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The Practice of Statistics for AP

David Moore,Daren Starnes,Dan Yates

$Math Studyset 91影视 Explanations$ Math

4th Edition 路 809 pages

ISBN: 9781319113339

Chapter 7: Q. 49 (page 454)

Songs on an iPod David鈥檚 iPod has about $10000$ songs. The distribution of the playtimes for these songs is heavily skewed to the right with a mean of $225$ seconds and a standard deviation of $60$ seconds. Suppose we choose an SRS of 10 songs from this population and calculate the mean playtime x of these songs. What are the mean and the standard deviation of the sampling distribution of x? Explain.

Short Answer

Expert verified

The mean of the sampling distribution is $渭_{x} = 225 s e c o n d s$

The standard deviation of the sampling distribution is $蟽_{x} = 18.9737 s e c o n d s$ .

Step by step solution

01

Given information

Total songs $= 10000$

Mean of total songs $= 225 s e c o n d s$

The standard deviation of total songs $= 60 s e c o n d s$

the mean and the standard deviation of the sampling distribution of x

02

Explanation

The given data is written as

$渭 = 225 s e c o n d s$

$蟽 = 60 s e c o n d s$

$n = 10$

The sample mean $\overset{炉}{x}$ is equivalent to the population mean, which is the mean of the sampling distribution.

localid="1650109681905" $渭_{x} = 渭 = 225 s e c o n d s$

The standard deviation of the sampling distribution of the sample mean $x$ is calculated by dividing the population standard deviation by the square root of the sample size.

localid="1650109697363" $蟽_{x} = \frac{蟽}{\sqrt{n}} = \frac{60}{\sqrt{10}} 鈮� 18.9737$ .

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Most popular questions from this chapter

According to the U.S. census, the proportion of adults in a certain county who owned their own home was $0.71$ . An SRS of $100$ adults in a certain section of the county found that $65$ owned their home. Which one
of the following represents the approximate probability of obtaining a sample of $100$ adults in which fewer than $65$ own their home, assuming that this section of the county has the same overall proportion of adults who own their home as does the entire county?

(a) $(\begin{matrix} 100 \\ 65 \end{matrix}) (0.71)^{65} (0.29)^{35}$

(b) $(\begin{matrix} 100 \\ 65 \end{matrix}) (0.29)^{65} (0.71)^{35}$
(c) $P (z < \frac{0.65 - 0.71}{\sqrt{\frac{(0.71) (0.29)}{100}}})$

(d) $P (z < \frac{0.65 - 0.71}{\sqrt{\frac{(0.65) (0.35)}{100}}})$
(e) $P (z < \frac{0.65 - 0.71}{\frac{(0.71) (0.29)}{\sqrt{100}})}$

A grocery chain runs a prize game by giving each customer a ticket that may win a prize when the box is scratched off. Printed on the ticket is a dollar value $(\(500, \) 100, \(10)$ or the statement, 鈥淭his ticket is not a winner.鈥� Monetary prizes can be redeemed for groceries at the store. Here are the distribution of the prize values and the associated probabilities for each prize:

Which of the following are the mean and standard deviation, respectively, of the winnings?
$\) 15.00, \(2900.00$
(b) $\) 15.00, \(53.85$
(c) $\) 15.00, \(26.9$

(d) $\) 156.25, \(53.85$

(e) $\) 156.25, $ 26.93$

On a New York鈥搕o鈥揇enver flight, $8 %$ of the $125$ passengers were selected for random security screening before boarding. According to the Transportation Security Administration, $10 %$ of passengers at this airport are chosen for random screening.

Sale of eggs that are contaminated with salmonella can cause food poisoning in consumers. A large egg producer takes an SRS of $200$ eggs from all the eggs shipped in one day. The laboratory reports that $9$ of these eggs had salmonella contamination. Unknown to the producer, $0.1 %$ (one-tenth of 1%) of all eggs shipped had salmonella. Identify the population, the parameter, the sample, and the statistic.

Records from a random sample of dairy farms yielded the information below on the number of male and female calves born at various times of the day.

What is the probability that a randomly selected calf was
born in the night or was a female?
(a) $\frac{369}{513}$

(b) $\frac{485}{513}$

(c) $\frac{116}{513}$

(d) $\frac{116}{252}$

(e) $\frac{116}{233}$

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