Answered: One year, demand for sperm whale oil…

College Algebra

10th Edition

ISBN:9781337282291

Author:Ron Larson

Publisher:Ron Larson

Chapter8: Sequences, Series,and Probability

Section8.7: Probability

Problem 11ECP: A manufacturer has determined that a machine averages one faulty unit for every 500 it produces....

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Question

One year, demand for sperm whale oil was particularly high and Captain Ahab decided to hunt only
those whales. The length of a sperm whale in the 1840s was normally distributed with mean 16.1
meters and standard deviation 1.5 meters.
(a) What is the probability that the first four whales they catch are all longer than 16.4
meters?
(b) If the selection and capture of the whales is random, what is the probability that the
mean length of the first 25 whales they capture is at least 16.3 meters.
(c) Upon return to Nantucket, Captain Ahab reports how many whales they caught, and
that the mean length of sperm whales they caught was 16.3 meters. His colleague Captain Bildad
exclaims, “But there’s less than a 1% chance of getting a mean that high or higher if the true
population mean and standard deviation are what we think they are!”
Approximately how many whales did Ahab tell Bildad they caught?
(d) The next year, they catch 123 whales with a mean length of 16.4 meters, and a
standard deviation of 1.4 meters. Based in this sample, find a 95%-confidence interval for the true
population mean length of sperm whales

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Expert Solution

Step 1

"Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for you. To get remaining sub-part solved please repost the complete question and mention the sub-parts to be solved."

(a)

The probability that the first four whales they catch are all longer than 16.4 meters is,

Thus, the probability that the first four whales they catch are all longer than 16.4 meters is 0.3446.

Step 2

(b)

The probability that the mean length of first 25 whales they capture is at least 16.3 meters is,

Thus, the probability that the mean length of first 25 whales they capture is at least 16.3 meters is 0.2524.

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