Stats Formulas and Calculator, Cheat Sheet of Statistics

Download Stats Formulas and Calculator and more Cheat Sheet Statistics in PDF only on Docsity! 1 | M o s t l y H a r m l e s s P r o b a b i l i t y & S t a t i s t i c s - R a c h e l L . W e b b Chapter 3 Formulas Sample Mean = ?̅? = ∑ 𝑥 𝑛 Population Mean = 𝜇 = ∑ 𝑥 𝑁 Weighted Mean = ∑(𝑥𝑤) ∑ 𝑤 Range = Max – Min Sample Standard Deviation = 𝑠 = √ ∑(𝑥−?̅?)2 𝑛−1 Population Standard Deviation = 𝜎 = √ ∑(𝑥−𝜇)2 𝑁 Sample Variance = 𝑠2 = ∑(𝑥−?̅?)2 𝑛−1 Population Variance = 𝜎2 = ∑(𝑥−𝜇)2 𝑁 Coefficient of Variation = CVar = ( 𝑠 ?̅? ∙ 100) % Z-Score = 𝑧 = 𝑥−?̅? 𝑠 Percentile Index = 𝑖 = (𝑛+1)∙𝑝 100 The interquartile range = IQR = Q3 – Q1 Outlier Lower Limit = Q1 – (1.5·IQR) Outlier Upper Limit = Q3 + (1.5·IQR) Empirical Rule: z = 1, 2, 3  68%, 95%, 99.7% Chebyshev’s Inequality = ((1 − 1 (𝑧)2 ) ∙ 100) % TI-84 Enter the data in a list and then press [STAT]. Use cursor keys to highlight CALC. Press 1 or [ENTER] to select 1:1-Var Stats. Press [2nd], then press the number key corresponding to your data list. Press Enter to calculate the statistics. Note: the calculator always defaults to L1 if you do not specify a data list. TI-89 Press [APPS], select FlashApps then press [ENTER]. Highlight Stats/List Editor then press [ENTER]. Press [ENTER] again to select the main folder. To clear a previously stored list of data values, arrow up to the list name you want to clear, press [CLEAR], then press enter. Make sure the cursor is in the list, not on the list name and type the data pressing [ENTER] after each one. Enter all x-values in one list. Press [F4], select 1: 1-Var Stats. To get the list name to the List box, press [2nd] [Var-Link], arrow down to list1 and press [Enter]. This will bring list1 to the List box. Select [Enter] to enter the list name and then enter again to calculate. Use the down arrow key to see all the statistics. Chapter 4 Formulas Sum of Two Dice Standard Deck of Cards + Second Die 1 2 3 4 5 6 F ir st D ie 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 2 | M o s t l y H a r m l e s s P r o b a b i l i t y & S t a t i s t i c s - R a c h e l L . W e b b Complement Rules: 𝑃(𝐴) + 𝑃(𝐴𝐶) = 1 𝑃(𝐴) = 1 − 𝑃(𝐴𝐶) 𝑃(𝐴𝐶) = 1 − 𝑃(𝐴) Mutually Exclusive Events: P(A ∩ B) = 0 Union Rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) Independent Events: P(A ∩ B) = P(A) ∙ P(B) Intersection Rule: P(A ∩ B) = P(A) ∙ P(B|A) Conditional Probability Rule: 𝑃(𝐴|𝐵) = 𝑃(𝐴∩𝐵) 𝑃(𝐵) Fundamental Counting Rule: m1·m2···mn Factorial Rule: n! = n·(n – 1)·(n – 2)···3·2·1 Combination Rule: nCr = 𝑛! (𝑟!(𝑛−𝑟)!) Permutation Rule: nPr = 𝑛! (𝑛−𝑟)! TI-84 Combinations/Permutations: Enter the number “trials” (n) on the home screen. Press [MATH]. Use cursor keys to move to the PRB menu. Press 2 for permutation (2: nPr), 3 for combination (3: nCr). Enter the number of “successes” (r). Press [ENTER] to calculate. TI-89 Combinations/Permutations: Press [2nd] Math > 7:Probability > Press 2 for permutation (2: nPr), 3 for combination (3: nCr). Enter the sample size on the home screen, then a comma, then enter the number of “successes” then end the parenthesis. Press [ENTER] to calculate. Chapter 5 Formulas Discrete Distribution Table: 0 ≤ 𝑃(𝑥) ≤ 1 ∑ P(x) = 1 Discrete Distribution Mean: 𝜇 = ∑(𝑥 ∙ 𝑃(𝑥)) Discrete Distribution Variance: 𝜎2 = ∑(𝑥2 ∙ 𝑃(𝑥)) − 𝜇2 Discrete Distribution Standard Deviation: 𝜎 = √𝜎2 TI-84 Calculator Instructions for Mean, Variance & Standard Deviation Select [STAT] then (1:Edit). Make sure the cursor is in the list, not on the list name and type the desired values pressing [ENTER] after each one. For x and P(x) data pairs, enter all x-values in one list. Enter all corresponding P(x)-values in a second list. Press [STAT]. Use cursor keys to highlight CALC. Press 1 or [ENTER] to select 1:1-Var Stats. Press [2nd], then press the number key corresponding to your x list, then a comma, then [2nd] and the number key corresponding to your P(x) values. (Should look like this 1-Var Stats L1,L2) Press Enter to calculate the statistics. Where the calculator says ?̅? this is µ the population mean and σx is the population standard deviation (square this number to get the population variance). 