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The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset Vaia Explanations$ Math

6th Edition · 837 pages

ISBN: 9781319113339

Chapter 6: Q. 7. (page 367)

Get on the boat! Refer to Exercise 3. Find the mean of Y. Interpret this value.

Short Answer

Expert verified

$µ = 19.35$

Step by step solution

Step 1. Given information.

Money collected	0	5	10	15	20	25
Probability	0.02	0.05	0.08	0.16	0.27	0.42

Step 2. The mean of Y.

The expected value (or mean) is calculated by multiplying each possibility y by its probability P(Y = y):

$µ = Σ y P (Y = y) = 0 x 0.02 + 5 x 0.05 + 10 x 0.08 + 15 x 0.16 + 20 x 0.27 + 25 x 0.42 µ = 19.35$

The average amount of money collected on a random ferry trip is $19.35.

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Most popular questions from this chapter

Take a spin An online spinner has two colored regions_blue and yellow. According to the website, the probability that the spinner lands in the blue region on any spin is $0.80$ . Assume for now that this claim is correct. Suppose we spin the spinner 12 times and let $X$ $=$ the number of times it lands in the blue region.

a. Explain why $X$ is a binomial random variable.

b. Find the probability that exactly 8 spins land in the blue region.

Get on the boat! A small ferry runs every half hour from one side of a large river to the other. The probability distribution for the random variable Y= money collected on a randomly selected ferry trip is shown here. From Exercise 7, $μ Y = $ 19.35$ .

(a) Find the median of Y.

(b) Compare the mean and median. Explain why this relationship makes sense based on the probability distribution.

Exercises 21 and 22 examine how Benford’s law (Exercise 9) can be used to detect fraud.

Benford’s law and fraud A not-so-clever employee decided to fake his monthly expense report. He believed that the first digits of his expense amounts should be equally likely to be any of the numbers from 1 to 9. In that case, the first digit Yof a randomly selected expense amount would have the probability distribution shown in the histogram.

(a) What’s $P (Y < 6)$ ? According to Benford’s law (see Exercise 9), what proportion of first digits in the employee’s expense amounts should be greater than 6? How could this information be used to detect a fake expense report?

(b) Explain why the mean of the random variable Yis located at the solid red line in the figure.

(c) According to Benford’s law, the expected value of the first digit is $μ X = 3.441$ . Explain how this information could be used to detect a fake expense report.

During the winter months, the temperatures at the Starneses’ Colorado cabin can stay well below freezing $(32 ° F o r 0 ° C)$ for weeks at a time. To prevent the pipes from freezing, Mrs. Starnes sets the thermostat at $50 ° F .$ She also buys a digital thermometer that records the indoor temperature each night at midnight. Unfortunately, the thermometer is programmed to measure the temperature in degrees Celsius. Based on several years’ worth of data, the temperature $T$ in the cabin at midnight on a randomly selected night can be modeled by a Normal distribution with mean $8.5 ° C$ and standard deviation $2.25 ° C$ . Let $Y =$ the temperature in the cabin at midnight on a randomly selected night in degrees Fahrenheit (recall that $F = (9 / 5) C + 32)$ .

a. Find the mean of $Y$ .

b. Calculate and interpret the standard deviation of $Y .$

c. Find the probability that the midnight temperature in the cabin is less than $40 ° F$ .

Quick, click! An Internet reaction time test asks subjects to click their mouse button as soon as a light flashes on the screen. The light is programmed to go on at a randomly selected time after the subject clicks “Start.” The density curve models the amount of time Y (in seconds) that the subject has to wait for the light to flash.

a) Find and interpret $P (Y > 3.75)$

b) What is $μ Y$ ? Explain your answer.

c) Find the value of k that makes this statement true: $P (Y \leq k) = 0.38$

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Get on the boat! Refer to Exercise 3. Find the mean of Y. Interpret this value.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The mean of Y.

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Most popular questions from this chapter

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