Section 10.1 Estimating with Confidence

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Section 10.1Estimating with Confidence

• AP Statistics
• www.toddfadoir.com/apstats

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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 100.
• 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 unbiased.
• There are approximately 350,000 California high
  school seniors. 350,000gt10500.
• We can estimate sigma-x-bar as s/v(n)4.5.
• The sample mean 461 one value in the distribution
  of sample means.

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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 ngt25, 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
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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 biased data?
• If SRS, were good. Otherwise PWC.
• Do we independent sampling?
• If popgt10n, were good. Otherwise PWC.
• Do we have a normal distribution?
• If pop is normal or ngt25, were 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

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Confidence interval behavior

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

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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.

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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 10.25 every other odd