Section 10.1 Estimating with Confidence

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# Section 10.1 Estimating with Confidence - PowerPoint PPT Presentation

Section 10.1 Estimating with Confidence. AP Statistics www.toddfadoir.com/apstats. An introduction to statistical inference. Statistical Inference provides methods for drawing conclusions about a population from sample data.

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## Section 10.1 Estimating with Confidence

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### Section 10.1Estimating with Confidence

AP Statistics

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.
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 100.
• What can you say about the mean of all California high school seniors on this exam?
Example (What we know)
• Data comes from SRS, therefore unbiased.
• There are approximately 350,000 California high school seniors. 350,000>10*500.
• We can estimate sigma-x-bar as σ/√(n)=4.5.
• The sample mean 461 one value in the distribution of sample means.
Example (What we know)
• The mean of the distribution of sample means is the same as the population mean.
• Because the n>25, the distribution of sample means is approximately normal. (Central Limit Theorem)
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.
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
Four Step Process (Inference Toolbox)
• Step 1 (Pop and para)
• Define the population and parameter you are investigating
• Step 2 (Conditions)
• Do we biased data?
• If SRS, we’re good. Otherwise PWC.
• Do we independent sampling?
• If pop>10n, we’re good. Otherwise PWC.
• Do we have a normal distribution?
• If pop is normal or n>25, we’re good. Otherwise, PWC.
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)”
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
Confidence interval behavior
• If you know a particular confidence level and ME, you can solve for your sample size.
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.

Example
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
Assignment
• Exercises 10.1 to 10.25 every other odd