Question

Let X have a binomial distribution with parameters

n = 25

and p. Calculate each of the following probabilities using the normal approximation (with the continuity correction) for the cases

p = 0.5, 0.6, and 0.8

and compare to the exact binomial probabilities calculated directly from the formula for

b(x; n, p).

(Round your answers to four decimal places.)

(a)

P(15 ≤ X ≤ 20)

p	P(15 ≤ X ≤ 20)	P(14.5 ≤ Normal ≤ 20.5)
0.5	1	2
0.6	3	4
0.8	5	6

(b)

P(X ≤ 15)

p	P(X ≤ 15)	P(Normal ≤ 15.5)
0.5	10	11
0.6	12	13
0.8	14	15

(c)

P(20 ≤ X)

p	P(20 ≤ X)	P(19.5 ≤ Normal)
0.5	19	20
0.6	21	22
0.8	23	24

Answer 1

Mean and standard deviation for normal:

For 0.5:

Mean, µ = n*p = 25 * 0.5 = 12.5

Standard deviation, σ = √(n*p*(1-p)) = √(25 * 0.5 * 0.5) = 2.5

For 0.6:

Mean, µ = n*p = 25 * 0.6 = 15

Standard deviation, σ = √(n*p*(1-p)) = √(25 * 0.6 * 0.4) = 2.4495

For 0.8:

Mean, µ = n*p = 25 * 0.8 = 20

Standard deviation, σ = √(n*p*(1-p)) = √(25 * 0.8 * 0.2) = 2

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(a) using Binomial:

For 0.5:

Probability between 15 and 20, P(15 ≤ X ≤ 20) = P(X ≤ 20) - P(X < 15) =

= P(X ≤ 20) - P(X ≤ 14)

= BINOM.DIST(20, 25, 0.5, 1) - BINOM.DIST(14, 25, 0.5, 1)

= 0.2117

For 0.6:

Probability between 15 and 20, P(15 ≤ X ≤ 20) = P(X ≤ 20) - P(X < 15) =

= P(X ≤ 20) - P(X ≤ 14)

= BINOM.DIST(20, 25, 0.6, 1) - BINOM.DIST(14, 25, 0.6, 1)

= 0.5763

For 0.8:

Probability between 15 and 20, P(15 ≤ X ≤ 20) = P(X ≤ 20) - P(X < 15) =

= P(X ≤ 20) - P(X ≤ 14)

= BINOM.DIST(20, 25, 0.8, 1) - BINOM.DIST(14, 25, 0.8, 1)

= 0.5738

Using the normal approximation (with the continuity correction)

For 0.5:

P(15 ≤ X ≤ 20)

Using continuity correction :

P(15-0.5 ≤ X ≤ 20+0.5)

= P((14.5 - 12.5)/2.5 ≤ (X - µ)/σ ≤ (20.5 - 12.5)/2.5)

= P( 0.8 ≤ z ≤ 3.2)

= P(z < 3.2) - P(z < 0.8)

Using excel function:

= NORM.S.DIST(3.2, 1) - NORM.S.DIST(0.8, 1)

= 0.2112

For 0.6:

P(15 ≤ X ≤ 20)

Using continuity correction :

P(15-0.5 ≤ X ≤ 20+0.5)

= P((14.5 - 15)/2.4495 ≤ (X - µ)/σ ≤ (20.5 - 15)/2.4495)

= P( -0.2041 ≤ z ≤ 2.2454)

= P(z < 2.2454) - P(z < -0.2041)

Using excel function:

= NORM.S.DIST(2.2454, 1) - NORM.S.DIST(-0.2041, 1)

= 0.5685

For 0.8:

P(15 ≤ X ≤ 20)

Using continuity correction :

P(15-0.5 ≤ X ≤ 20+0.5)

= P((14.5 - 20)/2 ≤ (X - µ)/σ ≤ (20.5 - 20)/2)

= P( -2.75 ≤ z ≤ 0.25)

= P(z < 0.25) - P(z < -2.75)

Using excel function:

= NORM.S.DIST(0.25, 1) - NORM.S.DIST(-2.75, 1)

= 0.5957

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(b) for binomial

For 0.5:

Probability of less than or equal to 15, P(X ≤ 15) =

= BINOM.DIST(15, 25, 0.5, 1)

= 0.8852

For 0.6:

Probability of less than or equal to 15, P(X ≤ 15) =

= BINOM.DIST(15, 25, 0.6, 1)

= 0.5754

For 0.8:

Probability of less than or equal to 15, P(X ≤ 15) =

= BINOM.DIST(15, 25, 0.8, 1)

= 0.0173

Using the normal approximation (with the continuity correction)

For 0.5:

P(X ≤ 15)

Using continuity correction :

P(X ≤ 15+0.5)

= P((X - µ)/σ ≤ (15.5 - 12.5)/2.5)

= P(z ≤ 1.2)

Using excel function:

= NORM.S.DIST(1.2, 1)

= 0.8849

For 0.6:

P(X ≤ 15)

Using continuity correction :

P(X ≤ 15+0.5)

= P((X - µ)/σ ≤ (15.5 - 15)/2.4495)

= P(z ≤ 0.2041)

Using excel function:

= NORM.S.DIST(0.2041, 1)

= 0.5809

For 0.8:

P(X ≤ 15)

Using continuity correction :

P(X ≤ 15+0.5)

= P((X - µ)/σ ≤ (15.5 - 20)/2)

= P(z ≤ -2.25)

Using excel function:

= NORM.S.DIST(-2.25, 1)

= 0.0122

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(c) For 0.5:

Probability of greater than or equal to 20, P(X ≥ 20) =

= 1 - BINOM.DIST(19, 25, 0.5, 1)

= 0.002

For 0.6:

Probability of greater than or equal to 20, P(X ≥ 20) =

= 1 - BINOM.DIST(19, 25, 0.6, 1)

= 0.0294

For 0.8:

Probability of greater than or equal to 20, P(X ≥ 20) =

= 1 - BINOM.DIST(19, 25, 0.8, 1)

= 0.6167

Using the normal approximation (with the continuity correction)

For 0.5:

P(X ≥ 20)

Using continuity correction :

P(X ≥ 20-0.5)

= P((X - µ)/σ ≥ (19.5 - 12.5)/2.5)

= P(z ≥ 2.8)

= 1 - P(z < 2.8)

Using excel function:

= 1 - NORM.S.DIST(2.8, 1)

= 0.0026

For 0.6:

P(X ≥ 20)

Using continuity correction :

P(X ≥ 20-0.5)

= P((X - µ)/σ ≥ (19.5 - 15)/2.4495)

= P(z ≥ 1.8371)

= 1 - P(z < 1.8371)

Using excel function:

= 1 - NORM.S.DIST(1.8371, 1)

= 0.0331

For 0.8:

P(X ≥ 20)

Using continuity correction :

P(X ≥ 20-0.5)

= P((X - µ)/σ ≥ (19.5 - 20)/2)

= P(z ≥ -0.25)

= 1 - P(z < -0.25)

Using excel function:

= 1 - NORM.S.DIST(-0.25, 1)

= 0.5987