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figts, inclusive, are on time.22. Morality In a recent poll, the Gallup Organization foundthat 45% of adult Americans believe that the overall state ofmoral values in the United States is poor. Supposea random sample of 500 adult Americans is conducted in whichthey are asked to disclose their feelings on the overall state ofmoral values in the United States. Use the normal approximationto the binomial to approximate the probability that(a) exactly 250 of those surveyed feel the state of morals isa survey ofpoor.(b) no more than 220 of those surveyed feel the state of moralsis poor.(c) more than 250 of those surveyed feel the state of morals ispoor.

Question
figts, inclusive, are on time.
22. Morality In a recent poll, the Gallup Organization found
that 45% of adult Americans believe that the overall state of
moral values in the United States is poor. Suppose
a random sample of 500 adult Americans is conducted in which
they are asked to disclose their feelings on the overall state of
moral values in the United States. Use the normal approximation
to the binomial to approximate the probability that
(a) exactly 250 of those surveyed feel the state of morals is
a survey of
poor.
(b) no more than 220 of those surveyed feel the state of morals
is poor.
(c) more than 250 of those surveyed feel the state of morals is
poor.
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figts, inclusive, are on time. 22. Morality In a recent poll, the Gallup Organization found that 45% of adult Americans believe that the overall state of moral values in the United States is poor. Suppose a random sample of 500 adult Americans is conducted in which they are asked to disclose their feelings on the overall state of moral values in the United States. Use the normal approximation to the binomial to approximate the probability that (a) exactly 250 of those surveyed feel the state of morals is a survey of poor. (b) no more than 220 of those surveyed feel the state of morals is poor. (c) more than 250 of those surveyed feel the state of morals is poor.

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Step 1

Normal approximation to the binomial:

From the given information, it is clear that n = 500

The probability of success, p = 0.45.

The probability of failure, q = 1 – 0.45 = 0.55

Since np (225) and nq (275) are greater than 5, normal approximation to the binomial can be done.

Step 2

Mean and standard deviation:

The mean and standard deviation are calculated as follows.

Mean, u %3D пхр
= 500 x 0.45
- 225
Standard deviation, a = Vnx px q
= V500 x0.45x 0.55
=11.124
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Mean, u %3D пхр = 500 x 0.45 - 225 Standard deviation, a = Vnx px q = V500 x0.45x 0.55 =11.124

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Step 3

(a) Computation of probability that exactly 250 of those surveyed feed the state of morals is poor:

The probability that exactly 250 of those surveyed...

P(X 250) P(249.5 <X <250.5) [By continuity correction factor]
249.5-225
250.5 225
<Z <
11.124
11.124
= P(2.2024 < Z <2.2923)
=P(Z < 2.2923)- P(Z <2.2024)
using EXCEL formula
P(Z 2.2923)= 0.9890
"=NORM.S.DIST(2.2923, TRUE)"
P(Z<2.2024)0.9861 using EXCEL formula
"=NORM.S.DIST(2.2024, TRUE)"
P(X 250) 0.9890 0.9861
=0.0029
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P(X 250) P(249.5 <X <250.5) [By continuity correction factor] 249.5-225 250.5 225 <Z < 11.124 11.124 = P(2.2024 < Z <2.2923) =P(Z < 2.2923)- P(Z <2.2024) using EXCEL formula P(Z 2.2923)= 0.9890 "=NORM.S.DIST(2.2923, TRUE)" P(Z<2.2024)0.9861 using EXCEL formula "=NORM.S.DIST(2.2024, TRUE)" P(X 250) 0.9890 0.9861 =0.0029

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