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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Homer Montgomery
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1 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 Value. The entire area denoted by can be placed in the right tail and produce a Critical Value. The entire area denoted by can split in half, denoted by, and an area of can be placed in both the left tail and the right tail. This will produce both a Critical Value and a Critical Value. The Critical Value for a Chi Square Distribution is a value on the axis that is the vertical boundary separating the area in one tail of the graph from the remaining area. Critical t Value The entire area denoted by is in the left tail. The left tail critical value is the score that has Critical t Value The entire area denoted by is in the right tail. The right tail critical value is the χr score that has an area of to the left of the value. has an area of to the right of the χr value. area = area = =? Critical Value 0 χ R =? Critical Value Section 8 4B Page 1 013Eitel

The entire area denoted by can split in half, denoted by, and an area of can be placed in both the left tail and the right tail. This will produce both a Critical Value and a Critical Value.

2 Two Tail Critical Values The area denoted by is split in half, denoted by, and an area of is be placed in both the left tail and the right tail. This will produce both a Critical Value and a χr Critical Value. Critical Area Critical Area The left tail has an area of The right tail has an area of The remaining area between the two tails is 1 1 Critical Value The left tail critical value is the score that has an area of to the left of the value. Critical Value The right tail critical value is the χr score that has an area of to the left of the χr value. 1 =? χ R =? Critical Value Critical Value Note: In a two tail graph the left tail area and the right tail area both have a value of. 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 is tall and narrow and the right tail is short and wide. In fact the right tail continues indefinitely without end. Section 8 4B Page 013Eitel

Critical Area Critical Area The left tail has an area of The right tail has an area of The remaining area between the two tails is 1 1 Critical Value The left tail critical value is the score that

3 Deg. of Freedom This is only a portion of the entire Chi Square Table Section 8 4B Page 3 013Eitel

645 1.59 14.449 16.81 18.548 7 0.989 1.39 1.690.167.833 1.017 14.067 16.013 18.475 0.78 8 1.344 1.647.180.733 3.490 13.36 15.507 17.535 0.090 1.955 9 1.735.088.700 3.35 4.168 14.684 16.919 19.03 1.

4 Finding The χ R Critical Value for the χ Distribution for a One Tail ( ) Hypothesis Test Example 1 Find the Critical χ R Value for an area of =.05 in the right tail and n = 41 If n = 41 then the Degrees of Freedom = n 1= 41 1= 40 area =.05 0 χ R = If the area in the right tail is 0.05 for DF = 40 then then χ R = D of F Section 8 4B Page 4 013Eitel

05 0 χ R = 55.758 If the area in the right tail is 0.05 for DF = 40 then then χ R = 55.758 D of F 0.995 0.99 0.975 0.

5 Finding The χ R Critical Value for the χ Distribution for a One Tail ( ) Hypothesis Test Example Find the Critical χ R Value for an area of =.01 in the right tail and n = 51 If n = 51 then the Degrees of Freedom = n 1= 51 1= 50 area =.01 0 χ R = If the area in the right tail is 0.01 for DF = 50 then then χ R = D of F Section 8 4B Page 5 013Eitel

01 0 χ R = 76.154 If the area in the right tail is 0.01 for DF = 50 then then χ R = 76.154 D of F 0.995 0.99 0.975 0.

6 Finding The Critical Value for the χ Distribution for a One Tail ( ) Hypothesis Test Example 3 Find the Critical Value for an area of =.05 in the Left tail and n = 41 If n = 41 then the Degrees of Freedom = n 1= 41 1= 40 area =.05 area =.95 = If the area in the left tail is.05 then the right tail area is 0.95 for DF = 40 then then χ R = D of F Section 8 4B Page 6 013Eitel

509 If the area in the left tail is.05 then the right tail area is 0.95 for DF = 40 then then χ R = 6.509 D of F 0.995 0.99 0.

7 Finding The Critical Value for the χ Distribution for a One Tail ( ) Hypothesis Test Example 4 Find the Critical Value for an area of =.01 in the Left tail and n = 51 If n = 51 then the Degrees of Freedom = n 1= 51 1= 50 area =.01 area =.99 = If the area in the left tail is.01 then the f the area in the right tail is 0.99 for DF = 50 then then χ R = D of F Section 8 4B Page 7 013Eitel

707 If the area in the left tail is.01 then the f the area in the right tail is 0.99 for DF = 50 then then χ R = 9.707 D of F 0.

8 Finding The Critical Value and the χr Critical Value for the Distribution in a Two Tail Hypothesis Test Example 5 Find the Critical Value and the Critical χr Value for =. 10 in in a Two Tail Hypothesis Test if the total area for two tails is =.10 then the area of the left and right tails is / =.05 If n = 41 then the Degrees of Freedom = n 1= 41 1= 40 area =.05 area =.05 = χ R = If the right tail area 0.05 for DF = 40 then χ R = D of F If the area in the left tail is.05 then the area to itʼs right is 0.95 for DF = 40 so χ R = D of F Section 8 4B Page 8 013Eitel

05 = 9.707 χ R = 55.758 If the right tail area 0.05 for DF = 40 then χ R = 55.758 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.

9 Finding The Critical Value and the χr Critical Value for the Distribution in a Two Tail Hypothesis Test Example 6 Find the Critical Value and the Critical χr Value for =. 05 in in a Two Tail Hypothesis Test if the total area for two tails is =.10 then the area of the left and right tails is / =.05 If n = 71 then the Degrees of Freedom = n 1= 71 1= 70 area =.05 area =.05 = χ R = If the right tail area 0.05 for DF = 70 then χ R = D of F If the area in the left tail is.05 then the area to itʼs right is for DF = 70 so χ R = D of F Section 8 4B Page 9 013Eitel

05 = 48.758 χ R = 95.03 If the right tail area 0.05 for DF = 70 then χ R = 95.03 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.

November 08, 2010. 155S8.6_3 Testing a Claim About a Standard Deviation or Variance

November 08, 2010. 155S8.6_3 Testing a Claim About a Standard Deviation or Variance Chapter 8 Hypothesis Testing 8 1 Review and Preview 8 2 Basics of Hypothesis Testing 8 3 Testing a Claim about a Proportion 8 4 Testing a Claim About a Mean: σ Known 8 5 Testing a Claim About a Mean: σ