Download FORMULA SHEET for- SMAM 319 1. Complement Rule: For any

FORMULA SHEET for- SMAM 319
1. Complement Rule: For any event A, P(Ac )= 1- P(A).
2. Addition Rule: If events A and B are disjoint, then
P(A or B) = P(A ∪ B)= P(A) + P(B)
3. Multiplication Rule: If events A and B are independent, then
P(A and B) = P(∩ B) = P(A)P(B)
4. The probability distribution of a discrete random variable is the pair(x,P(X=x)) where x is a value
of the random variable and P(X=x) is the probability assigned to that value.
5. The mean µX of a discrete random variable X is given by
µX = x1 P (X = x1 ) + x2 P (X = x2 ) + · · · + xn P (X = xn )
where x1 , x2 , · · · , xn are the possible values of the random variable and P (X = xi ) are the corresponding probabilities.
2
, is given by
6. The variance of a discrete random variable σX
2
= (x1 − µX )2 P (X = x1 ) + (x2 − µX )2 P (X = x2 ) + · · · + (xn − µX )2 P (X = xn ).
σX
The standard deviation of a random variable is the square root of the variance.
7. The means and variances obey the following general rules:
µa+bX = a + bµX
2
2
σa+bX
= b2 σX
;
For any two random variables X and Y,
µX+Y = µX + µY ;
For independent random variables X and Y,
2
2
σX+Y
= σX
+ σY2
2
2
σX−Y
= σX
+ σY2
8. If X is the sample count of the number of successes in n independent trials and p is the sample
proportion of the number of successes in the same trials and if π is the probability of success in a
single trial, then
(a) X has a binomial distribution with parameters n and π; X ∼ B(n,π);
r
p
π(1 − π)
(b) µX = nπ; µp = π; σX = nπ(1 − π); σp =
.
n
n!
and n! = n × (n −
k!(n − k)!
1) × (n − 2) × · · · × 1 for any number n. For selected values of n and π, you may also use the
Table of binomial probabilities compute these probabilities.
r
π(1 − π)
9. When nπ is at least 10 and n(1-π) is at least 10, p is approximately N(π,
).
n
(c) P (X = k) =
n
k
π k (1 − π)n−k , for k = 0, 1, · · · n. where
n
k
=
10. The sample mean X̄ of a simple random sample of size n drawn from a population with mean µ
σ
and standard deviation σ has a distribution with mean µX̄ = µ and s.d. σX̄ = √ .
n
11. For large n, X̄ ≈ Normal.(Central limit theorem).
12. Given data x1 , x2 , · · · , xn , the (sample) mean is
x̄ =
x1 + x2 + · · · + xn
n
and the sample variance is
s2 =
(x1 − x̄)2 + (x2 − x̄)2 + · · · + (xn − x̄)2
.
n−1
The sample standard deviation is the square root of the sample variance.
13. A confidence interval for the mean of a normal population with a known standard deviation σ,
based on a random sample of size n and with a confidence level of C is given by
!
σ
σ
x̄ − z ∗ √ , x̄ + z ∗ √
n
n
where z ∗ is the percentile corresponding to the confidence level of C. The probability is C that a
standard normal variable takes values between −z ∗ and z ∗ . If the population standard deviation
is unknown, we could use the above equation with the sample standard deviation replacing the
population standard deviation as long as the sample size is large.
14. The sample size required to obtain a confidence interval of specified margin of error B for a normal
mean is
z ∗ σ 2
n=
B
∗
where z the percentile value for the specified confidence and B is the margin of error. You will
always round-up the above to get the required sample size.
15. When a small sample of size n is drawn from a normal population whose standard deviation is
x̄ − µ
unknown, the statistic √ has a t-distribution with (n-1) degrees of freedom. The percentile z ∗
s/ n
above is replaced by the percentile t∗ from a t distribution with (n-1) degrees of freedom.
16. The confidence interval for a population proportion π is given by
!
r
r
p(1
−
p)
p(1
−
p)
p − z∗
, p + z∗
n
n
where p is the sample proportion in a random sample of size n.
17. The sample size required to get a level C confidence interval for π with a margin of error of B, is
given by
(z ∗ )2 (p∗ (1 − p∗ ))
n=
,
B2
rounded up, where p∗ is a assumed value for the population proportion based on a prior study. If no
such prior information is available, a conservative value for the sample size is obtained by replacing
p∗ with 1/2 to get
(z ∗ )2 (1/4)
,
n=
B2
rounded up.
18. Significance tests for the hypothesis H0 : µ = µ0 concerning the unknown mean µ of a population
are based on the standard normal test statistic
z=
x̄ − µ0
√
σ/ n
where the population s.d. σ is assumed to be known; if the population s.d. is unknown but the
sample is large, the sample s.d., s, may be used in place of σ. Depending on the type of the alternate
hypothesis, P-value is computed as the probability of the test statistic being “extreme”, the extreme
direction determined by the type of alternate hypothesis. P-value is computed using the standard
normal distribution. If a fixed level of significance α is specified, it is compared with the P-value
to decide.
19. If the population s.d. is unknown, and the sample size is not large enough, then the test statistic
to use is
x̄ − µ0
√
t=
s/ n
which has a t distribution with (n-1) degrees of freedom, provided that population has, at least
approximately, a normal distribution. The P-value in this case is computed using the t-distribution
with (n-1) degrees of freedom.
20. For testing the hypothesis H0 : π = π0 about a population proportion, the test statistic to be used
is the standard normal statistic:
p − π0
z=r
π0 (1 − π0 )
n
where again p is the sample proportion. The P-value is computed using the standard normal
distribution.