# 6.4 Confidence Intervals for Variance and Standard Deviation - PowerPoint PPT Presentation

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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

• 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 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 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 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

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### 6.4 Confidence Intervals for Variance and Standard Deviation

• How do we build confidence intervals using this information?

and use algebra to get to:

### 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 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:

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