Yes, and I’m ready to learn Yes, and I need a nap No Are you here? Yes, and I’m ready to learn Yes, and I need a nap No

Apple Problem When a truck load of apples arrives at a packing plant, a random sample of 125 is selected and examined for bruises, discoloration, and other defects. The whole truckload will be rejected if more than 5% of the sample is unsatisfactory. Suppose that in fact 9% of the apples on the truck do not meet the desired standard. What is the probability that the shipment will be accepted anyway. Z=1.54 %=.9382

Standard Deviation Both of the sampling distributions we’ve looked at are Normal. For proportions For means

What is the probability that the shipment will be accepted anyway? 0.062 1-0.062 1 -1.54

Standard Deviation vs. Standard Error We don’t know p, μ, or σ, we’re stuck, right? Nope. We will use sample statistics to estimate these population parameters. Sample statistics are notated as: s, Whenever we estimate the standard deviation of a sampling distribution, we call it a standard error.

Standard Error For a sample proportion, the standard error is For the sample mean, the standard error is

Exciting Statistics about ISU Students -2011 data 69.1% of sexually active students use condoms American College Health Association n=272

A Confidence Interval

A Confidence Interval By the 68-95-99.7% Rule, we know about 68% of all samples will have ’s within 1 SE of p about 95% of all samples will have ’s within 2 SEs of p about 99.7% of all samples will have ’s within 3 SEs of p

Certainty vs. Precision To be more confident, we wind up being less precise. Because of this, every confidence interval is a balance between certainty and precision.

Certainty vs. Precision The choice of confidence level is somewhat arbitrary, but keep in mind this tension between certainty and precision when selecting your confidence level. The most commonly chosen confidence levels are 90%, 95%, and 99% (but any percentage can be used).

What Does “95% Confidence” Really Mean? Each confidence interval uses a sample statistic to estimate a population parameter. But, since samples vary, the statistics we use, and thus the confidence intervals we construct, vary as well.

What Does “95% Confidence” Really Mean? (cont.) The figure to the right shows that some of our confidence intervals capture the true proportion (the green horizontal line), while others do not:

Sales Problem A catalog sales company promises to deliver orders placed on the Internet within 3 days. Follow-up calls to a few randomly selected customers show that a 95% CI for the proportion of all orders that arrive on time is 81% ± 4%

Which of the following statements is correct? Between 77% and 85% of all orders arrive on time. One can be 95% confident that the true population percentage of orders place on the Internet that arrive within 3 days is between 77% and 85% One can be 95% confident that all random samples of customers will show that 81% of orders arrive on time 95% of all random samples of customers will show that between 77% and 85% of orders arrive on time.

One-Proportion z-Interval When the conditions are met, we are ready to find the confidence interval for the population proportion, p. The confidence interval is where The critical value, z*, depends on the particular confidence level, C, that you specify.

Z* is the Critical Value Using a table or technology, we find that a more exact value for our 95% confidence interval is 1.96 instead of 2. We call 1.96 the critical value and denote it z*.

Critical Values (cont.) Example: For a 90% confidence interval, the critical value is 1.645:

Biased Survey Problem Often, on surveys there are two ways of asking the same question. 1) Do you believe the death penalty is fair or unfairly applied? 2) Do you believe the death penalty is unfair or fairly applied?

Biased Survey Problem Survey 1) n=597 2) n=597 For the second phrasing, 45% said the death penalty is fairly applied.

Suppose 54% of the respondents in survey #1 said the death penalty was fairly applied. Does this fall within a 95% confidence interval for survey #2? Yes, it falls within my CI No, it does not fall within my CI

Margin of Error: Certainty vs. Precision The more confident we want to be, the larger our z* has to be But to be more precise (i.e. have a smaller ME and interval), we need a larger sample size, n. We can claim, with 95% confidence, that the interval contains the true population proportion. The extent of the interval on either side of is called the margin of error (ME). In general, confidence intervals have the form estimate ± ME.

Margin of Error - Problem Suppose the truth is that 56% of ISU student drink every weekend. We want to create a 95% confidence interval, but we also want to be as precise as possible. How many people should we sample to get a ME of 1%?

How many people should we sample to get a ME of 1%? 1,000 Between 1,000 and 4,000 Between 4,000 and 8,000 Between 8,000 and 16,000

Upcoming work Quiz #4 in class today HW #8 due Sunday Part 3 of Data Project due Oct. 28th