Transcript Lecture 12 - Statistics

Statistics 400 - Lecture 12
 Today: Finish 8.4; begin Chapter 9
 Mid-Term Next Thursday
 Review Next Tuesday
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
Relationships Between Tests and CI’s
 Confidence interval gives a plausible range of values for a
population parameter based on the sample data
 Hypothesis Test assesses whether data gives evidence that a
hypothesized value of the population parameter is plausible or
implausible
 Seem to be doing something similar
 For testing:
H 0 :   0 vs. H1 :   0
 If the test reject the null hypothesis, then
 If the null hypothesis is not rejected,
Example (3.96)
 Based on a random sample of size 18 from a normal population, an
investigator computes a 95% confidence interval for the mean and
gets [27.1, 39.3]
 What is the conclusion of the t-test at the 5% level for:

H 0 :   29 vs. H1 :   29

H 0 :   26.8 vs. H1 :   26.8
 Suppose we reject the second null hypothesis at the 5% level
 Another experimenter wishes to perform the test at the 10%
level…would they reject the null hypothesis
 Another experimenter wishes to perform the test at the 1%
level…would they reject the null hypothesis
 What does changing the significance level do to the range of values
for which we would reject the null hypothesis
Large Sample Inferences for Proportions
Example:
 Consider 2 court cases:
 Company hires 40 women in last 100 hires
 Company hires 400 women in last 1000 hires
 Is there evidence of discrimination?
 Can view hiring process as a Bernoulli distribution:
 Want to test:
Situation:
 Want to estimate the population proportion (probability of a
“success”), p
 Select a random sample of size n
 Record number of successes, X
 Estimate of the sample proportion is:
 If n is large, what is distribution of p̂
 Can use this distribution to test hypotheses about proportions
Large Sample Hypothesis Test for the
Population Proportion
 Have a random sample of size n
 H 0 : p  p0
 pˆ 
X
n
 Test Statistic:
Z
pˆ  p0
p0 q0 / n
 P-value depends on the alternative hypothesis:
H1 : p  p0 : p - value  P(Z  z)
H1 : p  p0 : p - value  P(Z  z)
H1 : p  p0 : p - value  2P(Z  | z |)
 Where Z represents the standard normal distribution
 What assumptions must we make when doing large sample
hypotheses tests about proportions?
 Example revisited:
Large Sample Confidence Intervals for
the Population Proportion
 Large sample confidence interval for a population proportion:
Example
 For both court cases, find a 95% confidence interval for the
probability that the company hires a woman