Recap: Sample statistic may not exactly equal population parameter because of sampling variability. The sampling distribution a statistic tells us about the variability we get "by chance".

Proportion of heads in 20 Tosses: 0.30 0.30 0.30 0.30 0.35 0.35 0.40 0.40 0.40 0.45 0.45 0.45 0.45 0.50 0.50 0.50 0.50 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.60 0.60 0.60 0.60 0.60 0.70 0.70 0.80

Average =.4988 Standard deviation=.125 (Lect 9-10)

Theoretically, these 's should follow a N(.5, sqrt(.5(.5)/20)) distribution

How many values are within 2 Std Devs of .5?

In reality, just know one value, by constructing a confidence interval, taking sampling variability into account, can estimate the parameter, p.

Confidence Interval: estimate margin of error

Estimate=guess of parameter, statistic (e.g. or )

Margin of error=how accurate we think the guess is

how much random sampling error/sampling variability is present

1. Find the z* corresponding to level C (upper (1-C)/2 critical value)

2. Margin of error = z*(std dev of estimate)

Means: z*

Proportions: z* (approximate interval)

Example Packages of pretzels are automatically weighed at the end of the production line. For the last 50, = 11.92 ounces. Assume we know that the weights of packages from this machine have a standard deviation of .11 ounce.

(a) Construct a 95% confidence interval for the mean package weight m from this production line. What about a 99% confidence interval?

(b) What if the last 100 had a mean of 11.92 ounces. How does the confidence interval change?

• You specify the confidence level, C

determines the critical value

• Margin of error=z*(std dev). When does margin of error decrease?

1.

2.

3.

• A level C confidence interval for a parameter is an interval computed from sample data by a method that has probability C of producing an interval containing the true value of the parameter.

• The data must be an SRS

- Info available for more complicated sampling methods

- Haphazardly collected data can’t be saved.

• Interval for p is approximate

- Better for large n (Quite accurate for n=100, OK for n=25,30)

• If sample size is small or the population not Normal, then Confidence Interval for m is approximate and actual confidence level (% of intervals capturing m) will be different from C.

- Check data for skewness, nonnormality.

- Usually ok if n >15 - robustness property

• Outliers can have a big effect on Confidence Intervals for m

Example A member of Congress you advise receives 1310 pieces of mail on pending gun control legislation. Of these, 86% oppose the legislation. He asks you for an analysis of these opinions. What will you tell him?

Example Any cautions when interpreting the results of the pretzel measurements?