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Chapter 8: Problem 2

Consider the one-sided confidence interval expressions for a mean of a normal population. a. What value of \(z_{\alpha}\) would result in a \(90 \%\) CI? b. What value of \(z_{\alpha}\) would result in a \(95 \%\) CI? c. What value of \(z_{\alpha}\) would result in a \(99 \%\) CI?

Short Answer

Expert verified
For 90% CI, \( z_{\alpha} \approx 1.28\); for 95% CI, \( z_{\alpha} \approx 1.645\); for 99% CI, \( z_{\alpha} \approx 2.33\).

Step by step solution

01

Understanding the Z-score

The Z-score, \( z_{\alpha} \), is a critical value that corresponds to the desired confidence level in standard normal distribution tables. It denotes the number of standard deviations an element is from the mean.
02

Identify the Significance Level \( \alpha \) for 90% CI

For a one-sided 90% confidence interval, the area in the tail is 0.10. Thus, \( \alpha = 0.10 \), and we look up \( z_{0.10} \) in the Z-table.
03

Find \( z_{0.10} \) in Z-table

Consult the standard normal (Z) table to find the Z-score that corresponds to a cumulative probability of 0.90 (since 1 - 0.10). \( z_{0.10} \) is approximately 1.28.
04

Repeat for 95% CI

For a 95% confidence interval, the tail area is 0.05. Hence, \( \alpha = 0.05 \), and we need \( z_{0.05} \).
05

Find \( z_{0.05} \) in Z-table

Look in the Z-table for a cumulative probability of 0.95 (since 1 - 0.05). The \( z_{0.05} \) value is approximately 1.645.
06

Repeat for 99% CI

For a 99% confidence interval, the tail area is 0.01. Hence, \( \alpha = 0.01 \), and we need \( z_{0.01} \).
07

Find \( z_{0.01} \) in Z-table

Look up the Z-score for a cumulative probability of 0.99 (since 1 - 0.01). The \( z_{0.01} \) value is approximately 2.33.

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Key Concepts

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

Z-score
The Z-score is a statistical measurement that describes a value's relation to the mean of a group of values, expressed in terms of standard deviations. It tells us how many standard deviations away a particular data point is from the mean. In the context of confidence intervals, the Z-score is crucial because it helps determine the range in which we expect the true population parameter to lie.
  • The Z-score is symbolized as \( z_{\alpha} \), where \( \alpha \) is the significance level.
  • It is derived from the standard normal distribution, which has a mean of 0 and a standard deviation of 1.
  • A higher Z-score indicates a greater confidence that a sample statistic accurately reflects the population mean.
Understanding how to use the Z-score is essential when calculating confidence intervals, as it essentially determines how "widely" we cast our net around our sample mean.
Significance Level
The significance level, denoted as \( \alpha \), serves as a threshold for determining statistical significance. It represents the probability that a parameter lies outside the confidence interval, assuming the null hypothesis is true.
  • In hypothesis testing and confidence intervals, \( \alpha \) is the probability of rejecting the null hypothesis when it is true. This is known as a Type I error.
  • Common significance levels are 0.10, 0.05, and 0.01, which correspond to 90%, 95%, and 99% confidence intervals, respectively.
  • A lower significance level indicates stricter criteria for rejecting the null hypothesis, thus leading to a wider confidence interval and a higher confidence level.
For a one-sided confidence interval, the significance level helps determine the tail area of the distribution that lies beyond the cutoff point represented by the Z-score.
One-sided Confidence Interval
A one-sided confidence interval is a range of values that estimates a parameter's value with a certain level of confidence, but only in one direction鈥攅ither above or below the sample estimate.
  • It is particularly useful when researchers have a specific direction in hypothesis testing or when regulatory standards specify limits.
  • For example, calculating a one-sided 90% confidence interval focuses on either the upper or lower tail of the distribution and uses 90% of the probability to include the true mean.
  • It uses a different Z-score threshold than a two-sided interval because all the significance level is allocated to one tail.
Unlike two-sided confidence intervals, where the interval extends in both directions, a one-sided interval places all probabilistic weight on exceeding or not exceeding a particular threshold.
Standard Normal Distribution
The standard normal distribution is a probability distribution that has a mean of 0 and a standard deviation of 1. It is a special case of the normal distribution and is also known as the Z-distribution.
  • In this context, it is used to convert individual data points, such as sample means, into Z-scores.
  • The symmetry of the standard normal distribution simplifies calculations and helps statisticians predict the probability of observing any given Z-score.
  • The total area under the standard normal curve is 1, allowing for easy probability calculations鈥攊mportant for defining confidence levels.
Z-tables, often used in statistics, provide the cumulative probability associated with each Z-score in this distribution, allowing researchers to quickly determine the significance of their findings.

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Most popular questions from this chapter

Information on a packet of seeds claims that \(93 \%\) of them will germinate. Of the 200 seeds that were planted, only 180 germinated. a. Find a \(95 \%\) confidence interval for the true proportion of seeds that germinate based on this sample. b. Does this seem to provide evidence that the claim is wrong? w

The U.S. Postal Service (USPS) has used optical character recognition (OCR) since the mid-1960s. In 1983 , USPS began deploying the technology to major post offices throughout the country (www.britannica.com). Suppose that in a random sample of 500 handwritten zip code digits, 466 were read correctly. a. Construct a \(95 \%\) confidence interval for the true proportion of correct digits that can be automatically read. b. What sample size is needed to reduce the margin of error to \(1 \% ?\) c. How would the answer to part (b) change if you had to assume that the machine read only one-half of the digits correctly?

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 \(95 \% \mathrm{CI}\) 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.

The yield of a chemical process is being studied. From previous experience, yield is known to be normally distributed and \(\sigma=3 .\) The past 5 days of plant operation have resulted in the following percent yields: \(91.6,88.75,90.8,89.95,\) and 91.3. Find a \(95 \%\) two-sided confidence interval on the true mean yield.

A random sample has been taken from a normal distribution and the following confidence intervals constructed using the same data: (38.02,61.98) and (39.95,60.05) a. What is the value of the sample mean? b. One of these intervals is a \(95 \% \mathrm{CI}\) and the other is a \(90 \%\) CI. Which one is the \(95 \%\) CI and why?
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