χ L = χ R =

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1 Chapter 7 3C: Examples of Constructing a Confidence Interval for the true value of the Population Standard Deviation σ for a Normal Population. Example 1 Construct a 95% confidence interval for the true value of the population standard deviation σ if the population is normal. A random sample of size 31( n = 31) results in a sample mean ( x ) of x =1.8 and a sample standard deviation of s x =1.6 The population is given as normal so we can use the Chi Square χ Table. The confidence level is 95% so α =.05 If α =.05 then α =.05 DF = n 1 = 31 1 = 30 x =1.8 s x =1.6 n = 31 DF = 30 α =.05 = χr = The Confidence Interval for the population standard deviation α is given by (n 1)(s x ) (n 1)(s x ) (30)(1.6) (30)(1.6) Confidence Interval Statement: I am 95% confident that the interval contains the value of the true population standard deviation α Finding and χr right of is.975 α =.05 α =.05 right of is.05 χ = = Stat C Examples Page 1 of Eitel

2 Example 1 detailed explanation for finding To find the value for a right tail area of 0.05 and Degrees of Freedom 30 use the part of the χ table shown below. the value is + tα = DF = 30 right of is.05 right tail area α =.05 0 = χ Area to the right of the Chi-Square value D of F Stat C Examples Page of Eitel

3 Example 1 detailed explanation for finding To find you must use the AREA TO THE RIGHT of if the left tail has an area of.05 then RIGHT of is.975 To find the value for a right tail area of and Degrees of Freedom 30 use the part of the χ table shown below the value is + tα = Degrees of Freedom = 30 right of is.975 α =.05 = χ Area to the right of the Chi-Square value D of F Stat C Examples Page 3 of Eitel

4 Example Construct a 99 % confidence interval for the true value of the population standard deviation σ if the population is normal. A random sample of size 71( n = 71) results in a sample mean ( x ) of x = 7.1 and a sample standard deviation of s x = 5.8 The population is given as normal so we can use the Chi Square χ Table. The confidence level is 99 % so α =.01 and α =.005 DF = n 1 = 71 1 = 70 x = 7.1 s x = 5.8 n = 71 DF = 70 α =.005 = χr = The Confidence Interval for the population standard deviation σ is given by (n 1)(s x ) (n 1)(s x ) (70)(5.8) (70)(5.8) Confidence Interval Statement: I am 99 % confident that the interval contains the value of the true population standard deviation α Finding and χr right of is.995 right of is.005 α = α = χ = = Area to the RIGHT of the Chi-Square value D of F Stat C Examples Page 4 of Eitel

5 Example detailed explanation for finding Finding To find the value for a right tail area of and Degrees of Freedom 70 use the part of the χ table shown below. the value is + tα = DF = 70 right of is.005 right tail area α = = χ Area to the RIGHT of the Chi-Square value D of F Stat C Examples Page 5 of Eitel

6 Example detailed explanation for finding To find you must use the AREA TO THE RIGHT of if the left tail has an area of.005 then RIGHT of is.995 To find the value for a right tail area of and Degrees of Freedom 70 use the part of the χ table shown below the value is + tα = DF = 70 left tail area α =.005 right of is.995 χ = Area to the right of the Chi-Square value D of F Stat C Examples Page 6 of Eitel

7 Example 3 The California Consumer Protection Agency decides to conduct an investigation into the quality of car tires. They sample 41 randomly selected tires and find an average tread wear of 79,000 and a standard deviation of,300 miles. Construct a 90% confidence interval for the true value of the population standard deviation σ. Assume that if the population of car tire tread wear is normally distributed. The population is given as normal so we can use the Chi Square χ Table. The confidence level is 90% so α =.10 and α =.05 DF = n 1 = 41 1 = 40 x = 79,000 s x =,300 n = 41 DF = 40 α =.05 = χr = The Confidence Interval for the population standard deviation α is given by (n 1)(s x ) (n 1)(s x ) (40)(300) (40)(300) Confidence Interval Statement: I am 99% confident that the interval contains the value of the true population standard deviation α Finding and χr right of is.95 right of is.05 α = α = χ = = Stat C Examples Page 7 of Eitel

8 Area to the RIGHT of the Chi-Square value D of F Example 3 detailed explanation for finding Finding To find the value for a right tail area of α = 0.05 and Degrees of Freedom 40 use the part of the χ table shown below. the value is + tα = DF = 40 right of is.05 right tail area α =.05 0 = χ Area to the RIGHT of the Chi-Square value D of F Stat C Examples Page 8 of Eitel

9 Example 3 detailed explanation for finding To find you must use the AREA TO THE RIGHT of if the left tail has an area of.05 then RIGHT of is.95 To find the value for a right tail area of 0.95 and Degrees of Freedom 40 use the part of the χ table shown below. the value is + tα = DF = 40 left tail area α =.05 right of is.95 χ = Area to the right of the Chi-Square value D of F Stat C Examples Page 9 of Eitel

10 Stat C Examples Page 10 of Eitel