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Chapter 8: Problem 2

For a normal population with known variance \(\sigma^{2}\) : (a) What value of \(z_{\alpha / 2}\) in Equation \(8-5\) gives \(98 \%\) confidence? (b) What value of \(z_{\alpha / 2}\) in Equation \(8-5\) gives \(80 \%\) confidence? (c) What value of \(z_{\alpha / 2}\) in Equation \(8-5\) gives \(75 \%\) confidence?

Short Answer

Expert verified
For 98% confidence, \(z_{\alpha/2} = 2.33\); for 80%, \(z_{\alpha/2} = 1.28\); for 75%, \(z_{\alpha/2} = 1.15\).

Step by step solution

01

Understand the Confidence Level

The confidence level tells us how certain we are about our estimate. For a confidence level of X%, we look for the value of \(z_{\alpha / 2}\), which corresponds to the area in the tails of the standard normal distribution equals to \(1 - \frac{X}{100}\).
02

Calculate Alpha (\(\alpha\))

Alpha (\(\alpha\)) represents the total area in the two tails of the distribution. For a confidence level of X%, \(\alpha = 1 - \frac{X}{100}\). Then, \(\alpha / 2\) is the area in each tail.
03

Step 3a: Determine \(z_{\alpha/2}\) for 98% confidence

For 98% confidence, \(\alpha = 0.02\) and \(\alpha/2 = 0.01\). We look up 0.01 in the standard normal distribution table to find \(z_{\alpha/2}\). The value is approximately 2.33.
04

Step 3b: Determine \(z_{\alpha/2}\) for 80% confidence

For 80% confidence, \(\alpha = 0.20\) and \(\alpha/2 = 0.10\). Find the \(z\)-value for 0.10 in each tail. The \(z_{\alpha/2}\) is approximately 1.28.
05

Step 3c: Determine \(z_{\alpha/2}\) for 75% confidence

For 75% confidence, \(\alpha = 0.25\) and \(\alpha/2 = 0.125\). Look for 0.125 in the standard normal distribution table. The value 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 normal distribution, often referred to as the bell curve, is a fundamental concept in statistics. It's a continuous probability distribution defined by two parameters: the mean (\(\mu\)) and the variance (\(\sigma^2\)). The curve is symmetric and centered around the mean, with most data points falling close to this central value. As you move away from the mean, the probability of occurrence decreases.
  • Symmetry: The curve is perfectly symmetrical around the mean.
  • Peakedness: The highest point on the curve corresponds to the mean.
  • Area Under the Curve (AUC): The total area under the curve represents probability and it equals 1.
Understanding the normal distribution is crucial because many natural phenomena and statistical processes follow this pattern, making it a powerful tool for statistical analysis.
Standard Normal Distribution
The standard normal distribution is a special form of the normal distribution. It has a mean of 0 and a standard deviation of 1. The process of converting a normal distribution to a standard normal distribution is called standardization. This is done by transforming each value using the formula:\[z = \frac{X - \mu}{\sigma}\]
  • This transformation is useful as it allows for the comparison of scores or data points across different datasets or distributions.
  • Each point in the standard normal distribution is represented by a z-value, which indicates how many standard deviations away a given value is from the mean.
All standard normal distributions look alike because they follow the same curve. This uniformity enables statisticians to use standard normal distribution tables to find probabilities, making calculations easier and more consistent.
z-value
The z-value, also known as the z-score, is a statistical measurement that indicates how many standard deviations a data point is from the mean of the standard normal distribution. This score is crucial for interpreting the position of a data point within a distribution. Here鈥檚 how z-values are utilized:
  • Significance: A z-value of 0 indicates the data point is exactly at the mean, while positive and negative z-values indicate positions above or below the mean respectively.
  • Use in Confidence Intervals: In constructing confidence intervals, z-values determine the margin of error by providing the appropriate critical values from the standard normal distribution.
  • The wider the confidence interval, the larger the corresponding z-value would be for heightened accuracy.
Understanding z-values helps in assessing the probability and reliability of our data estimates and their relative importance in statistical analysis.
Statistical Confidence Levels
Statistical confidence levels express our certainty about the accuracy of an estimate. It鈥檚 denoted by a percentage, such as 95%, which implies how sure we are that our parameter lies within the specified interval. Here's how confidence levels work:
  • Determining Confidence Intervals: Confidence intervals give a range of values within which a parameter is expected to lie. Higher confidence levels indicate a wider interval to account for greater uncertainty.
  • Alpha (\(\alpha\)) and Tails: The alpha represents the total probability of an error occurring outside the confidence interval. For a given confidence level, alpha is split equally between both tails of the distribution.
  • Critical z-values: For given confidence levels, the z-values that correspond to the cutoff points (tails) can be obtained from standard normal distribution tables.
Grasping statistical confidence levels helps in making informed decisions based on data, by providing a measure of the reliability of those decisions.

