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Unit 7

Unit 7. Section 7- 5. 7-5: Confidence Intervals for Variances and Standard Deviations. In 7-2 to 7-4 we determined confidence intervals for means and proportions. Confidence intervals can also be determined for standard deviations and variance .

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Unit 7

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  1. Unit 7 Section 7-5

  2. 7-5: Confidence Intervals for Variances and Standard Deviations • In 7-2 to 7-4 we determined confidence intervals for means and proportions. Confidence intervals can also be determined for standard deviations and variance. • For example: If we were manufacturing pipes, you would want to keep the variation of diameters down in order for the pieces to fit together and avoid being scrapped. • In order to find a confidence interval for standard deviation and variance, we use a new distribution known as a chi-square distribution.

  3. Section 7-5 • A chi-square distribution is family of curves like the t distribution (based on degrees of freedom). • Chi-square variable can not be negative. • The chi-square distribution is positively skewed • The area under a chi-square distribution is 1 or 100% • Values for variance and standard deviation are assumed to be normally distributed. • Two sets of values will be used in the formula for chi-square (left and right side of the table).

  4. Section 7-5 Finding the left and right chi-squared value First determine what confidence interval we wish to find (in decimal form). Subtract the percentage from 1. Divide this value by two to determine value of X2right. Subtract the value of X2right from 1 to get X2left.

  5. Section 7-5 • Example 1: Find the values for X2right and X2left for a 90% confidence interval when n = 25.

  6. Section 7-5 Formula for Confidence Interval for a Variance • where • Rounding Rule: Round to one more decimal place than the original data (or same as sample standard deviation).

  7. Section 7-5 Formula for Confidence Interval for a Standard Deviation • where • Note: if the problem gives sample variance you do not need to square it! (remember, s2 is sample variance)

  8. Section 7-5 • Example 2: Find the 95% confidence interval for the variance and standard deviation of the nicotine content of cigarettes manufactured if a sample of 20 cigarettes has a standard deviation of 1.6 milligrams.

  9. Section 7-5 • Example 3: Find the 90% confidence interval for the variance and standard deviation for the price in dollars of an adult single-day ski lift ticket. The data represents a selected sample of nationwide ski resorts. Assume the variable is normally distributed. 59 54 53 52 51 39 49 46 49 48

  10. Section 7-5 Homework: • Pg 381: #’s 1 - 7

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