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 σ= $0.337$ .
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

Given Information

Given in the question, the value as

The population standard deviation is σ= 0.337.

Explanation

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

$\bar{x} - z_{\frac{α}{2}} \frac{σ}{\sqrt{n}} \leq μ \leq \bar{x} + z_{\frac{α}{2}} \frac{σ}{\sqrt{n}}$ (1)

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

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

$\bar{x} = \frac{1.48 + 0.8 + \dots + 0.869}{20} = 0.936$

We need to find a

93

% confidence interval estimate for the population mean. Therefore,

$\frac{α}{2} = \frac{1 - 0.93}{2} = 0.035 \Rightarrow z_{\frac{α}{2}} = z_{0.035} = 1.81$ (2)

Explanation

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

From equations (1) and (2) we get

$0.936 - 1.81 \frac{0.337}{\sqrt{20}} \leq μ \leq 0.936 + 1.81 \frac{0.337}{\sqrt{20}}$

$Therefore, 93 % Cl for μ is$

$0.7996 \leq μ \leq 1.0724$

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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 $79$	$0.869$