Question

Find the z-values needed to calculate large-sample confidence intervals for the confidence levels given. Confidence coefficient \(1-\alpha=.98\)

Short answer

Answer: The z-value needed is approximately 2.33.

Step-by-step solution

Identify the Confidence Level and Confidence Coefficient

The given confidence level is 98%, which means there is a 98% chance that the true population parameter lies within the confidence interval. Confidence coefficient \(1-\alpha\) is given as 0.98.

Calculate \(\alpha\)

Calculate alpha as \(\alpha = 1 - (1-\alpha)\) which is \(\alpha = 1 - 0.98 = 0.02\). This value indicates the remaining probability outside the confidence interval on both tails.

Divide \(\alpha\) by 2

As the standard normal distribution table is a two-tailed distribution, divide \(\alpha\) by 2 to distribute the probability equally across both tails. Thus, the probability in each tail would be \(\frac{\alpha}{2} = \frac{0.02}{2} = 0.01\).

Find the Probability for One Tail

Now, we need to find the probability for one tail, which is \(1-\frac{\alpha}{2}\). Calculate it as \(1-0.01 = 0.99\). This is the value we will look for in the body of the z-table.

Lookup Z-Value in Standard Normal Distribution Table

Now, search in the standard normal distribution table (z-table) for the value closest to 0.99. The number with the closest probability value in the z-table is 2.33.

State the Answer

The z-value needed to calculate large-sample confidence intervals for the confidence level of 98% (confidence coefficient \(1-\alpha=.98\)) is approximately 2.33.

Key concepts

Z-values in Statistics

In statistics, z-values, also known as z-scores, are a crucial part of understanding data distribution. They represent the number of standard deviations a data point is from the mean. A z-value provides a way to determine the probability of a score occurring within a normal distribution and is commonly used in the context of hypothesis testing.

A positive z-value indicates a score above the mean, while a negative z-value signifies a score below the mean. To put it in context regarding confidence intervals, a z-value is used to represent the cut-off points, or critical values, that help define the range within which we expect a certain percentage of the data to fall.

For example, a 98% confidence level which translates to a confidence coefficient of 0.98, requires finding the z-value that includes 98% of the data within the center of the distribution. As our exercise shows, this can be found using the standard normal distribution table, leading to a z-value of approximately 2.33 for a two-tailed test.

Probability and Statistics

Probability is the foundation upon which the field of statistics is built. It deals with the likelihood of events occurring and enables statisticians to make inferences about populations based on sample data. The principles of probability are applied to calculate confidence intervals, test hypotheses, and make predictions.

In the context of confidence intervals, the probability tells us how certain we can be that our interval estimate includes the population parameter. The confidence level (98%, in our exercise) reflects this probability. This level of confidence means that if we were to take many samples and construct intervals in the same way, we expect about 98% of those intervals to contain the true population parameter.

The probability helps to determine the critical values (like our z-value of 2.33) needed to set the range of these intervals. It is important to grasp that while probability provides a percent certainty about the intervals, it does not guarantee that a particular interval calculated from a sample contains the population parameter.

Large-Sample Estimation

Large-sample estimation refers to statistical techniques used to estimate population parameters, such as the mean or proportion, when sample sizes are sufficiently large. According to the Central Limit Theorem, when the sample size is large, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution.

This property allows us to use z-values from the standard normal distribution to create confidence intervals around large sample estimates. As samples get larger, the margin of error within a confidence interval becomes smaller, resulting in more precise estimates. In practical terms, with a large sample, we can be more confident that our sample statistics are close to the true population parameters.

The example shown in the exercise uses the large-sample estimation to determine the confidence interval for a population mean. The z-value found through the exercise is a critical component in calculating this interval.

Confidence Coefficient

The confidence coefficient is a key part of constructing confidence intervals. It is numerically represented by the symbol \(1 - \alpha\) and provides the probability that the interval estimate contains the population parameter. This value is tied directly to the confidence level. For example, a confidence coefficient of 0.98 implies a 98% confidence level, meaning there is a 98% probability that the confidence interval includes the true parameter.

The confidence coefficient helps determine the critical z-values needed to set the upper and lower bounds of the confidence interval. By calculating the alpha level (\(\alpha\)) as the complement of the confidence coefficient, and dividing it by 2 for a two-tailed test, we establish the boundaries needed to find the appropriate z-values from a z-table.

Understanding the confidence coefficient is essential for interpreting confidence intervals correctly and knowing how much faith we can put in the interval estimates we compute from our sample data, as demonstrated in our exercise.