441
Chapter 9
Estimating the Value of a Parameter Using Confidence Intervals
9.1
The Logic in Constructing Confidence Intervals about a Population
Mean Where the Population Standard Deviation is Known
1.
The margin of error of a confidence interval of a parameter depends on the level of
confidence, the sample size, and the standard deviation of the population.
2.
The margin of error increases as the level of confidence increases because, if we want to be
more confident that the interval contains the population mean, then we need to make the
interval wider.
3.
The margin of error decreases as the sample size increases because the Law of Large
Numbers states that as the sample size increases the sample mean approaches the value of
the population mean.
4.
The sample mean is the midpoint of the interval:
1
(10 18)
14
2
x
=
+=
.
The margin of error
is the distance from the midpoint to the bounds:
18 14
4
E
=
−=
.
5.
The mean age of the population is a fixed value (i.e., constant), so it is not probabilistic.
The 95% level of confidence refers to confidence in the method by which the interval is
obtained, not the specific interval.
A better interpretation would be:
“We are confident
that the interval 21.4 years to 28.8 years, obtained by using our method, is one of the 95%
of confidence intervals that contains the mean.”
6.
Since the margin of error increases as the sample size decreases, rounding down would
have the effect of slightly increasing the margin of error beyond its desired value.
So, we
round up to give a slightly smaller margin of error.
7.
No, a
Z
interval should not be constructed because the data are not normal since a point is
outside the bounds of the normal probability plot.
Also, the data and contain outliers
which can be seen in the boxplot.
8.
No, a
Z
interval should not be constructed because the data are not normal since points are
outside the bounds of the normal probability plot.
Also, the data contain outliers which can
be seen in the boxplot.
9.
No, a
Z
interval should not be constructed because the data are not normal since points are
outside the bounds of the normal probability plot.
From the boxplot, the data are skewed
right.
10.
No, a
Z
interval should not be constructed because the data are not normal since points are
outside the bounds of the normal probability plot.
From the box plot, the data are skewed
left.
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Estimating the Value of a Parameter Using Confidence Intervals
442
11.
Yes, a
Z
interval can be constructed.
The plotted points are all within the bounds of the
normal probability plot, which also has a generally linear pattern.
The boxplot shows that
there are no outliers.
12.
Yes, a
Z
interval can be constructed.
The plotted points are all within the bounds of the
normal probability plot, which also has a generally linear pattern.
The boxplot shows that
there are no outliers.
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 Spring '11
 ahmad
 Normal Distribution, Confidence, lower bound

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