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6.4 Confidence Intervals for Variance and Standard DeviationPowerPoint Presentation

6.4 Confidence Intervals for 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

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- 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

- 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

- The variation we see in the sample will be our best guess.
- To build confidence intervals for σ2 and σ, we start with the sampling distribution of a modified version of s2.

- 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.

- 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

- How do we build confidence intervals using this information?
We can start with:

and use algebra to get to:

- 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:

- 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)