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Chapter 8: Problem 23

Determine the value of the confidence coefficient \(z(\alpha / 2)\) for each situation described: a. \(98 \%\) confidence b. \(99 \%\) confidence

Short Answer

Expert verified
The value of the confidence coefficient \(z(\alpha / 2)\) for a 98% confidence level is approximately 2.33 and for a 99% confidence level is approximately 2.58.

Step by step solution

01

Identify the significance level

First, find \(\alpha\) which is the significance level. For a 98% confidence level, \(\alpha = 1 - 0.98 = 0.02\). For a 99% confidence level, \(\alpha = 1 - 0.99 = 0.01\).
02

Calculate the value of \(\alpha / 2\)

The next step involves calculating \(\alpha / 2\). For a 98% confidence level, \(\alpha / 2 = 0.01\). For a 99% confidence level, \(\alpha / 2 = 0.005\).
03

Identify the z-value corresponding to \(\alpha / 2\)

The final step is to find the z-value corresponding to \(\alpha / 2\) from a standard normal distribution table or other statistical tool. For a two-tailed test, these values for \(\alpha / 2 = 0.01\) and \(\alpha / 2 = 0.005\) correspond to z-values of approximately 2.33 and 2.58 respectively.

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Key Concepts

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

Significance Level
The significance level, commonly denoted by \( \alpha \), is a crucial concept in hypothesis testing and statistics. It represents the probability of rejecting the null hypothesis if it is actually true. In simpler terms, it is the level of risk you are willing to accept when making a statistical decision. This probability is often expressed as a percentage.
Here are some quick points to remember about significance level:
  • A lower significance level means a more stringent criteria for rejecting the null hypothesis.
  • Common significance levels include 0.05, 0.01, and 0.1, corresponding to 95%, 99%, and 90% confidence levels.
  • The significance level complements the confidence level, with \( \alpha = 1 - \text{confidence level}\).
For example, if the confidence level is 98%, the significance level \( \alpha \) would be 2%, or \( 0.02 \). This tells us that there is a 2% chance of making a Type I error, which occurs when we wrongly reject a true null hypothesis.
Confidence Level
The confidence level is a parameter that provides a range of values which is designed to contain the true population parameter a certain percentage of the time. It reflects how confident you are that this range, or confidence interval, will capture the true parameter.
Some important aspects of confidence level include:
  • It is represented as a percentage indicating the degree of certainty one has in the result.
  • The higher the confidence level, the more accurate and reliable the results are perceived to be, although it may sometimes result in a wider confidence interval.
  • Common confidence levels are 90%, 95%, and 99%.
  • A 98% confidence level means you expect the true parameter to fall within your confidence interval 98% of the time in repeated samples.
For instance, if calculating a confidence interval for a mean with a 98% confidence level, it implies that if we repeat the study 100 times, we anticipate the true mean to fall within the calculated range 98 out of those 100 times.
Standard Normal Distribution
The standard normal distribution is a fundamental concept in statistics, depicting a continuous probability distribution that is symmetric about the mean, represented by a bell curve. This distribution has a mean of 0 and a standard deviation of 1.
Key points about the standard normal distribution:
  • It is a standardized form (Z distribution), applicable to many scenarios, especially when dealing with standard scores or Z-scores.
  • Z-scores refer to the number of standard deviations a data point is from the mean.
  • These scores allow for the comparison of data points from different normal distributions.
  • Often used in calculating probabilities and identifying confidence intervals, especially when exact distribution parameters are unknown.
In the context of confidence intervals, the Z-score is used to determine the number of standard deviations from the mean that a particular value lies. For instance, a 98% confidence interval corresponds to a Z-score of 2.33, meaning that values deviate from the mean by 2.33 standard deviations in a typical normal distribution.

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Most popular questions from this chapter

Nationally, the ratio of nurses to students falls short of the recommended federal standard, according to the USA Today article "School nurses in short supply" (August \(11,2009) .\) The recommendation from the Centers for Disease Control and Prevention (CDC) is 1 nurse per 750 students. Use the sample below from 38 randomly selected schools in the state of New York to test the statement "The New York mean number of students per school nurse is significantly higher than the CDC standard of \(750 . "\) Assume \(\sigma=540\) $$\begin{array}{lllllllllll} \hline 1062 & 1070 & 353 & 675 & 1557 & 1374 & 459 & 302 & 1946 & 487 & 295 \\\ 1047 & 1751 & 784 & 480 & 377 & 883 & 1035 & 332 & 330 & 989 & 1098 \\ 1241 & 778 & 1691 & 963 & 1645 & 1594 & 2125 & 338 & 1380 & 885 & 707 \\ 1267 & 1412 & 1037 & 1603 & 915 & & & & & & \\ \hline \end{array}$$ a. Describe the parameter of interest. b. State the null and alternative hypothesis. c. Calculate the value for \(z \star\) and find the \(p\) -value d. State your decision and conclusion using \(\alpha=0.01\)

Determine the critical region and critical values for \(z\) that would be used to test the null hypothesis at the given level of significance, as described in each of the following: a. \(\quad H_{o}: \mu=20, H_{a}: \mu \neq 20, \alpha=0.10\) b. \(\quad H_{o}: \mu=24(\leq), H_{a}: \mu>24, \alpha=0.01\) c. \(\quad H_{o}: \mu=10.5(\geq), H_{a}: \mu<10.5, \alpha=0.05\) d. \(\quad H_{o}: \mu=35, H_{a}: \mu \neq 35, \alpha=0.01\)

The standard deviation of a normally distributed population is equal to \(10 .\) A sample size of 25 is selected, and its mean is found to be \(95 .\) a. Find an \(80 \%\) confidence interval for \(\mu\) b. What would the \(80 \%\) confidence interval be for a sample of size \(100 ?\) c. What would be the \(80 \%\) confidence interval for a sample of size 25 with a standard deviation of 5 (instead of 10 )?

Use a computer or calculator to select 40 random single-digit numbers. Find the sample mean and \(z\) Using \(\alpha=0.05,\) state the decision for testing \(H_{o}: \mu=4.5\) against a two-tailed alternative. Repeat it several times as in Table \(8.12 .\) Describe your findings after several tries.

Waiting times (in hours) at a popular restaurant are believed to be approximately normally distributed with a variance of 2.25 during busy periods. a. A sample of 20 customers revealed a mean waiting time of 1.52 hours. Construct the \(95 \%\) confidence interval for the population mean. b. Suppose that the mean of 1.52 hours had resulted from a sample of 32 customers. Find the \(95 \%\) confidence interval. c. What effect does a larger sample size have on the confidence interval?
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