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 = 16.781 χr = 46.979 The Confidence Interval for the population standard deviation α is given by (n 1)(s x ) (n 1)(s x ) (30)(1.6) 46.979 (30)(1.6) 16.791 1.8.14 Confidence Interval Statement: I am 95% confident that the interval 1.8.14 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 = 16.791 = 46.979 Stat 300 7 3C Examples Page 1 of 10 013 Eitel
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 = 46.979 DF = 30 right of is.05 right tail area α =.05 0 = 46.979 χ Area to the right of the Chi-Square value D of F 0.995 0.99 0.975 0.95 0.9 0.10 0.05 0.05 0.01 0.005 30 13.787 14.953 16.791 18.493 0.599 40.56 43.773 46.979 50.89 53.67 Stat 300 7 3C Examples Page of 10 013 Eitel
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 0.975 and Degrees of Freedom 30 use the part of the χ table shown below the value is + tα = 16.791 Degrees of Freedom = 30 right of is.975 α =.05 = 16.791 χ Area to the right of the Chi-Square value D of F 0.995 0.99 0.975 0.95 0.9 0.10 0.05 0.05 0.01 0.005 30 13.787 14.953 16.791 18.493 0.599 40.56 43.773 46.979 50.89 53.67 Stat 300 7 3C Examples Page 3 of 10 013 Eitel
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 = 43.75 χr = 104.15 The Confidence Interval for the population standard deviation σ is given by (n 1)(s x ) (n 1)(s x ) (70)(5.8) 104.15 (70)(5.8) 43.75 4.33 6.7 Confidence Interval Statement: I am 99 % confident that the interval 4.33 6.7 contains the value of the true population standard deviation α Finding and χr right of is.995 right of is.005 α = α =.005.005 χ = 43.75 = 104.15 Area to the RIGHT of the Chi-Square value D of F 0.995 0.99 0.975 0.95 0.9 0.10 0.05 0.05 0.01 0.005 70 43.75 45.44 48.758 51.739 55.39 85.57 90.531 95.03 100.45 104.15 Stat 300 7 3C Examples Page 4 of 10 013 Eitel
Example detailed explanation for finding Finding To find the value for a right tail area of 0.005 and Degrees of Freedom 70 use the part of the χ table shown below. the value is + tα = 104.15 DF = 70 right of is.005 right tail area α =.005 0 = 104.15 χ Area to the RIGHT of the Chi-Square value D of F 0.995 0.99 0.975 0.95 0.9 0.10 0.05 0.05 0.01 0.005 70 43.75 45.44 48.758 51.739 55.39 85.57 90.531 95.03 100.45 104.15 Stat 300 7 3C Examples Page 5 of 10 013 Eitel
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 0.995 and Degrees of Freedom 70 use the part of the χ table shown below the value is + tα = 43.75 DF = 70 left tail area α =.005 right of is.995 χ = 43.75 Area to the right of the Chi-Square value D of F 0.995 0.99 0.975 0.95 0.9 0.10 0.05 0.05 0.01 0.005 70 43.75 45.44 48.758 51.739 55.39 85.57 90.531 95.03 100.45 104.15 Stat 300 7 3C Examples Page 6 of 10 013 Eitel
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 = 6.509 χr = 55.758 The Confidence Interval for the population standard deviation α is given by (n 1)(s x ) (n 1)(s x ) (40)(300) 55.758 (40)(300) 6.509 1948.1 85.3 Confidence Interval Statement: I am 99% confident that the interval 1948.1 85.3 contains the value of the true population standard deviation α Finding and χr right of is.95 right of is.05 α = α =.05.05 χ = 6.509 = 55.758 Stat 300 7 3C Examples Page 7 of 10 013 Eitel
Area to the RIGHT of the Chi-Square value D of F 0.995 0.99 0.975 0.95 0.9 0.10 0.05 0.05 0.01 0.005 40 0.707 1.164 4.433 6.509 9.051 51.805 55.758 59.34 63.691 66.766 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 = 55.758 DF = 40 right of is.05 right tail area α =.05 0 = 55.758 χ Area to the RIGHT of the Chi-Square value D of F 0.995 0.99 0.975 0.95 0.9 0.10 0.05 0.05 0.01 0.005 40 0.707 1.164 4.433 6.509 9.051 51.805 55.758 59.34 63.691 66.766 Stat 300 7 3C Examples Page 8 of 10 013 Eitel
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 = 6.509 DF = 40 left tail area α =.05 right of is.95 χ = 6.509 Area to the right of the Chi-Square value D of F 0.995 0.99 0.975 0.95 0.9 0.10 0.05 0.05 0.01 0.005 40 0.707 1.164 4.433 6.509 9.051 51.805 55.758 59.34 63.691 66.766 Stat 300 7 3C Examples Page 9 of 10 013 Eitel
Stat 300 7 3C Examples Page 10 of 10 013 Eitel