Estimating a Population Standard Deviation. Chi-Square Distribution.

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Presentation transcript:

Estimating a Population Standard Deviation

Chi-Square Distribution

Characteristics of the Chi- Square Distribution 1.It is not symmetric 2.The shape of the chi-square distribution depends on the degrees of freedom, just like the Student’s t-distribution 3.As the number of degrees of freedom increases, the chi-square distribution becomes more nearly symmetric 4.The values of χ 2 are nonnegative

Note We use chi-square distribution when finding confidence intervals of population variance and standard deviation

Critical Values

Finding Critical Values (left chi square and right chi square) 1. Find Degree of freedom (DF) = n – 1 2. If given the percentage, find the amount in each tail: example, if I have 90% degree of confidence, I would have 5% in each tail.

Finding Critical Values (left chi square and right chi square) - Continued 3. Find the lookup value: For the right tail, I would look at the area to the right of the 90% in the middle, in this case it would be 5% or 0.05 For the left tail, I would look at the area to the right of the 5% on the left side, in this case it would be 95% or 0.95

Finding Critical Values (left chi square and right chi square) - Continued 4. Looking at the Chi Square Distribution: using the DF and the lookup value, find the appropriate value of the right and left critical values

1. Critical Values Given 90% confidence desired and n = 15, find the critical values

2. Critical Values Given 95% confidence desired and n = 21, find the critical values

Confidence Interval for Population Standard Deviation

Confidence Interval for Population Variance

3. Confidence Intervals Given grades are normally distributed at a college, if a sample of 18 grades has a sample standard deviation of 2.5, construct a 95% confidence interval for the population standard deviation for the grades

4. Confidence Intervals Given ages are normally distributed in a particular town, if a sample of 25 people in the town have a sample standard deviation of 10.1, construct a 99% confidence interval for the population variance for the ages in this town

5. Confidence Intervals A sample of stock prices in a particular industry revealed the following stock prices: Construct a 90% confidence interval for the population standard deviation of stock prices in this industry

Comparison in Notation