bell shaped
 Symmetric
 They are not exactly like normal distributions. Why?
Probabilities for the t-distribution
 How many different t-distributions are there?
 Table 4 can be used for computing probabilities for various
t-distributions
 The probability being computed is a “greater than” probability
Example
 Suppose T has a Student’s t-distribution with 10 degrees of
freedom
 P(T>2.228)=
 P(T<2.764)=
 P(T>0)=
 P(T>2.5)=
 What is the 90th percentile of this distribution?
 What is the 10th percentile of this distribution?
Small Sample Confidence Interval for the
Population Mean
 If x1, x2, …, xn is a random sample from a normal population with
mean  , and standard deviation  , then a 100(1   )% confidence
interval for the population mean is:
S 

 X  t / 2 n 
 If you have  use a
distribution instead!
Example:
 Heights of males are believed to be normally distributed
 Random sample of 25 adult males is taken and the sample mean &
standard deviation are 69.72 and 4.15 inches respectively
 Find a 95% confidence interval for the mean
Small Sample Hypothesis Test for the
Population Mean
 Have a random sample of size n ; x1, x2, …, xn

H 0 :   0
 Test Statistic:
t
X 
S/ n
Small Sample Hypothesis Test for the
Population Mean (cont.)
 P-value depends on the alternative hypothesis:
H1 : 
 0 : p - value  P(T  t )
H1 : 
 0 : p - value  P(T  t )
H1 : 
 0 : p - value  2P(T  | t |)
 Where T represents the t-distribution with (n-1 ) degrees of
freedom
Example:
 An ice-cream company claims its product contains 500 calories per
pint on average
 To test this claim, 24 of the company’s one-pint containers were
randomly selected and the calories per pint measured
 The sample mean and standard deviation were found to be 507 and
21 calories
 At the 0.01 level of significance, test the company’s claim
 What assumptions do we make when using a t-test?
 How can we check assumptions?
 Can use t procedures even when population distribution is not
normal. Why?
Practical Guidelines for t-Tests
 n<15: Use t procedures if the data are normal or close to normal
 n<15: If the data are non-normal or outliers are present DO NOT
use t procedures
 n>15: t procedures can be used except in the presence of outliers
or strong skewness
 t>30: t procedures tend to perform well