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I will provide these cheat sheets on the day of the final exam. Do not bring this copy to the final. Also, it is your responsibility to verify that the formulas are correct and notify me of any necessary corrections.
Formulas Data Distribution
𝑥𝑥̅ = ∑ 𝑥𝑥𝑛𝑛 sd = �∑(𝑥𝑥−𝑥𝑥̅)
2 𝑛𝑛−
Probability Distribution 𝜇𝜇 = ∑^ 𝑥𝑥 ∙ 𝑝𝑝(𝑥𝑥)^ 𝜎𝜎 = �∑(𝑥𝑥 − 𝜇𝜇)^2 ∙ 𝑝𝑝(𝑥𝑥) 𝑍𝑍 = 𝑥𝑥−𝜇𝜇𝜎𝜎 , 𝑥𝑥 = 𝜇𝜇 + 𝑧𝑧 ∙ 𝜎𝜎 Empirical Rule = 68-95-99.7 Rule See also the common probability rules below.
Regression Predicted 𝑦𝑦 = 𝑎𝑎 + 𝑏𝑏 ∙ 𝑥𝑥 𝑏𝑏 = 𝑟𝑟 ∙
𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥^ ,^ 𝑎𝑎^ =^ 𝑦𝑦� − 𝑏𝑏 ∙ 𝑥𝑥̅ Residual ( predicted error ) = 𝑦𝑦𝑜𝑜𝑜𝑜𝑠𝑠𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑦𝑦𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜𝑝𝑝𝑝𝑝𝑝𝑝𝑜𝑜𝑜𝑜
S𝑜𝑜 = � (^) 𝑛𝑛SSE− 2
Distribution of Sample Proportions
Mean = P, SE = �𝑃𝑃∙(1−𝑃𝑃 𝑛𝑛 ), 𝑧𝑧 = 𝑝𝑝�−𝑝𝑝𝑆𝑆𝑆𝑆
Normal if 𝑛𝑛𝑝𝑝 ≥ 10 and ( 1 − 𝑝𝑝) ∙ 𝑛𝑛 ≥ 10
CI = 𝑝𝑝�^ ± 𝑍𝑍𝑐𝑐 ∙ SE
Distribution of Sample Means Mean = 𝜇𝜇, 𝑧𝑧 = 𝑥𝑥 𝑆𝑆𝑆𝑆̅^ −𝜇𝜇
𝝈𝝈 known & population variable is normally distributed OR n > 30: Use Z-test SE = (^) √𝑛𝑛𝜎𝜎 , CI = 𝑥𝑥̅ ± 𝑍𝑍𝑝𝑝 ∙ SE
𝝈𝝈 not known & population variable is normally distributed OR n > 30: Use T-test One sample : SE = (^) √𝑛𝑛𝑠𝑠 , 𝑑𝑑𝑑𝑑 = 𝑛𝑛 − 1 , CI = 𝑥𝑥̅ ± 𝑇𝑇𝑝𝑝 ∙ 𝑆𝑆𝑆𝑆
Two independent samples : SE = � 𝑠𝑠^1
2 𝑛𝑛 1 +^
𝑠𝑠 22 𝑛𝑛 2 ,^ 𝑑𝑑𝑑𝑑^ (use technology) CI = (𝑥𝑥̅ 1 − 𝑥𝑥̅ 2 ) ± 𝑇𝑇𝑝𝑝 ∙ 𝑆𝑆𝑆𝑆
Chi-Square Tests All chi-square curves are skewed right; mean = 𝑑𝑑𝑑𝑑. Test statistic for all Chi-Square tests: 𝜒𝜒 2 = (𝑜𝑜𝑜𝑜𝑠𝑠𝑜𝑜𝑜𝑜 � 𝑜𝑜𝑜𝑜−𝑜𝑜𝑥𝑥𝑝𝑝𝑜𝑜𝑝𝑝𝑝𝑝𝑜𝑜𝑜𝑜)
2 𝑜𝑜𝑥𝑥𝑝𝑝𝑜𝑜𝑝𝑝𝑝𝑝𝑜𝑜𝑜𝑜 One-way table: 𝑑𝑑𝑑𝑑 = (𝑟𝑟 − 1) where r = number of categories. Two-way table: 𝑑𝑑𝑑𝑑 = (𝑟𝑟 − 1 )(𝑐𝑐 − 1 )^ where r = number of categories for one variable and c = number of categories for the other variable.
