Today

• Today: Chapter 9

• Assignment: 9.2, 9.4, 9.42 (Geo(p)=“geometric distribution”), 9-R9(a,b)

• Recommended Questions: 9.1, 9.8, 9.20, 9.23, 9.25

Example

• Suppose X=(X1, X2,…,Xn) represents random sample from a population

• Suppose the population has pdf

• Find the MLE for θ

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Example

• Suppose X=(X1, X2,…,Xn) represents random sample from a normal population (N(μ,σ2) )

• Find the MLE for μ and σ2

Large Sample Confidence Intervals

• Can estimate a parameter (e.g., the mean, μ) of a population using sample statistics (e.g., the sample mean)

• For example, in a sample of 100 males at Umich, we estimate the mean height of males at the school to be 5 foot, 10 inches

• The point estimate of 5 foot, 10 inches is found using the sample mean

• Can also present a plausible range of values for the mean

Large Sample Confidence Intervals

• Suppose X=(X1, X2,…,Xn) is a random sample from a N(μ,σ2) population

• Then the sample mean has distribution:

• Find probability that the sample mean lies within 2 standard deviations of the true mean

• Find probability that the sample mean lies within 3 standard deviations of the true mean

Large Sample Confidence Intervals

• Confidence Interval for the population mean of a N(μ,σ2) population:

• Approximate large sample confidence interval for the population mean of a population with mean, μ, and variance, σ2 :

Large Sample Confidence Intervals

• Common confidence levels (Table II.d):

Confidence Level k 68% 1 90% 1.645 95% 1.960 99% 2.580

Example

• DDTs is a very persistent pesticide. Once applied, it remains in the environment for many years and tends to accumulate up the food chain.

• For example, birds which eat rodents which eat insects which ingest DDT contaminated plants can have very high levels of DDT and this can interfere with reproduction. [This is similar to what is happening in the Great Lakes where herring gulls have very high levels of pesticides …so high they are considered hazardous waste if they die and wash up on shore.]

• Scientists took a random sample of gulls from Triangle Island (off the coast of BC) and measured the DDT levels in 100 gulls

• The sample mean and variance were 100.8 and 11.25 ppm• Find a 95% confidence interval for the mean

Large Sample Confidence Intervals

• Interpretation of the interval:

• DDT Example:

Large Sample Confidence Intervals for Proportions

• Can also do confidence intervals for proportions

• Require a large sample for the CLT to apply (npq>5)

• Note: the variance of sample proportion is a function of true population proportion

• Have to estimate the variance:

Large Sample Confidence Intervals for Proportions

• Suppose X=(X1, X2,…,Xn) is a random sample from a Ber(p) population

• Suppose have a large sample (npq>5)

• Then the sample proportion has distribution:

• Approximate large sample confidence interval for the population proportion:

Example

• A Time/CNN poll was conducted to estimate p, the proportion of all adult Americans who feel that the affirmative action program for minorities and women should be continued at some level

• A random sample of n = 100 adult Americans was taken and 80 people supported affirmative action

• Find a 95% confidence interval for the population proportion 

Example

• A recent CNN/USA Today/Gallup poll suggests that while Americans are clearly upset with France, one might characterize current Franco-American tensions more as a spat between siblings than the beginning of a war between mortal enemies

• The poll, conducted March 14-15, shows that 64% of Americans currently express an unfavorable view of France, while only about half that number, 34%, have a favorable view

• Results are based on telephone interviews with 1,007 national adults, aged 18+. For results based on the total sample of national adults, the margin of sampling error is ±3 percentage points

• What is missing?

How to Change the Width of a C.I.

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Small Sample Confidence Intervals for Means

• When constructing large sample confidence intervals, have assumed:– The sample mean is approximately normal

– The sample variance is close to the population variance

• What is the sample size is not large?

• If the parent population is close to normal, the sample mean is likely to be close to normally distributed

• Z-score is almost distributed as a standard normal…just a little more spread out

Small Sample Confidence Intervals for Means

Example

• A poultry processing company has received a shipment of 2000 Cornish hens, and the firm's quality manager wishes to estimate the true average weight of the hens

•

• The firms goal is that the average weight of each hen should be at least 1 kg

• They take a sample of 20 hens, and the mean weight of hens from the sample is 985 g with a sample standard deviation of 200g.

• An approximate 95% confidence interval for the true mean weight of the hens is: