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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 ̂ :
( )