MATH 115,

PROBABILITY & STATISTICS

Ch. 5.1 - HOMEWORK (SOLUTIONS)

Section 5.1

5.9 Generating a sampling distribution.the case of a very small sample from a very small population. The population is the 10 scholarship players currently on your men’s basketball team. For convenience, the 10 players have been labeled with the integers 0 to 9. For each player, the total amount of time spent (in minutes) on Facebook during the last month is recorded in the table below.

The parameter of interest is the average amount of time on Facebook. The sample is an SRS of size n = 3 drawn from this population of players. Because the players are labeled 0 to 9, a single random digit from Table B chooses one player for the sample.

(a) Find the mean of the 10 p(b) Use Table B to draw an SRS of size 3 from this population (Note: you may sample the

same player’s time more than once).calculate the sample mean

(c) Repeat this process 10 times using different parts of values of . You are constructing the sampling distribution of

(d) Is the center of your histogram close to repeated this sampling process? Explain.

Solution

(a) Mean time for ten players =338.8

)

distribution. Let’s illustrate the idea of a sampling distribution in the case of a very small sample from a very small population. The population is the 10 scholarship players currently on your men’s basketball team. For convenience, the 10 players

ith the integers 0 to 9. For each player, the total amount of time spent (in minutes) on Facebook during the last month is recorded in the table below.

The parameter of interest is the average amount of time on Facebook. The sample is an SRS of 3 drawn from this population of players. Because the players are labeled 0 to 9, a single

chooses one player for the sample.

Find the mean of the 10 players in the population. This is the population mean to draw an SRS of size 3 from this population (Note: you may sample the

same player’s time more than once). Write down the three times in your sample and calculate the sample mean . This statistic is an estimate of µ. Repeat this process 10 times using different parts of Table B. Make a histogram of the 10

. You are constructing the sampling distribution of . Is the center of your histogram close to µ? Would it get closer to µ the more times you repeated this sampling process? Explain.

8.8

N. PSOMAS

Let’s illustrate the idea of a sampling distribution in the case of a very small sample from a very small population. The population is the 10 scholarship players currently on your men’s basketball team. For convenience, the 10 players

ith the integers 0 to 9. For each player, the total amount of time spent (in

The parameter of interest is the average amount of time on Facebook. The sample is an SRS of 3 drawn from this population of players. Because the players are labeled 0 to 9, a single

layers in the population. This is the population mean µ. to draw an SRS of size 3 from this population (Note: you may sample the

Write down the three times in your sample and

ake a histogram of the 10

the more times you

(b) , (c), (d)

Average of 10 sample means =338.0

5.10 Total sleep time of college college students was approximately Normally distributed with mean deviation σ = 1.15 hours. Suppose you plan to take an SRS of size average total sleep time.

(a) What is the standard deviation for the average time?(b) Use the 95 part of the 68–(c) What is the probability that your average will be below 6.9 hours?

Solution

(a) 1.15/sqrt(200) = 0.078842

(b) 95% of sample means fall in the interval

µ±2σ/sqrt(200) or 7.02 ± 2*(0.078842) = (6.8623, 7.1777)

(c) P[x-bar < 6.9] = 0.064002 (why?)

Sample 1 Sample 2 Sample 3 Sample 4

370 319 319

366 358 327

309 370 327

348.33 349.00 324.33

338.0

students. In Example 5.1, the total sleep time per night among college students was approximately Normally distributed with mean µ = 7.02 hours and standard

= 1.15 hours. Suppose you plan to take an SRS of size n = 200 and compute the

the standard deviation for the average time? –95–99.7 rule to describe the variability of this sample mean.

What is the probability that your average will be below 6.9 hours?

ple means fall in the interval

0.078842) = (6.8623, 7.1777)

(why?)

Sample 4 Sample 5 Sample 6 Sample 7 Sample 8 Sample 9

358 290 319 327 323

358 370 368 327 309

358 323 319 358 290

358.00 327.67 335.33 337.33 307.33

, the total sleep time per night among = 7.02 hours and standard

= 200 and compute the

99.7 rule to describe the variability of this sample mean.

Sample 9 Sample 10

309 370

309 366

358 366

325.33 367.33

5.11 Determining sample size. Recall the previous exercise. Suppose you want to use a sample size such that about 95% of the averages fall within ±5 minutes of the true mean µ = 7.02.

(a) Based on your answer to part (b) in Exercise 5.8, should the sample size be larger or smaller than 200? Explain.

