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Chapter 8: Problem 1

Consider a normal population distribution with the value of \(\sigma\) known. a. What is the confidence level for the interval \(\pi \pm 2.81 \sigma / \sqrt{n}\) ? b. What is the confidence level for the interval \(\pi \pm 1.44 \sigma / \sqrt{n}\) ? c. What value of \(z_{x / 2}\) in the CI formula (8.5) results in a confidence level of \(99.7 \%\) ? d. Answer the question posed in part (c) for a confidence level of \(75 \%\).

Short Answer

Expert verified
a. 99.5% confidence level. b. 85.5% confidence level. c. z = 3. d. z 鈮� 1.15.

Step by step solution

01

Understanding the confidence interval formula

A confidence interval for a population mean with known standard deviation \( \sigma \) is given by \( \pi \pm z_{x / 2} \times \left(\frac{\sigma}{\sqrt{n}}\right) \). Here, \( z_{x / 2} \) is the z-score that corresponds to half of the remaining probability in the tails for a given confidence level.
02

Identify z-score for given intervals (2.81)

For part a, the confidence interval is given by \( \pi \pm 2.81 \frac{\sigma}{\sqrt{n}} \). Here, \( z_{x / 2} = 2.81 \). Use z-score tables or calculators to find that a z-score of 2.81 corresponds to approximately 99.5\% of data under the normal distribution.
03

Calculate confidence level for z-score 1.44

For part b, the confidence interval is \( \pi \pm 1.44 \frac{\sigma}{\sqrt{n}} \). Check standard normal distribution tables to see that the cumulative probability for \( z = 1.44 \) corresponds to about 85.5\%, hence the confidence level is roughly 85.5\%.
04

Determine z-score for 99.7% confidence level

For part c, the problem asks for the \( z_{x / 2} \) value associated with a 99.7\% confidence level. From the empirical rule (68-95-99.7), a 99.7\% confidence level corresponds to \( z_{x / 2} = 3 \).
05

Determine z-score for 75% confidence level

Part d requires finding the \( z_{x / 2} \) for a 75\% confidence level. Divide 25\% (the remaining area in the tails) by two, resulting in 12.5\% in each tail. A z-score corresponding to a cumulative probability of 87.5\% (100\% - 12.5\%) is approximately 1.15.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution
The concept of normal distribution is foundational in statistics, representing how data points are spread around a central mean. When plotted, it takes the shape of a symmetric bell curve, indicating that most observations cluster around the mean value.

This curve has distinct properties:
  • It is symmetric around the mean.
  • The total area under the curve is equal to 1, representing the entirety of the data set.
  • Most data falls within certain standard deviations from the mean, specifically, 68% within one standard deviation, 95% within two, and 99.7% within three.

Normal distribution is crucial when calculating confidence intervals for a population mean. Knowing the normal distribution helps in determining the z-scores needed for specific levels of confidence.
Z-Score
A z-score plays a crucial role in statistics by indicating how many standard deviations a data point is from the population mean.

In the context of confidence intervals, the z-score helps to define the margin of error. Here's how it works:
  • Z-score tables, or calculators, are used to find the z-value that corresponds to a certain confidence level, such as 99.5% or 85.5%.
  • This z-value represents the distance from the mean, encapsulating the data within the desired probability limits under the normal curve.

Practically, a z-score helps in understanding whether a particular data point is typical or unusual compared to the overall data. For example, in part c of the exercise, determining the z-score for a 99.7% confidence level demands looking at the empirical rule, identifying that a z-value of 3 places data within nearly three standard deviations.
Population Mean
The population mean is a central point around which the normal distribution is centered. It represents the average of all data points in a population. In practice, it helps illustrate a typical value within a given data set.

In the context of statistical inference, the population mean (\(\pi\)) is critical when forming confidence intervals.
Here are important points about the population mean:
  • It serves as the expected value around which confidence intervals are built.
  • Confidence intervals give a range where we can expect the true population mean to fall, given a specific confidence level.
  • Fluctuations in sample data impact the sample mean, but the population mean remains a constant value.