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Most popular questions from this chapter

An article in Computers \& Electrical Engineering ['Parallel Simulation of Cellular Neural Networks" (1996, Vol. 22, pp. \(61-84\) ) considered the speed- up of cellular neural networks (CNN) for a parallel general-purpose computing architecture based on six transputers in different areas. The data follow: \(\begin{array}{llllll}3.775302 & 3.350679 & 4.217981 & 4.030324 & 4.639692 \\\ 4.139665 & 4.395575 & 4.824257 & 4.268119 & 4.584193 \\ 4.930027 & 4.315973 & 4.600101 & & \end{array}\) (a) Is there evidence to support the assumption that speed-up of CNN is normally distributed? Include a graphical display in your answer. (b) Construct a \(95 \%\) two-sided confidence interval on the mean speed-up. (c) Construct a \(95 \%\) lower confidence bound on the mean speed-up.

A random sample has been taken from a normal distribution. Output from a software package is given below: \(\begin{array}{ccccccc}\text { Variable } & \mathrm{N} & \text { Mean } & \text { SE Mean } & \text { StDev } & \text { Variance } & \text { Sum } \\\ \mathrm{x} & 10 & ? & 0.507 & 1.605 & ? & 251.848\end{array}\) (a) Fill in the missing quantities. (b) Find a \(95 \% \mathrm{CI}\) on the population mean.

An article in Cancer Research ["Analyses of LitterMatched Time-to-Response Data, with Modifications for Recovery of Interlitter Information" (1977, Vol. 37, pp. \(3863-3868\) ) ] tested the tumorigenesis of a drug. Rats were randomly selected from litters and given the drug. The times of tumor appearance were recorded as follows: \(101,104,104,77,89,88,104,96,82,70,89,91,39,103,93,\) \(85,104,104,81,67,104,104,104,87,104,89,78,104,86,\) \(76,103,102,80,45,94,104,104,76,80,72,73,\) Calculate a \(95 \%\) confidence interval on the standard deviation of time until a tumor appearance. Check the assumption of normality of the population and comment on the assumptions for the confidence interval.

Students in the industrial statistics lab at ASU calculate a lot of confidence intervals on \(\mu\). Suppose all these CIs are independent of each other. Consider the next one thousand \(95 \%\) confidence intervals that will be calculated. How many of these CIs do you expect to capture the true value of \(\mu\) ? What is the probability that between 930 and 970 of these intervals contain the true value of \(\mu\) ?

Suppose that \(n=100\) random samples of water from a freshwater lake were taken and the calcium concentration (milligrams per liter) measured. A \(95 \%\) CI on the mean calcium concentration is \(0.49 \leq \mu \leq 0.82\) (a) Would a \(99 \%\) CI calculated from the same sample data be longer or shorter? (b) Consider the following statement: There is a \(95 \%\) chance that \(\mu\) is between 0.49 and \(0.82 .\) Is this statement correct? Explain your answer. (c) Consider the following statement: If \(n=100\) random samples of water from the lake were taken and the \(95 \%\) CI on \(\mu\) computed, and this process were repeated 1000 times, 950 of the CIs would contain the true value of \(\mu\). Is this statement correct? Explain your answer.
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