(b) What standard deviation of the average do you need such that about 95% of all samples will have a mean within 5 minutes of µ?

(c) Using the standard deviation calculated in part (b), determine the number of students you need to sample.

Solution

(a) Larger

(b) 5' = (5/60) hr = 0.083333 hrs ==> sample mean SD should be 1/2 of 0.083333

(c) 2*(1.15)/sqrt(n) = 0.083333 == > sqrt(n) = 2*(1.15)/ 0.083333 ==> n = (27.6)^2 = 761.8 or 762

5.12 Songs on an iPod. An iPod has about 10,000 songs. The distribution of the play time for these songs is highly skewed. Assume that the standard deviation for the population is 280 seconds.

(a) What is the standard deviation of the average time when you take an SRS of 10 songs from this population?

(b) How many songs would you need to sample if you wanted the standard deviation of to be 15 seconds?

Solution

(a) 280/sqrt(10) = 88.544 sec

(b) 280/sqrt(n) = 15 sec ==> sqrt(n) = 280/15 = 18.66667 ==> n = (18.7)^2 = 348.4444 or 349 songs

5.13 Bottling an energy drink. A bottling company uses a filling machine to fill cans with an energy drink. The cans are supposed to contain 250 milliliters (ml). The machine, however, has some variability, so the standard deviation of the size is σ = 3 ml. A sample of 6 cans is inspected each hour for process control purposes, and records are kept of the sample mean volume. If the process mean is exactly equal to the target value, what will be the mean and standard deviation of the numbers recorded?

Solution

mean of x-bar = 250 ml & SD of x-bar = 3/sqrt(6) = 1.224745

5.14 Play times for songs on anindividual measurements. Suppose the true mean duration of the play time for the songs in the iPod of Exercise 5.12 is 350 seconds.

(a) Sketch on the same graph the two Normal curves, for sampling a single song and for the mean of 10 songs.

(b) What is the probability that the sample mean differs from the population mean by more than 19 seconds when only 1 song is sampled?

(c) How does the probability tha10 songs?

Solution

(a)

(b) Find P[ x < 350 - 19 OR x > 350 + 19]

= P[ x < 350 - 19] + P[ x > 350 + 19]

(c) Find P[ < 350 - 19 OR > 350 + 19]

= P[ < 350 - 19] + P[ > 350 + 19]

an iPod. Averages of several measurements are less variable than individual measurements. Suppose the true mean duration of the play time for the songs in the

is 350 seconds.

n the same graph the two Normal curves, for sampling a single song and for the

What is the probability that the sample mean differs from the population mean by more than 19 seconds when only 1 song is sampled? How does the probability that you calculated in part (b) change for the mean of an SRS of

19 OR x > 350 + 19] ; Here X=N(350, 280)

19] + P[ x > 350 + 19] = ..... = 0.94586 or 94.6%

> 350 + 19] ; Here X-bar = N(350, 280/sqt(10))

> 350 + 19] = ..... =0 .8301 or 83.0%

Averages of several measurements are less variable than individual measurements. Suppose the true mean duration of the play time for the songs in the

n the same graph the two Normal curves, for sampling a single song and for the

What is the probability that the sample mean differs from the population mean by more

t you calculated in part (b) change for the mean of an SRS of

5.15 Can volumes. Averages are less variable than individual observations. Suppose that the can volumes in Exercise 5.13 vary according to a Normal distribution. In that case, the mean of an SRS of cans also has a Normal distribution.

(a) Make a sketch of the Normal curve for a single can. Add the Normal curve for the mean of an SRS of 6 cans on the same sketch.

(b) What is the probability that the volume of a single randomly chosen can differs from the target value by 1 ml or more?

(c) What is the probability that the mean volume of an SRS of 6 cans differs from the target value by 1 ml or more?

Solutions

Work like in the previous exercise (5.14)

5.16 Number of friends on Facebook. Facebook provides a variety of statistics on their Web site that detail the growth and popularity of the site. One such statistic is that the average user has 130 friends. This distribution only takes integer values, so it is certainly not Normal. We’ll also assume it is skewed to the right with a standard deviation σ = 85. Consider an SRS of 30 Facebook users.

(a) What are the mean and standard deviation of the total number of friends in this sample? (b) What are the mean and standard deviation of the mean number of friends per user? (c) Use the central limit theorem to find the probability that the average number of friends in

30 Facebook users is greater than 140.

Solutions

Use the results from the Central Limit theorem and work like in the previous exercises

5.18 ACT scores of high school seniors. The scores of high school seniors on the ACT college entrance examination in a recent year had mean µ = 19.2 and standard deviation σ = 5.1. The distribution of scores is only roughly Normal.