5 | M o s t l y H a r m l e s s P r o b a b i l i t y & S t a t i s t i c s - R a c h e l L . W e b b Chapter 7 Formulas TI-84: Press the [STAT] key, arrow over to the [TESTS] menu, arrow down to the [A:1-PropZInterval] option and press the [ENTER] key. Then type in the values for X, sample size and confidence level, arrow down to [Calculate] and press the [ENTER] key. The calculator returns the answer in interval notation. Note: sometimes you are not given the x value but a percentage instead. To find the x to use in the calculator, multiply ?̂? by the sample size and round off to the nearest integer. The calculator will give you an error message if you put in a decimal for x or n. For example if ?̂? = 0.22 and n = 124 then 0.22*124 = 27.28, so use x = 27. TI-89: Go to the [Apps] Stat/List Editor, then select 2nd then F7 [Ints], then select 5: 1-PropZInt. Type in the values for X, sample size and confidence level, and press the [ENTER] key. The calculator returns the answer in interval notation. Note: sometimes you are not given the x value but a percentage instead. To find the x value to use in the calculator, multiply ?̂? by the sample size and round off to the nearest integer. The calculator will give you an error message if you put in a decimal for x or n. For example if ?̂? = 0.22 and n = 124 then 0.22*124 = 27.28, so use x = 27. Confidence Interval for One Proportion ?̂? ± 𝑧𝛼/2√( ?̂??̂? 𝑛 ) ?̂? = 𝑥 𝑛 ?̂? = 1 − ?̂? TI-84: 1-PropZInt Sample Size for Proportion: 𝑛 = 𝑝∗ ∙ 𝑞∗ ( 𝑧𝛼 2⁄ 𝐸 ) 2 Always round up to whole number. If p is not given use p* = 0.5. E = Margin of Error Confidence Interval for One Mean Use z-inteval when σ is known. Use t-interval when s is known. If n < 30, population needs to be normal. z-interval ?̅? ± 𝑧𝛼/2 ( 𝜎 √𝑛 ) TI-84: ZInterval Z-Critical Values TI-84: 𝑧𝛼/2 = invNorm(1–area/2,0,1) T-Critical Values TI-84: 𝑡𝛼/2 = invT(1–area/2,df) t-interval, df = n – 1 ?̅? ± 𝑡𝛼 2⁄ ,𝑛−1 ( 𝑠 √𝑛 ) TI-84: TInterval Sample Size for Mean 𝑛 = ( 𝑧𝛼 2⁄ ∙ 𝜎 𝐸 ) 2 Always round up to whole number. TI-84: Press the [STAT] key, arrow over to the [TESTS] menu, arrow down to the [7:ZInterval] option and press the [ENTER] key. Arrow over to the [Stats] menu and press the [ENTER] key. Then type in the population or sample standard deviation, sample mean, sample size and confidence level, arrow down to [Calculate] and press the [ENTER] key. The calculator returns the answer in interval notation. 6 | M o s t l y H a r m l e s s P r o b a b i l i t y & S t a t i s t i c s - R a c h e l L . W e b b TI-89: Go to the [Apps] Stat/List Editor, then select 2nd then F7 [Ints], then select 1: ZInterval. Choose the input method, data is when you have entered data into a list previously or stats when you are given the mean and standard deviation already. Type in the population standard deviation, sample mean, sample size (or list name (list1), and Freq: 1) and confidence level, and press the [ENTER] key to calculate. The calculator returns the answer in interval notation. TI-84: Press the [STAT] key, arrow over to the [TESTS] menu, arrow down to the [8:TInterval] option and press the [ENTER] key. Arrow over to the [Stats] menu and press the [ENTER] key. Then type in the mean, sample standard deviation, sample size and confidence level, arrow down to [Calculate] and press the [ENTER] key. The calculator returns the answer in interval notation. Be careful, if you accidentally use the [7:ZInterval] option you would get the wrong answer. Alternatively (If you have raw data in list one) Arrow over to the [Data] menu and press the [ENTER] key. Then type in the list name, 1L , leave Freq:1 alone, enter the confidence level, arrow down to [Calculate] and press the [ENTER] key. TI-89: Go to the [Apps] Stat/List Editor, then select 2nd then F7 [Ints], then select 2:TInterval. Choose the input method, data is when you have entered data into a list previously or stats when you are given the mean and standard deviation already. Type in the mean, standard deviation, sample size (or list name (list1), and Freq: 1) and confidence level, and press the [ENTER] key. The calculator returns the answer in interval notation. Be careful, if you accidentally use the [1:ZInterval] option you would get the wrong answer. Chapter 8 Formulas Hypothesis Test for One Mean If n < 30, population needs to be normal. Type I Error-Reject H0 when H0 is true. Type II Error-Fail to reject H0 when H0 is false. t-Test: 𝐻0: 𝜇 = 𝜇0 𝐻1: 𝜇 ≠ 𝜇0 𝑡 = ?̅?−𝜇0 ( 𝑠 √𝑛 ) TI-84: T-Test Hypothesis Test for One Proportion 𝐻0: 𝑝 = 𝑝0 𝐻1: 𝑝 ≠ 𝑝0 𝑧 = 𝑝−𝑝0 √( 𝑝0𝑞0 𝑛 ) TI-84: 1-PropZTest p-value rejection rule: Reject H0 when the p-value ≤ alpha Do not reject H0 when the p-value > alpha