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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

Section 10.1Estimating with Confidence

AP Statistics

www.toddfadoir.com/apstats

an introduction to statistical inference
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
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
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 know6
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
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
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
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 toolbox15
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
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 behavior17
Confidence interval behavior
  • If you know a particular confidence level and ME, you can solve for your sample size.
example18
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
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
Assignment
  • Exercises 10.1 to 10.25 every other odd