Distribution is a χ 2 value on the χ 2 axis that is the vertical boundary separating the area in one tail of the graph from the remaining area.

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1 Section 7 3B Finding Two Tail Critical Values for a Chi Square χ Distribution A Confidence Interval is based on the Confidence Level we require The Confidence Level is 1 α and represents the area between the Left and Right Tail Areas. If the area between the two tails is 1 α then the total area in both tails is α. Two Tail Critical Values If the total area in both tails is α and this area is divided equally between the the left and right tails then the left and right tails will each have an area of α A Critical Value for a Chi Square χ Distribution is a χ value on the χ axis that is the vertical boundary separating the area in one tail of the graph from the remaining area. The vertical boundary separating the area in the Left Tail of the graph from the remaining area is the left tail critical value or The vertical boundary separating the area in the Right Tail of the graph from the remaining area is the right tail critical value or χ R left tail area = α right tail The area area between = α the tails α is 1 α α χ =?? left tail critical value the area to the left of is α χ R =?? right tail critical value the area to the right of χ R is α Note: In a two tail graph the left tail area and the right tail area both have a value of α. The two tail areas do not look equal because the distribution is not normal. It is skewed left causing the tails with equal area to look different. The left tail area is tall and narrow shaped and the right tail area is short and wide. In fact the right tail continues indefinitely without end. Section 7 3B Page 1 of 1 13 Eitel

2 The χ distribution is NOT Normal. It is skewed to the Right As the the number of degrees of freedom increases (DF = n 1) increases the χ distribution becomes more normal. DF = 1 DF = 5 χ χ left tail area = α right tail The area area between = α the tails α is 1 α α =?? χ R =?? χ The χ starts at and increases in the positive direction. No χ values are negative. The left tail critical values will be less than the right tail critical value. The left tail critical value will be a different positive number than the right tail critical value. Section 7 3B Page of 1 13 Eitel

3 The χ table below is used to find the value for the LEFT TAIL Critical Value and the χ R value for the RIGHT TAIL Critical Value the right tail critical value χ R is based on the area in the RIGHT TAIL and the the degrees of freedom equal to n 1 The left tail critical value is based on RIGHT of the left tail area and the the degrees of freedom equal to n 1 Deg. of Area in the Right Tail (One Tail) Freedom This is only a portion of the entire Chi Square Table Section 7 3B Page 3 of 1 13 Eitel

4 Example 1 Finding and χr if the total tail area α =.5 with n = 31 If α=.5 then α =.5 and DF = 31 1 = 3 Example 1 detailed explanation for finding χ R To find the right tail critical value χ R for a right tail area of α =.5 and the Degrees of Freedom 3 use the part of the t table shown below. the χ R value is = DF = 3 right of χ R is.5 right tail area α =.5 χ R = χ Area to the right of the Chi-Square value D of F Section 7 3B Page 4 of 1 13 Eitel

5 Example 1 detailed explanation for finding To find the value for a left tail area of α =.5 and Degrees of Freedom 3 you must use the AREA TO THE RIGHT of if the left tail has an area of.5 then RIGHT of is.975 To find the left tail critical value for a right tail area of.975 and the Degrees of Freedom 3 use the part of the t table shown below. the value is = Degrees of Freedom = 3 right of is = χ Area to the right of the Chi-Square value D of F Section 7 3B Page 5 of 1 13 Eitel

6 Example 1 Finding χ R and χl at the same time The row of the χ table with the DF = 3 contains both the value for the LEFT TAIL Critical Value and the χ R value for the RIGHT TAIL Critical Value for DF = 3 The χ R value for the RIGHT TAIL Critical Value is read with a right tail area of.5 The value for the Left TAIL Critical Value is read with an area to the right of the left tail area of.5 which is 1.5 =.975 right of is.975 α =.5 α =.5 = χr = right of χ R is.5 χ Area to the right of the Chi-Square value D of F the value is the χ R value is Section 7 3B Page 6 of 1 13 Eitel

7 Example Finding and χr if the total tail area α =.1 with n = 71 If α=.1 then α =.5 and DF = 71 1 = 7 Example 1 detailed explanation for finding χ R To find the right tail critical value χ R for a right tail area of α =.5 and the Degrees of Freedom 7 use the part of the t table shown below. the χ R value is = DF = 7 right of χ R is.5 right tail area α =.5 χ R = χ Area to the RIGHT of the Chi-Square value D of F Section 7 3B Page 7 of 1 13 Eitel

8 Example detailed explanation for finding To find the value for a left tail area of α =.5 and Degrees of Freedom 7 you must use the AREA TO THE RIGHT of if the left tail has an area of.5 then RIGHT of is.995 To find the left tail critical value for a right tail area of.995 and the Degrees of Freedom 7 use the part of the t table shown below. the value is = DF = 7 left tail area α =.5 right of is.995 χ = Area to the right of the Chi-Square value D of F Section 7 3B Page 8 of 1 13 Eitel

9 Example Finding χ R and χl at the same time The row of the χ table with the DF = 7 contains both the value for the LEFT TAIL Critical Value and the χ R value for the RIGHT TAIL Critical Value for DF = 7 The χ R value for the RIGHT TAIL Critical Value is read with a right tail area of.5 The value for the Left TAIL Critical Value is read with an area to the right of the left tail area of.5 which is 1.5 =.995 right of is.995 α =.5 α =.5 = χr = right of χ R is.5 χ Area to the RIGHT of the Chi-Square value D of F the value is = the χ R value is = Section 7 3B Page 9 of 1 13 Eitel

10 Example 3 Finding and χr if the total tail area α =.1 with n = 41 If α=.1 then α =.5 and DF = 41 1 = 4 Example 1 detailed explanation for finding χ R To find the right tail critical value χ R for a right tail area of α =.5 and the Degrees of Freedom 4 use the part of the t table shown below. the χ R value is = DF = 4 right of χ R is.5 right tail area α =.5 χ R = χ Area to the RIGHT of the Chi-Square value D of F Section 7 3B Page 1 of 1 13 Eitel

11 Example 3 detailed explanation for finding To find the value for a left tail area of α =.5 and Degrees of Freedom 4 you must use the AREA TO THE RIGHT of if the left tail has an area of.5 then RIGHT of is.95 To find the left tail critical value for a right tail area of.95 and the Degrees of Freedom 4 use the part of the t table shown below. the value is = 6.59 left tail area α =.5 right of is.95 χ = 6.59 Area to the right of the Chi-Square value D of F Section 7 3B Page 11 of 1 13 Eitel

12 Example 3 Finding χ R and χl at the same time The row of the χ table with the DF = 4 contains both the value for the LEFT TAIL Critical Value and the χ R value for the RIGHT TAIL Critical Value for DF = 4 The χ R value for the RIGHT TAIL Critical Value is read with a right tail area of.5 The value for the Left TAIL Critical Value is read with an area to the right of the left tail area of.5 which is 1.5 =.95 right of is.95 α =.5 α =.5 = 6.59 χr = right of χ R is.5 χ Area to the RIGHT of the Chi-Square value D of F the value is 6.59 the χ R value is Section 7 3B Page 1 of 1 13 Eitel

Distribution is a χ 2 value on the χ 2 axis that is the vertical boundary separating the area in one tail of the graph from the remaining area.

Distribution is a χ 2 value on the χ 2 axis that is the vertical boundary separating the area in one tail of the graph from the remaining area. Section 8 4B Finding Critical Values for a Chi Square Distribution The entire area that is to be used in the tail(s) denoted by. The entire area denoted by can placed in the left tail and produce a Critical