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Statistics Formula Sheet for Chapters 4, 5, 6 Discrete Random Variables 1. For discrete random variable X, the mean (or expected value of X) is µ = E(X) = ∑ 2. The variance of X is σ2 = ∑ , and the standard deviation is σ = √ . 3. To find the mean and standard deviation of a discrete random variable with a calculator, put the values of X in L1 and the probabilities in L2. Go to STAT CALC 1:1-Var Stats. After 1-Var Stats, type L1, L2 and ENTER. 4. Binomial Random Variable X: (i) Only two possible outcomes, labeled “success” and “failure” (ii) n independent trials (iii) The probability of success for each trial is p; the probability of failure is q = 1 – p. 5. For Binomial Random Variable X: Calculator shortcut for P(x = r) = P(r): Go to 2nd [ binompdf(n, p, r). Calculator shortcut for : Go to 2nd [ ] , and choose 0 (or A): binompdf. Then enter ], and choose A (or B): binomcdf. Then enter binomcdf(n, p, r). Formula for Calculator shortcut for nCr : Type n, then go to MATH PRB 3:nCr, type r, and ENTER 6. The Binomial Random Variable X has mean µ = np, variance σ2 = npq, and standard deviation σ=√ . Normal Distributions 7. Areas under normal curves on a TI83/84: Go to 2nd 2: normalcdf( Normalcdf(left endpoint, right endpoint, mean, standard deviation) (You may use 10000 for infinity.) 8. Find a percentile (that is, find an X-value given the area to the left of X) on a TI83/84: Go to 2nd 3: invNorm( invNorm(area to left, mean, standard deviation) 9. Central Limit Theorem If samples are drawn from a population with population mean, µ, and population standard deviation, σ, and if either (i) the population itself is normally distributed or (ii) the sample size n 30, then the sampling distribution of the sample means ( ̅ ) is (approximately) normally distributed, and ̅ ̅ √ 10. Sampling Distribution of the sample proportion. If (where p is the population proportion, and q = 1 – p), then the sampling distribution of the sample proportions ( ̂ ) is (approximately) normally distributed, and ̂ Confidence Intervals 11. Critical values for the normal distribution: Confidence Level 0.80 0.90 0.95 0.99 zc 1.28 1.645 1.96 2.575 ̂ √ 12. Confidence Interval for the population mean: (a) If n 30, then the endpoints of the confidence interval are ̅ √ When σ is unknown, use the sample standard deviation, s On a TI83/84, use STAT TESTS 7: ZInterval (b) If n < 30 but the population is normally (or approximately normally) distributed, then the endpoints of the confidence interval are ̅ with n – 1 degrees of freedom. √ On a TI83/84, use STAT TESTS 8: TInterval 13. Minimum sample size needed to estimate the population mean for a given c-confidence level and margin of error, E. ( ) (You may use the sample standard deviation, s, if σ is unknown and you have a preliminary sample with n at least 30.) 14. Confidence Interval for the population proportion: If ̂ and ̂ , then the endpoints of the confidence interval are √ ̂ ̂̂ On a TI83/84, use STAT TESTS A:1-PropZInt 15. Minimum sample size to estimate the population proportion for a given c-confidence level and margin of error, E. (a) If you have preliminary estimates for ̂ and ̂: ̂̂( ) (b) If you don’t have preliminary estimates for ̂ and ̂ : ( )