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Introductory Statistics

Barbara Illowsky, Susan Dean

$Math Studyset 91影视 Explanations$ Math

OER 2018 Edition 路 902 pages

ISBN: 9781938168208

Chapter 8: Q.8.3 (page 452)

Table $8.2$ shows a different random sampling of $20$ cell phone models. Use this data to calculate a $93$ % confidence interval for the true mean SAR for cellphones certified for use in the United States. As previously, assume that the population standard deviation is 蟽= $0.337$ .
Phone Model SAR Phone Model SAR
Blackberry pearl $8120$ $1.48$
Nokia E71x $1.53$
HTC Evo Design 4G $0.8$
Nokia N $75$ $0.68$
HTC Freestyle $1.15$
Nokia N $79$ $1.4$
LG Ally $1.36$
Sagem Puma $1.24$
LG Fathom $0.77$
Samsung Fascinate $0.57$
LG Optimus Vu $0.462$
Samsung Infuse 4G
$0.2$
Motorola Cliq XT $1.36$
Samsung Nexus S $0.51$
Motorola Droid pro $1.39$
Samsung Replenish $0.3$
Motorola Droid Razr M $1.3$
Sony W5 $18$ a walkman $0.73$
Nokia $7705$ Twist $0.7$
ZTE C $79$ $0.869$

Short Answer

Expert verified

We estimate with $93$ % confidence that the true population mean is between $0.7996$ and $1.0724$

Step by step solution

01

Given Information

Given in the question, the value as

The population standard deviation is 蟽= 0.337.

02

Explanation

If $\bar{x}$ is the sample mean of a random sample of size n from a normal population with unknown variance $蟽$ ^$2$, a $100$ ( $1$ - $伪$ ) % Cl on $渭$ is given by

$\overset{炉}{x} 鈭� z_{\frac{伪}{2}} \frac{蟽}{\sqrt{n}} 鈮� 渭鈮� \overset{炉}{x} + z_{\frac{伪}{2}} \frac{蟽}{\sqrt{n}}$ (1)

$where z_{\frac{伪}{2}} is the upper 100 \frac{伪}{2} percentage point of the standard normal distribution.$

$We know that standard deviation is 蟽 = 0.337, a random sample n = 20 and a sample mean$

$\overset{炉}{x} = \frac{1.48 + 0.8 + 鈰� + 0.869}{20} = 0.936$

We need to find a

93

% confidence interval estimate for the population mean. Therefore,

$\frac{伪}{2} = \frac{1 鈭� 0.93}{2} = 0.035 鈬� z_{\frac{伪}{2}} = z_{0.035} = 1.81$ (2)

03

Explanation

The earlier implication was obtained on a probability table for the standard normal distribution.

From equations (1) and (2) we get

$0.936 鈭� 1.81 \frac{0.337}{\sqrt{20}} 鈮� 渭鈮� 0.936 + 1.81 \frac{0.337}{\sqrt{20}}$

$Therefore, 93 % Cl for 渭 is$

$0.7996 鈮� 渭鈮� 1.0724$

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Most popular questions from this chapter

The data in the Table are the result of a random survey of $39$ national flags (with replacement between picks) from various countries. We are interested in finding a confidence interval for the true mean number of colors on a national flag. Let $X =$ the number of colors on a national flag.

$\begin{matrix} X & Freq. \\ 1 & 1 \\ 2 & 7 \\ 3 & 18 \\ 4 & 7 \\ 5 & 6 \end{matrix}$

Construct a $95 %$ confidence interval for the true mean number of colors on national flags.

Calculate the following:

a. lower limit

b. upper limit

c. error bound

Suppose that a committee is studying whether or not there is waste of time in our judicial system. It is interested in

the mean amount of time individuals waste at the courthouse waiting to be called for jury duty. The committee randomly

surveyed $81$ people who recently served as jurors. The sample mean wait time was eight hours with a sample standard

deviation of four hours.

$a . i . x 虅 =__________i i . s x =__________i i i . n =__________i v . n 鈥� 1 =__________$

b. Define the random variables $X a n d X 虅$ in words.

c. Which distribution should you use for this problem? Explain your choice.

d. Construct a $95 %$ confidence interval for the population mean time wasted.

i. State the confidence interval.

ii. Sketch the graph.

iii. Calculate the error bound.

e. Explain in a complete sentence what the confidence interval means.

A telephone poll of $1000$ adult Americans was reported in an issue of Time Magazine. Another question in the poll was 鈥淸How much are] you worried about the quality of education in our schools?鈥� Sixty-three percent responded 鈥渁 lot鈥�. We are interested in the population proportion of adult Americans who are worried a lot about the quality of education in our schools.

a. Define the random variables $X$ and $P 鈥�$ in words.

b. Which distribution should you use for this problem? Explain your choice.

c. Construct a $95 %$ confidence interval for the population proportion of adult Americans who are worried a lot about the quality of education in our schools.

i. State the confidence interval.

ii. Sketch the graph.

iii. Calculate the error bound.

d. The sampling error given by Yankelevich Partners, Inc. (which conducted the poll) is $卤 3 %$ . In one to three complete sentences, explain what the $卤 3 %$ represents.

The data in Table 8.10 are the result of a random survey of 39 national flags (with replacement between picks) from various countries. We are interested in finding a confidence interval for the true mean number of colors on a national flag. Let X = the number of colors on a national flag.

Construct a 95% confidence interval for the true mean number of colors on national flags.

Using the same $\overset{炉}{x}$ , $s_{x}$ , and n = 39, how would the error bound change if the confidence level were reduced to 90%? Why?

A sample of $16$ small bags of the same brand of candies was selected. Assume that the population distribution of bag

weights is normal. The weight of each bag was then recorded. The mean weight was two ounces with a standard deviation

of $0.12$ ounces. The population standard deviation is known to be $0.1$ ounce.

$a . i . x 虅 =________i i . 蟽 =________i i i . s x =________$

b. In words, define the random variable $X .$

c. In words, define the random variable $X 虅 .$

d. Which distribution should you use for this problem? Explain your choice.

e. Construct a $90 %$ confidence interval for the population mean weight of the candies.

i. State the confidence interval.

ii. Sketch the graph.

iii. Calculate the error bound.

f. Construct a $98 %$ confidence interval for the population mean weight of the candies.

i. State the confidence interval.

ii. Sketch the graph.

iii. Calculate the error bound.

g. In complete sentences, explain why the confidence interval in part $f$ is larger than the confidence interval in part $e$ .

h. In complete sentences, give an interpretation of what the interval in part $f$ means.

See all solutions

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