Question

A certain flight arrives on time 86 percent of the time. Suppose 112 flights are randomly selected. Use the normal approximation to the binomial to approximate the probability that

(a) exactly 100 flights are on time.

(b) at least 100 flights are on time.

(c) fewer than 99 flights are on time.

(d) between 99 and 106, inclusive are on time.

Answer #1

a)

n= | 112 | p= | 0.8600 | |

here mean of distribution=μ=np= | 96.32 | |||

and standard deviation σ=sqrt(np(1-p))= | 3.6722 | |||

for normal distribution z score =(X-μ)/σx | ||||

therefore from normal approximation of binomial distribution and continuity correction: |

P(exactly 100 flights are on time):

probability = | P(99.5<X<100.5) | = | P(0.87<Z<1.14)= | 0.8729-0.8078= |
0.0651 |

b)

probability = | P(X>99.5) | = | P(Z>0.87)= | 1-P(Z<0.87)= | 1-0.8078= |
0.1922 |

c)

probability = | P(X<98.5) | = | P(Z<0.594)= |
0.7224 |

d)

probability = | P(98.5<X<106.5) | = | P(0.59<Z<2.77)= | 0.9972-0.7224= |
0.2748 |

A certain flight arrives on time
81
percent of the time. Suppose
112
flights are randomly selected. Use the normal approximation to
the binomial to approximate the probability that
(a) exactly
103
flights are on time.
(b) at least
103
flights are on time.
(c) fewer than
96
flights are on time.
(d) between
96
and
101
,
inclusive are on time.

A certain flight arrives on time 81 percent of the time. Suppose
122 flights are randomly selected. Use the normal approximation to
the binomial to approximate the probability that
(a) exactly 110 flights are on time.
(b) at least 110 flights are on time.
(c) fewer than 110 flights are on time.
(d) between 110 and 111, inclusive are on time.

A certain flight arrives on time
83 percent of the time. Suppose 130 flights are randomly
selected. Use the normal approximation to the binomial to
approximate the probability that
(a) exactly 117 flights are on time.
(b) at least 117
flights are on time.
(c) fewer than 96 flights are on time.
(d) between 96 and 108, inclusive are on time.

A certain flight arrives on time 85 percent of the time. Suppose
188 flights are randomly selected. Use the normal approximation to
the binomial to approximate the probability that (a) exactly 160
flights are on time. (b) at least 160 flights are on time. (c)
fewer than 155 flights are on time. (d) between 155 and 166,
inclusive are on time.

A certain flight arrives on time 90 percent of the time. Suppose
166 flights are randomly selected. Use the normal approximation to
the binomial to approximate the probability that (a) exactly 146
flights are on time. (b) at least 146 flights are on time. (c)
fewer than 139 flights are on time. (d) between 139 and 158,
inclusive are on time.

A certain flight arrives on time
81
percent of the time. Suppose
113
flights are randomly selected. Use the normal approximation to
the binomial to approximate the probability that
(a) exactly
89
flights are on time.
(b) at least
89
flights are on time.
(c) fewer than
87
flights are on time.
(d) between
87
and
97
,
inclusive are on time.

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118 flights are randomly selected. Use the normal approximation to
the binomial to approximate the probability that
(a) exactly 95 flights are on time.
(b) at least 95 flights are on time.
c) fewer than 107 flights are on time.
(d) between 107 and 108, inclusive are on time.

A certain flight arrives on time 90 percent of the time. Suppose
190 flights are randomly selected. Use the normal approximation to
the binomial to approximate the probability that
(a) exactly 163 flights are on time.
(b) at least 163 flights are on time.
(c) fewer than 182 flights are on time.
(d) between 182 and 183, inclusive are on time.

A certain flight arrives on time 89 percent of the time. Suppose
135 flights are randomly selected. Use the normal approximation to
the binomial to approximate the probability that (a) exactly 127
flights are on time. (b) at least 127 flights are on time. (c)
fewer than 117 flights are on time. (d) between 117 and 129,
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183 flights are randomly selected. Use the normal approximation to
the binomial to approximate the probability that
(a) exactly 147 flights are on time.
(b) at least 147 flights are on time.
(c) fewer than 166 flights are on time.
(d) between 166 and 173, inclusive are on time.

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