6 4 confidence intervals for variance and standard deviation
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6.4 Confidence Intervals for Variance and Standard Deviation. Key Concepts: Point Estimates for the Population Variance and Standard Deviation Chi-Square Distribution Building and Interpreting Confidence Intervals for the Population Variance and Standard Deviation.

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6.4 Confidence Intervals for Variance and Standard Deviation

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6 4 confidence intervals for variance and standard deviation

6.4 Confidence Intervals for Variance and Standard Deviation

  • Key Concepts:

    • Point Estimates for the Population Variance and Standard Deviation

    • Chi-Square Distribution

    • Building and Interpreting Confidence Intervals for the Population Variance and Standard Deviation


6 4 confidence intervals for variance and standard deviation1

6.4 Confidence Intervals for Variance and Standard Deviation

  • How do we estimate the population variance or the population standard deviation using sample data?

    • The variation we see in the sample will be our best guess.

      • the sample variance, s2, is used to estimate σ2

      • the sample standard deviation, s, is used to estimate σ

  • To build confidence intervals for σ2 and σ, we start with the sampling distribution of a modified version of s2.


6 4 confidence intervals for variance and standard deviation2

6.4 Confidence Intervals for Variance and Standard Deviation

  • If we find all possible samples of size n from a normal population of size N and then record the value of

    for each sample, it can be shown that follows a chi-square distribution with n – 1 degrees of freedom.


6 4 confidence intervals for variance and standard deviation3

6.4 Confidence Intervals for Variance and Standard Deviation

  • Properties of the chi-square distribution:

    • All chi-square vales are greater than or equal to zero.

    • The shape of a chi-square curve is determined by the number of degrees of freedom.

    • The area below a chi-square curve is 1.

    • All chi-square curves are positively skewed.

  • Practice working with chi-square curves

    #4 p. 341

    #6


6 4 confidence intervals for variance and standard deviation4

6.4 Confidence Intervals for Variance and Standard Deviation

  • How do we build confidence intervals using this information?

    We can start with:

    and use algebra to get to:


6 4 confidence intervals for variance and standard deviation5

6.4 Confidence Intervals for Variance and Standard Deviation

  • Fortunately, we can use the previous result for both confidence intervals.

    • To build a confidence interval for the populationvariance, we use:

    • To build a confidence interval for the populationstandard deviation, we use:


6 4 confidence intervals for variance and standard deviation6

6.4 Confidence Intervals for Variance and Standard Deviation

  • Guidelines for constructing these confidence intervals are provided on page 339.

    • Remember the population must be normal for us to apply these techniques.

    • When building our confidence intervals, we need the chi-square curve with n – 1 degrees of freedom.

  • Practice:

    #10 p. 341 (Cough Syrup)

    #16 p. 342 (Cordless Drills)

    #17 p. 342 (Pulse Rates)


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