Question

The normal approximation of the binomial distribution is appropriate when

*np* ≥ 5.

*n*(1 − *p*) ≥ 5.

*np* ≤ 5.

*n*(1 −
*p*) ≤ 5 and *np* ≤ 5.

*np* ≥ 5 and *n*(1 − *p*) ≥ 5.

Answer #1

np ≥ 5 and n(1 − p) ≥ 5.

By using some mathematics it can be shown that there are a few
conditions that we need to use a normal approximation to the
binomial distribution. The number of observations *n* must
be large enough, and the value of *p* so that both
*np* and *n*(1 - *p*) are greater than or
equal to 5. This is a rule of thumb, which is guided by statistical
practice. The normal approximation can always be used, but if these
conditions are not met then the approximation may not be that good
of an approximation.

The normal approximation of the binomial distribution is
appropriate when:
A. np 10
B. n(1–p) 10
C. np ≤ 10
D. np(1–p) ≤ 10
E. np 10 and n(1–p) 10

If np≥5 and nq≥5, estimate P(fewer than 6) with n=13 and p=0.6
by using the normal distribution as an approximation to the
binomial distribution; if np<5 or nq<5, then state that the
normal approximation is not suitable.

1. Normal Approximation to Binomial Assume
n = 10, p = 0.1.
a. Use the Binomial Probability function to compute the P(X =
2)
b. Use the Normal Probability distribution to approximate the
P(X = 2)
c. Are the answers the same? If not, why?

Suppose that x has a binomial distribution with n
= 202 and p = 0.47. (Round np and n(1-p) answers
to 2 decimal places. Round your answers to 4 decimal places. Round
z values to 2 decimal places. Round the intermediate value (σ) to 4
decimal places.)
(a) Show that the normal approximation to the
binomial can appropriately be used to calculate probabilities about
x
np
n(1 – p)
Both np and n(1 – p) (Click to select)≥≤
5
(b)...

Suppose that x has a binomial distribution with n = 199 and p =
0.47. (Round np and n(1-p) answers to 2 decimal places. Round your
answers to 4 decimal places. Round z values to 2 decimal places.
Round the intermediate value (σ) to 4 decimal places.) (a) Show
that the normal approximation to the binomial can appropriately be
used to calculate probabilities about x. np n(1 – p) Both np and
n(1 – p) (Click to select) 5 (b)...

5) If it is appropriate to do so, use the
normal approximation to the p^-distribution to calculate
the indicated probability:
n=60,p=0.40n=60,p=0.40
P( p̂ < 0.50)= ?
Enter 0 if it is not appropriate to do so.

If np greater than or equals 5(np≥5) and nq greater than or
equals 5(nq≥5), estimate P(fewer than 4) with nequals=13 and p
equals=0.5 by using the normal distribution as an approximation to
the binomial distribution; if np<5 or nq<5, then state that
the normal approximation is not suitable.

A binomial distribution has p? = 0.26 and n? = 76. Use the
normal approximation to the binomial distribution to answer parts
?(a) through ?(d) below.
?a) What are the mean and standard deviation for this?
distribution?
?b) What is the probability of exactly 15 ?successes?
?c) What is the probability of 14 to 23 ?successes?
?d) What is the probability of 11 to 18 ?successes

Suppose X is binomial random variable with n = 18 and p = 0.5.
Since np ≥ 5 and n(1−p) ≥ 5, please use binomial distribution to
find the exact probabilities and their normal approximations. In
case you don’t remember the formula, for a binomial random variable
X ∼ Binomial(n, p), P(X = x) = n! x!(n−x)!p x (1 − p) n−x . (a) P(X
= 14). (b) P(X ≥ 1).

Normal Approximation to Binomial
Assume n = 100, p = 0.4.
Use the Binomial Probability function to compute the P(X =
40)
Use the Normal Probability distribution to approximate the P(X
= 40)
Are the answers the same? If not, why?

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