Download Lecture 14 - Statistics

Statistics 400 - Lecture 14
 Today: Chapter 8.5
 Assign #5: 8.62, 8.70, 8.78, and
 In recent years, there has been growing concern about the health effects of video display
terminals (VDTs). It is known that the miscarriage rate of a pregnancy under normal
conditions is about 20%. In a survey of 15 part-time female employees who used VDTs at
least 20 hours per week, there were 5 reported miscarriages. At the 1% significance level,
is there evidence that exposure to VDTs is increasing the miscarriage rate?
 Due next Thursday (in 9 days)
 For testing:
H 0 :   0 vs. H1 :   0
 If the test reject the null hypothesis, then
 If the null hypothesis is not rejected,
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
 H0 : 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
Motivation for Large Sample Inference
for Proportions
 Can express sample proportion a sample mean
 Sampling distribution of sample proportion is approximately normal
(why?)
 What assumptions are we making?
Sample Size for a Desired Margin of Error
 Prior to sampling, one should have an idea of the required precision
for the experiment
 The margin of error for the 100(1   )% confidence interval is
 The required sample size is
 Since we do not have p or q, we substitute
Small Sample Hypotheses for Proportions
 Do not always have enough resources to take a large sample
 What is the sampling distributions for the number of successes in n
Bernoulli trials
Situation:
 Want to make inferences for the population proportion (probability
of a “success”), p
 Select a random sample of size n
 Record number of successes, X
 Distribution of X is:
 H 0 : p  p0 versus H 0 : p  p0
 If H0 is true, what is distribution of X
 P-value =
 H 0 : p  p0 versus H 0 : p  p0
 If H0 is true, what is distribution of X
 P-value =
Example:
 Group of 10 subjects with certain disease are given a new
treatment
 8 subjects showed improvement
 Test the claim: majority of disease sufferers using this treatment
show improvement with a 5% significance level