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# Are you here? - PowerPoint PPT Presentation

Are you here?. Yes, and I’m ready to learn Yes, and I need a nap No. HW - Problem 6. 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.

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• Yes, and I’m ready to learn

• Yes, and I need a nap

• No

• 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.

• 0.062

• 1-0.062

• 0

• 1

• -1.54

For proportions

For means

Standard Deviation

• 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.

For the sample mean, the standard error is

Standard Error

Exciting Statistics about ISU Normal.Students -2011 data

• 69.1% of sexually active students use condoms

• American College Health Association

• n=272

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

A Confidence Interval

Certainty vs. Precision Normal.

• 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).

• 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.

• The figure to the right shows that some of our confidence intervals capture the true proportion (the green horizontal line), while others do not:

Homework Problem Normal.

• 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%

• 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.

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.

One-Proportion z-Interval

Z* is the Critical Value confidence interval for the population proportion,

• 80%  z*=1.282

• 90% z*=1.645

• 95% z*=1.96

• 98%z*=2.326

• 99% z*=2.576

Critical Values (cont.) confidence interval for the population proportion,

• Example: For a 90% confidence interval, the critical value is 1.645:

HW – Problem 18 confidence interval for the population proportion,

• 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?

HW – Problem 18 confidence interval for the population proportion,

• 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 penalty was fairly applied. Does this fall within a 95% confidence interval for survey #2?

• 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 penalty was fairly applied. Does this fall within a 95% confidence interval for survey #2?

• 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?

• How large should our margin of error be?

How many people should we sample to get a ME of 1%? penalty was fairly applied. Does this fall within a 95% confidence interval for survey #2?

• 1,000

• Between 1,000 and 4,000

• Between 4,000 and 8,000

• Between 8,000 and 16,000

Upcoming work penalty was fairly applied. Does this fall within a 95% confidence interval for survey #2?

• Quiz #4 in class today

• HW #8 due next Sunday

• Part 3 of Data Project due April 2nd