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Section 10.1Estimating with ConfidenceAP StatisticsJanuary 2013

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ˆ

Sample Proportions?

Make sure the population size 10

so you may use p

n

pq

n

Sample Means?

Make sure the population size 10

so you may use x

n

n

10 or 10?

Use Binomial distribution tools.

np nq

10 and 10?

Use Normal distribution tools.

np nq

Is the population distribution normal?

Use Normal distribution tools.

Is the shape of population distribution

unknown or distinctly nonnormal?

If 30, the Central Limit Theorem

applies so you may use Normal distribution tools.

Otherwise, you need other tools.

n

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An introduction to statistical inference Statistical Inference provides methods for

drawing conclusions about a population from sample data.

In other words, from looking a sample, how much can we “infer” about the population.

We may only make inferences about the population if our samples unbiased. This happens when we get our data from SRS or well-designed experiments.

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Example

A SRS of 500 California high school seniors finds their mean on the SAT Math is 461. The standard deviation of all California high school seniors on this test 111.

What can you say about the mean of all California high school seniors on this exam?

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Example (What we know)

Data comes from SRS, therefore is unbiased.

There are approximately 350,000 California high school seniors. 350,000>10*500. We can estimate sigma-x-bar as

The sample mean 461 is one value in the distribution of sample means.

x

1114.5

n 500

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Example (What we know)

The mean of the distribution of sample means is the same as the population mean.

Because the n>30, the distribution of sample means is approximately normal. (Central Limit Theorem)

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Our sample is just one value in a distribution with unknown mean…

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Confidence Interval

A level C confidence interval for a parameter has two parts.An interval calculated from the data, usually in

the form (estimate plus or minus margin of error)

A confidence level C, which gives the long term proportion that the interval will capture the true parameter value in repeated samples.

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Conditions for Confidence Intervals

the data come from an SRS or well designed experiment from the population of interest

the sample distribution is approximately normal

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Confidence Interval Formulas

*

* *

*

,

where is the upper critical value

CI x zn

CI x z x zn n

z p

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Using the z table…

Confidence level

Tail Area z*

90% .05 1.645

95% .025 1.960

99% .005 2.576

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Four Step Process (Inference Toolbox) Step 1 (Pop and para)

Define the population and parameter you are investigating

Step 2 (Conditions) Do we have biased data?

If SRS, we’re good. Otherwise PWC (proceed with caution) Do we have independent sampling?

If pop>10n, we’re good. Otherwise PWC. Do we have a normal distribution?

If pop is normal or n>30, we’re good. Otherwise, PWC.

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Four Step Process (Inference Toolbox) Step 3 (Calculations)

Find z* based on your confidence level. If you are not given a confidence level, use 95%

Calculate CI. Step 4 (Interpretation)

“With ___% confidence, we believe that the true mean is between (lower, upper)”

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Confidence interval behavior

To make the margin of error smaller… make z* smaller,

which means you have lower confidence

make n bigger, which will cost more

*margin of error zn

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Confidence interval behavior

If you know a particular confidence level and ME, you can solve for your sample size.

*margin of error zn

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Example

Company management wants a report screen tensions which have standard deviation of 43 mV. They would like to know how big the sample has to be to be within 5 mV with 95% confidence?

You need a sample size of at least 285.

*

2

ME

435 1.96

431.96

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431.96 284.12

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zn

n

n

n

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Mantras

“Interpret 80% confidence interval of (454,467)” With 80% confidence we believe that the true mean of

California senior SAT-M scores is between 454 and 467.

“Interpret 80% confidence” If we use these methods repeatly, 80% of the time our

confidence interval captures the true mean. Probability

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Assignment

Exercises 10.1 to 10.8