Hypothesis Testing
𝐻𝐻 0 : ≤, ≥, = (then change to =)
𝐻𝐻𝑎𝑎: <, >, ≠ (< or > is a one-tailed test; ≠ is a two-tailed test)
Compare P-value to 𝛼𝛼 One-tailed test: 𝛼𝛼 = significance level
Two-tailed test: 𝛼𝛼 =
significance level 2 Z-test: SE = (^) √𝑛𝑛𝜎𝜎, 𝑧𝑧 = 𝑥𝑥̅−𝜇𝜇𝑆𝑆𝑆𝑆^0
1-sample T-test: SE = (^) √𝑛𝑛𝑠𝑠, 𝑇𝑇 = 𝑥𝑥̅−𝜇𝜇𝑆𝑆𝑆𝑆^0
2-sample T-test: SE = �^ 𝑠𝑠^1
2 𝑛𝑛 1 +^
𝑠𝑠 22 𝑛𝑛 2 𝑇𝑇 = (𝑥𝑥̅^1 −𝑥𝑥̅^2 ) 𝑆𝑆𝑆𝑆−(𝜇𝜇^1 −𝜇𝜇^2 )
Paired T-test: SE = (^) √𝑠𝑠𝑛𝑛, 𝑇𝑇 = 𝑥𝑥 𝑆𝑆𝑆𝑆̅^ −^0
Probability Rules
a) For any event A, 0 ≤ 𝑃𝑃(𝐴𝐴)^ ≤ 1. b) If S is a sample space, then P( S ) = 1. c) The sum of the probabilities of all possible disjoint events in a sample space is 1. d) If A and B are disjoint events (no outcomes in common), then P(A or B) = P(A) + P(B). e) If A and B are NOT disjoint events (share at least one outcome), then P(A or B) = P(A) + P(B) – P( A and B). f) For any event A, P(not A) = 1 – P(A). g) If P(B|A) ≈ P(B) , then A and B are independent events. h) P(A and B) = P(A)∙P(B|A). i) When A and B are independent events then P(A and B) = P(A)∙P(B)
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G ENERAL C ALCULATOR D IRECTIONS (TI-83 OR 84 C ALCULATOR )
Turn Diagnostics On To turn diagnostics on, access the catalog. Press the 2 nd^ button, and then press the number 0. Scroll down to Diagnostics On. Press ENTER. Press ENTER again. Enter Data into the STAT LIST Editor To access the list editor , press the STAT button (just to the left of the arrow pad). With option 1 (EDIT) highlighted, press ENTER. If your lists already contain data, see Clear Lists below. Enter the data into one of the lists (e.g. L1). Find the Descriptive Statistics for a Single Variable (one list) Input the data into the STAT LIST Editor. Press the STAT button. Select the CALC menu. With option 1 ( 1-Var Stats ) highlighted, press ENTER. Find the Least Squares Regression Line Enter the data into the STAT LIST editor (𝑥𝑥 data in L1 and 𝑦𝑦 data in L2). Press the STAT button. Scroll right to the CALC menu. Select the LinReg (a + b x ) option. Fill in the prompts on the screen. Clear Lists These instructions assume you need to clear data from L1. To access the list editor , press the STAT button. With option 1 (EDIT) highlighted, press ENTER. Scroll up until L1 is highlighted. Press the CLEAR button. Press ENTER.
H YPOTHESIS T ESTS & C ONFIDENCE I NTERVALS U SING A TI-83 OR TI-84 C ALCULATOR
Press the STAT button and scroll right to the TESTS menu. Select the appropriate test. Fill in the prompts on the screen. Hypothesis test hints : select the appropriate 𝜇𝜇 option based on your alternative hypothesis, 𝐻𝐻𝑎𝑎, and highlight Draw. Confidence interval hints: From the TESTS menu, make sure you select an “Interval” or “Int” test. Enter the confidence interval as a decimal number (area) NOT a percentage. For the two-sample T interval, make sure NO is highlighted for the Pooled: option.
F INDING THE S TANDARD E RROR : 𝑺𝑺𝒆𝒆 = �
𝐒𝐒𝐒𝐒𝐒𝐒
𝒏𝒏−𝟐𝟐 (TI-83^ OR^84 C^ ALCULATOR^ )
Find the least squares regression equation (see directions above).
Populate L3 with the predicted values from your regression equation, go to L3 and scroll up until L3 is highlighted at the top. Enter your regression equation and substitute L1 for 𝑥𝑥.
Populate L4 with the predicted errors, and populate L5 with the “squared errors” (also known as the “SE”).
L1 L2 L3 L4 L
Values of 𝑥𝑥. The observed values of 𝑦𝑦.
The predicted values of 𝑦𝑦.
The predicted errors (a.k.a. the residuals).
The SE , i.e. the squared errors.
To sum the squared errors in L5; return to the home screen, press the 2 nd^ button and then LIST to access the list menu. Select the MATH submenu. Then sum (. To enter L5, press 2 nd^ and then the number 5. Close the parentheses and press ENTER. You now have SSE (the sum of the squared errors).
You are now ready to use the remainder of the formula, 𝑆𝑆𝑜𝑜 = �^ 𝑛𝑛−2SSE, to find the standard error, 𝑆𝑆𝑜𝑜.