Working with the population mean and understanding its role helps ensure precise statistical conclusions.
Standard Deviation
Standard deviation is vital for assessing the spread of data in a distribution. It measures the typical deviation of data points from the mean, showing how much variation or dispersion exists in the data.

Key aspects of standard deviation include:
  • It helps to calculate the confidence interval size for a given sample, particularly when the population standard deviation (\(\sigma\)) is known.
  • Smaller standard deviations indicate data points are close to the mean, implying less variability.
  • Larger standard deviations suggest data points are spread out, showing greater variability.

Understanding and calculating the standard deviation allows statisticians to make informed predictions. In the context of the exercise, using the standard deviation alongside the sample size is essential for determining the width of the confidence interval, tailoring it to the data's variability.

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Most popular questions from this chapter

Chronic exposure to asbestos fiber is a wellknown health hazard. The article "The Acute Effects of Chrysotile Asbestos Exposure on Lung Function" (Envir. Res., 1978: 360-372) reports results of a study based on a sample of construction workers who had been exposed to asbestos over a prolonged period. Among the data given in the article were the following (ordered) values of pulmonary compliance \(\left(\mathrm{cm}^{3} / \mathrm{cm} \mathrm{H}_{2} \mathrm{O}\right)\) for each of 16 subjects 8 months after the exposure period (pulmonary compliance is a measure of lung elasticity, or how effectively the lungs are able to inhale and exhale): \(\begin{array}{llllll}167.9 & 180.8 & 184.8 & 189.8 & 194.8 & 200.2 \\ 201.9 & 206.9 & 207.2 & 208.4 & 226.3 & 227.7 \\ 228.5 & 232.4 & 239.8 & 258.6 & & \end{array}\) a. Is it plausible that the population distribution is normal? b. Compute a \(95 \%\) CI for the true average pulmonary compliance after such exposure.

A manufacturer of college textbooks is interested in estimating the strength of the bindings produced by a particular binding machine. Strength can be measured by recording the force required to pull the pages from the binding. If this force is measured in pounds, how many books should be tested to estimate the average force required to break the binding to within . \(1 \mathrm{lb}\) with \(95 \%\) confidence? Assume that \(\sigma\) is known to be .8.

A random sample of \(n=15\) heat pumps of a certain type yielded the following observations on lifetime (in years): \(\begin{array}{rrrrrrrr}2.0 & 1.3 & 6.0 & 1.9 & 5.1 & .4 & 1.0 & 5.3 \\ 15.7 & .7 & 4.8 & .9 & 12.2 & 5.3 & .6 & \end{array}\) a. Assume that the lifetime distribution is exponential and use an argument parallel to that of Example \(8.5\) to obtain a \(95 \%\) CI for expected (true average) lifetime. b. How should the interval of part (a) be altered to achieve a confidence level of \(99 \%\) ? c. What is a \(95 \%\) CI for the standard deviation of the lifetime distribution? [Hint: What is the standard deviation of an exponential random variable?]

The article "An Evaluation of Football Helmets Under Impact Conditions" (Amer.J. Sports Med., 1984: 233-237) reports that when each football helmet in a random sample of 37 suspension-type helmets was subjected to a certain impact test, 24 showed damage. Let \(p\) denote the proportion of all helmets of this type that would show damage when tested in the prescribed manner. a. Calculate a \(99 \%\) CI for \(p\). b. What sample size would be required for the width of a \(99 \%\) CI to be at most . 10, irrespective of \(\hat{p}\) ?

A random sample of 110 lightning flashes in a region resulted in a sample average radar echo duration of 81 s and a sample standard deviation of \(.34\) s ("Lightning Strikes to an Airplane in a Thunderstorm," J. Aircraft, 1984: 607-611). Calculate a \(99 \%\) (two-sided) confidence interval for the true average echo duration \(\mu\), and interpret the resulting interval.
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