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Estimating Population Standard Deviation and Confidence Intervals

This chapter discusses the estimation of population standard deviation using chi-square distribution and constructing confidence intervals for population variance and standard deviation.

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Estimating Population Standard Deviation and Confidence Intervals

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  1. Chapter 9 Estimating the Value of a Parameter

  2. Section 9.3 Estimating a Population Standard Deviation

  3. Objectives • Find critical values for the chi-square distribution • Construct and interpret confidence intervals for the population variance and standard deviation

  4. Objective 1 • Find Critical Values for the Chi-Square Distribution

  5. If a simple random sample of size n is obtained from a normally distributed population with mean μ and standard deviation σ, then has a chi-square distribution with n-1 degrees of freedom.

  6. Characteristics of the Chi-Square Distribution • It is not symmetric. • The shape of the chi-square distribution depends on the degrees of freedom, just like the Student’s t-distribution. • As the number of degrees of freedom increases, the chi-square distribution becomes more nearly symmetric. • The values of χ2 are nonnegative (greater than or equal to 0).

  7. Parallel Example 1: Finding Critical Values for the Chi-Square Distribution Find the chi-square values that separate the middle 95% of the distribution from the 2.5% in each tail. Assume 18 degrees of freedom.

  8. Solution Find the chi-square values that separate the middle 95% of the distribution from the 2.5% in each tail. Assume 18 degrees of freedom. χ20.975=8.231 χ20.025=31.526

  9. Objective 2 • Construct and Interpret Confidence Intervals for the Population Variance and Standard Deviation

  10. A (1–χ)·100% Confidence Interval for χ2 If a simple random sample of size n is taken from a normal population with mean μ and standard deviation σ, then a (1 – α)·100% confidence interval for χ2 is given by Lower bound: Upper bound: Note: To find a (1-)·100% confidence interval about σ, take the square root of the lower bound and upper bound.

  11. Parallel Example 2: Constructing a Confidence Interval for a Population Variance and Standard Deviation One way to measure the risk of a stock is through the standard deviation rate of return of the stock. The following data represent the weekly rate of return (in percent) of Microsoft for 15 randomly selected weeks. Compute the 90% confidence interval for the risk of Microsoft stock. 5.34 9.63 –2.38 3.54 –8.76 2.12 –1.95 0.27 0.15 5.84 –3.90 –3.80 2.85 –1.61 –3.31 Source: Yahoo!Finance

  12. Solution A normal probability plot and boxplot indicate the data is approximately normal with no outliers. • s = 4.6974; s2 = 22.0659 • χ20.95 = 6.571 and χ20.05 = 23.685 for 15 – 1 = 14 • degrees of freedom • Lower bound: • Upper bound: We are 90% confident that the population standard deviation rate of return of the stock is between 13.04 and 47.01.

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