Question

Find the z-values needed to calculate large-sample confidence intervals for the confidence levels given. A \(90 \%\) confidence interval

Short answer

Answer: The z-values corresponding to a 90% confidence interval in a standard normal distribution are approximately -1.645 and 1.645.

Step-by-step solution

Understand the Standard Normal Distribution

The z-values come from the standard normal distribution (also known as the z-distribution). The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Z-values represent the number of standard deviations from the mean.

Find the area in the tails corresponding to the confidence level

We need to find the z-values for a \(90\%\) confidence interval, which means that \(90\%\) of the area under the curve should be between the two z-values. To determine the area in the tails, subtract the given confidence level from 100% and divide the result by 2 (since we have two tails). In this case, the area in each tail would be: \(100\% - 90\% = 10\%\) in both tails Half of 10% in each tail: \(\frac{10}{2} = 5\%\) So, we need to find the z-values that correspond to having 5% of the area in each tail.

Use the z-table to find the z-value corresponding to the area

Using a z-table, we can look up the area within the tails to find the corresponding z-value. In this case, we want to find the z-value that has an area of \(0.05\) (5%) to the right of it. Looking at the z-table, we can find that a z-value of approximately \(1.645\) corresponds to \(0.95\) (or 95%) of the area to its left. Since we want \(5\%\) of the area to the right, this means that the z-value we need is \(1.645\).

Determine the z-values for both tails

Since the standard normal distribution is symmetrical, we can find the z-value for the other tail by taking the negative of the z-value we found in Step 3. In this case, the z-value for the left tail is \(-1.645\)

Present the z-values for the \(90\%\) confidence interval

The z-values needed to calculate a large-sample \(90\%\) confidence interval are approximately as follows: \(Z_{left} = -1.645\) \(Z_{right} = 1.645\) These z-values can now be used to calculate the desired confidence interval.

Key concepts

Standard Normal Distribution

In statistics, the standard normal distribution plays a critical role in understanding various data-related phenomena. A fundamental property of the standard normal distribution is that it is symmetrical around a mean of 0 with a standard deviation of 1. This specific distribution is used as a basis for finding probabilities connected to normal random variables.

To visualize this, imagine a bell-shaped curve perfectly centered over the zero on a horizontal axis. Each point on this curve represents a z-value, which indicates how many standard deviations a point is away from the mean. In practice, the standard normal distribution allows us to compare measurements from different normal distributions by translating them into a common scale – the z-scale.

Z-values

Z-values, also referred to as z-scores, are numerical measurements that describe a value's relationship to the mean of a group of values, measured in terms of standard deviations. In the realm of the standard normal distribution, a z-score tells us how many standard deviations an element is from the mean.

Interpreting Z-values

A positive z-score indicates the value lies above the mean, while a negative z-score signifies it is below the mean. The higher the absolute value of the z-score, the further away it is from the mean. Z-values are especially useful in the field of statistics for finding probabilities, comparing scores, and calculating confidence intervals, which brings us to our next topic.

Confidence Levels

Confidence levels are an integral part of inferential statistics. They provide a way to quantify the uncertainty in an estimation process. A confidence level, usually expressed as a percentage (e.g., 90%, 95%), corresponds to the degree of certainty in a statistical inference.

Understanding Confidence Levels

For example, a 90% confidence level suggests that if we were to take numerous random samples and compute the confidence interval for each sample, about 90% of those intervals would be expected to contain the actual population parameter. It reflects how sure we can be that our estimations capture the true value we're interested in. Choosing a confidence level depends on the risk one is willing to take on estimating incorrectly – the higher the confidence level, the wider the confidence interval.

Z-table

The z-table, also known as the standard normal probability table, is a reference table that provides the probabilities of a z-score occurring in a standard normal distribution. These tables are essential tools for statisticians to determine the proportion of values that lie below a certain z-score.

Using the Z-table

To use a z-table, you first need to calculate the z-score of interest. Once you have this score, you can then reference the table to find the corresponding probability. Z-tables come in a few variations – some show the cumulative probability from the left up to a given z-score, while others show the probability that remains to the right of the z-score. Understanding how to read one is paramount for tasks such as determining the critical values for confidence intervals.