(a) What is the approximate probability that a single student randomly chosen from all those taking the test scores 23 or higher?

(b) Now take an SRS of 25 students who took the test. What are the mean and standard deviation of the sample mean score of these 25 students?

(c) What is the approximate probability that the mean score of these students is 23 or higher?

(d) Which of your two Normal probability calculations in parts (a) and (c) is more accurate? Why?

Solution

Use the results from the Central Limit theorem and work like in the previous exercises

5.20 Grades in an English course.

for its courses online. Students in one section of English 210 in the Fall 2008 semester received

33% A’s, 24% B’s, 18% C’s, 16% D’s, and 9% F’s.

(a) Using the common scale A = 4, B = 3, C = 2, D = 1, F = 0, take randomly chosen English 210 student. Use the definitions of the mean (standard deviation (page 269standard deviation σ of grades in this course.

(b) English 210 is a large course. We can take the grades of an SRS of 50 students to be independent of each other. If standard deviation of ?

(c) What is the probability P(better? What is the approximate probability randomly chosen English 210 students is a B or better?

Solution

(a)

Letter

Grade

Point

equivalent

(X)

A 4

B 3

C 2

D 1

F 0

µX = Σ (x) * p(x) = 2.56

σ2

X = Σ(x - µ)2 * p(x) = 1.7664

σX = sqrt(11.568) = 1.32906

(c)

P(X ≥ 3) = 0.24 + 0.33 = 0.57 (Why?)

= normalcdf(3, ∞, 2.56, 1.32906/sqrt(50)

course. North Carolina State University posts the grade distributions

Students in one section of English 210 in the Fall 2008 semester received

33% A’s, 24% B’s, 18% C’s, 16% D’s, and 9% F’s.

Using the common scale A = 4, B = 3, C = 2, D = 1, F = 0, take X to be the grade of a randomly chosen English 210 student. Use the definitions of the mean (page 261

page 269) for discrete random variables to find the mean of grades in this course.

English 210 is a large course. We can take the grades of an SRS of 50 students to be ch other. If is the average of these 50 grades, what are the mean and

(X ≥ 3) that a randomly chosen English 210 student gets a B or

better? What is the approximate probability that the grade point averagrandomly chosen English 210 students is a B or better?

(b)

Probability

P(x)

0.33

0.24

0.18

0.16

0.09

= sample mean in a sample of 50 students

��̅ = 2.56

��̅ = 3.401176/√50 = 0.480999

1.7664

1.32906

57 (Why?)

1.32906/sqrt(50)) = .0096

North Carolina State University posts the grade distributions

Students in one section of English 210 in the Fall 2008 semester received

to be the grade of a page 261) and

) for discrete random variables to find the mean µ and the

English 210 is a large course. We can take the grades of an SRS of 50 students to be is the average of these 50 grades, what are the mean and

≥ 3) that a randomly chosen English 210 student gets a B or that the grade point average for 50

= sample mean in a sample of 50 students

480999

5.22 A lottery payoff. A $1 bet in a state lottery’s Pick 3 game pays $500 if the three-digit number you choose exactly matches the winning number, which is drawn at random. Here is the distribution of the payoff X:

Each day’s drawing is independent of other drawings.

(a) What are the mean and standard deviation of X? (b) Joe buys a Pick 3 ticket twice a week. What does the law of large numbers say about the

average payoff Joe receives from his bets? (c) What does the central limit theorem say about the distribution of Joe’s average payoff

after 104 bets in a year? (d) Joe comes out ahead for the year if his average payoff is greater than $1 (the amount he

spent each day on a ticket). What is the probability that Joe ends the year ahead?

5.25 Weights of airline passengers. In response to the increasing weight of airline passengers, the Federal Aviation Administration told airlines to assume that passengers average 190 pounds in the summer, including clothing and carry-on baggage. But passengers vary: the FAA gave a mean but not a standard deviation. A reasonable standard deviation is 35 pounds. Weights are not Normally distributed, especially when the population includes both men and women, but they are not very non-Normal. A commuter plane carries 25 passengers. What is the approximate probability that the total weight of the passengers exceeds 5200 pounds? (Hint: To apply the central limit theorem, restate the problem in terms of the mean weight.)