Question

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 48% of strikes (while trolling) resulted in a catch. Suppose that on a given day a fleet of fishing boats got a total of 26 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 3.4% of the general population will live past their 90th birthday. In a graduating class of 730 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

Suppose the heights of 18-year-old men are approximately normally distributed, with mean 73 inches and standard deviation 2 inches.

(a) What is the probability that an 18-year-old man selected at random is between 72 and 74 inches tall? (Round your answer to four decimal places.)
  

(b) If a random sample of eleven 18-year-old men is selected, what is the probability that the mean height x is between 72and 74 inches? (Round your answer to four decimal places.)


(c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?

The probability in part (b) is much higher because the standard deviation is larger for the x distribution.

The probability in part (b) is much higher because the mean is smaller for the x distribution.

The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.

The probability in part (b) is much lower because the standard deviation is smaller for the x distribution.The probability in part (b) is much higher because the mean is larger for the x distribution.

Answer 1

a)
Mean = Expected Value = μx̅ = n•p = (26)(0.48) = 12.48

n•p•(1 - p) = (26)(0.48)(1 - 0.48) = 6.4896

Standard Deviation = σx̅ = √ n•p•(1 - p) = √6.4896 = 2.5474

Finding probability of getting less than 12 success(es):

Using Binomial Distribution: P(X < 12) = 0.3515

b)

Mean = Expected Value = μx̅ = n•p = (26)(0.48) = 12.48

n•p•(1 - p) = (26)(0.48)(1 - 0.48) = 6.4896

Standard Deviation = σx̅ = √ n•p•(1 - p) = √6.4896 = 2.5474

Finding probability of getting at most 5 success(es):

Using Binomial Distribution: P(X ≤ 5) = 0.0024

c)

Mean = Expected Value = μx̅ = n•p = (26)(0.48) = 12.48

n•p•(1 - p) = (26)(0.48)(1 - 0.48) = 6.4896

Standard Deviation = σx̅ = √ n•p•(1 - p) = √6.4896 = 2.5474

Find probability of getting a number of success(es) between 5 and 12:

Using Binomial Distribution: P(5 ≤ X ≤ 12) = 0.5036

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