Question

Find to 4 decimal places the following binomial probabilities using the normal approximation.

a. n = 130, p = 0.42, P(x = 77)

P(x = 77) =

b. n = 100, p = 0.57, P(52 ≤ x ≤ 61)

P(52 ≤ x ≤ 61) =

c. n = 90, p = 0.41, P(x ≥ 38)

P(x ≥ 38) =

d. n = 103, p = 0.75, P(x ≤ 75)

P(x ≤ 75) =

Answer 1

a)
This is a binomial distribution question with
n = 130
p = 0.42
q = 1 - p = 0.58
This binomial distribution can be approximated as Normal distribution since
np > 5 and nq > 5

Since we know that


P(X = 77.0) = ?
For a continous the probability is the integration of probability density function in an given interval. Since if we give a particular point as an interval the integration comes out as 0.
P(X = 77.0) = 0

b)
This is a binomial distribution question with
n = 100
p = 0.57
q = 1 - p = 0.43
This binomial distribution can be approximated as Normal distribution since
np > 5 and nq > 5
Since we know that

c)
This is a binomial distribution question with
n = 90
p = 0.41
q = 1 - p = 0.59
This binomial distribution can be approximated as Normal distribution since
np > 5 and nq > 5

Since we know that




The z-score at x = 37.5 is,

This implies that

d)
This is a binomial distribution question with
n = 103
p = 0.75
q = 1 - p = 0.25

This binomial distribution can be approximated as Normal distribution since
np > 5 and nq > 5

Since we know that

The z-score at x = 75.5 is,

This implies that

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