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Magazine Chapter 8: Problem 31
98% confidence Find z* for a 98% confidence interval using Table A or your calculator. Show your method.
Short Answer
Expert verified
The critical z-value for a 98% confidence interval is approximately 2.33.
Step by step solution
01
Understand the Problem
To find the critical value \( z^* \) for a 98% confidence interval, we need to determine the \( z \) score such that the middle 98% of the standard normal distribution is captured between \( -z^* \) and \( z^* \). This involves finding the \( z \) value that leaves 1% in each tail of the distribution.
02
Identify Tail Probabilities
For a 98% confidence interval, 2% of the distribution falls outside this interval, with 1% in each tail (as 100% - 98% = 2%). Thus, the cumulative probability in the left tail is 0.01.
03
Find Critical z-Value
Use a standard normal distribution table (often called Table A) or a calculator with inverse normal functions to find the z-value that corresponds to a cumulative probability of 0.99 (since we want the area to the left of \( z^* \) and it's symmetric for 98%).
04
Use Calculator or Table A
If using a calculator, use the function \( \text{invNorm}(0.99, 0, 1) \) to find that \( z^* \approx 2.33 \). If using Table A, find the value nearest to a cumulative probability of 0.99, which typically shows \( z^* \approx 2.33 \).
05
Verify Symmetry
To verify, remember that the standard normal distribution is symmetric. So, the \( z^* \) value for a confidence level of 98% should be approximately equal to the \( z \) score at which 99% of values lie below: \( z^* = -z_{0.01} \approx -(-2.33) = 2.33 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Critical Value
A critical value is a key concept in inferential statistics. It defines the boundary in a distribution beyond which events are considered statistically unlikely or extreme. Critical values are essential when constructing confidence intervals. For a 98% confidence interval, the critical value ensures that 98% of the data falls within that range, leaving 1% in each tail of the standard normal distribution.
This boundary is determined by the chosen confidence level. The higher the confidence level, the further away the critical value is from the mean.
This boundary is determined by the chosen confidence level. The higher the confidence level, the further away the critical value is from the mean.
- In our case, for a 98% confidence interval, the critical value is about 2.33.
- This means that we consider points that fall more than 2.33 standard deviations away from the mean as extreme.
Standard Normal Distribution
The standard normal distribution is a special case of the normal distribution. It is a continuous probability distribution that is symmetric about the mean. The mean is zero and the standard deviation is one. These qualities make it a cornerstone in statistical theory.
In practice, the standard normal distribution is used to understand probabilities related to normally distributed variables. It allows comparisons across different datasets and simplifies calculations.
In practice, the standard normal distribution is used to understand probabilities related to normally distributed variables. It allows comparisons across different datasets and simplifies calculations.
- Because it has a mean of zero, data can be standardized and transformed through z-scores.
- The area under the curve represents probabilities, which are used extensively in hypothesis testing and confidence interval construction.
Z-Score
A z-score is a statistical measurement that describes a value's position within a normal distribution. It tells how many standard deviations a point is from the mean. If it is positive, the value is above the mean; if negative, below the mean.
The formula for calculating a z-score is: \[ z = \frac{(X - \mu)}{ \sigma} \]where:
Z-scores are also crucial for finding critical values. For example, a z-score of 2.33 accounts for the 98% confidence interval in our original exercise.
The formula for calculating a z-score is: \[ z = \frac{(X - \mu)}{ \sigma} \]where:
- \(X\) is the value.
- \(\mu\) is the mean of the dataset.
- \(\sigma\) is the standard deviation.
Z-scores are also crucial for finding critical values. For example, a z-score of 2.33 accounts for the 98% confidence interval in our original exercise.
Cumulative Probability
Cumulative probability refers to the likelihood that a random variable is less than or equal to a value. In the context of the standard normal distribution, it signifies the total area under the curve to the left of a given z-score.
It provides insight into how values are distributed across the spectrum of possibilities. For example, in our exercise, a cumulative probability of 0.99 means 99% of the data falls to the left of the critical value.
It provides insight into how values are distributed across the spectrum of possibilities. For example, in our exercise, a cumulative probability of 0.99 means 99% of the data falls to the left of the critical value.
- The cumulative probability helps in locating critical values using tables or calculators.
- For a 98% confidence interval, we look at the cumulative probability of 0.99 to identify the corresponding z-score.
