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In the following problem, check that it is appropriate to use the normal approximation to the...

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

Ocean fishing for billfish is very popular in the Cozumel region of Mexico. In the Cozumel region about 43% of strikes (while trolling) resulted in a catch. Suppose that on a given day a fleet of fishing boats got a total of 30 strikes. Find the following probabilities. (Round your answers to four decimal places.)

(a) 12 or fewer fish were caught
(b) 5 or more fish were caught

(c) between 5 and 12 fish were caught

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

It is estimated that 3.9% of the general population will live past their 90th birthday. In a graduating class of 741 high school seniors, find the following probabilities. (Round your answers to four decimal places.)

(a) 15 or more will live beyond their 90th birthday

(b) 30 or more will live beyond their 90th birthday

(c) between 25 and 35 will live beyond their 90th birthday

(d) more than 40 will live beyond their 90th birthday

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Answer #1

1)

p = 0.43, n = 30

mean = np = 30 * 0.43 = 12.90

sd = sqrt(npq) = sqrt( 30 * 0.43 * 0.57) = 2.71

a)
P(x <12)

z = (x -mean)/s
= ( 12 - 12.9)/2.71
= -0.33

P(x <12) = P(z < -0.33) = 0.3707

b)
P(x >5)

z = (x -mean)/s
= ( 5- 12.9)/2.71
= -2.92

P(x > 5) = P(z > -2.92) = 0.9982

c)

P(5 < x < 12)

= P(( 5- 12.9)/2.71 < z < ( 12 - 12.9)/2.71)
= P(-0.33 < z < -2.92)
= 0.3689

2)

p = 0.039 , n = 741

mean = np = 741 * 0.039 = 28.899

sd = sqrt(npq) = sqrt(741 *0.039 *0.961) = 5.2699

a)

P(x >15)

z = (x -mean)/s
= (15 - 28.899)/5.2699
= -2.6374

P(x >15) = P(z > -2.6374) = 0.9958

b)
P(x >30)

z = (x -mean)/s
= (30 - 28.899)/5.2699
= 0.2089

P(x >30) = P(z > 0.2089) = 0.4173

c)

P(25 < x <35)

= P((25 - 28.899)/5.2699 <z < (35 - 28.899)/5.2699)
= P(-0.7399 < z < 1.1577)
= 0.6468

d)

P(x >40)

z = (x -mean)/s
= (40 - 28.899)/5.2699
= 2.1065

P(x >40) = P(z > 2.1065) = 0.0176

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