Problem 6 Find the critical value \(z_{\al... [FREE SOLUTION] (2024)

Short Answer

Confidence Level

The confidence level, often expressed as a percentage, indicates how certain we can be that a parameter lies within the confidence interval derived from our sample data.
For example, a confidence level of 99% means that if we repeated an experiment or study 100 times, we would expect the true parameter to fall within the calculated interval 99 times out of 100.
Higher confidence levels provide more certainty but result in wider confidence intervals, while lower confidence levels generate narrower intervals.
This is essential when making decisions based on statistical data as it provides a measure of reliability for our estimates.

Alpha Level

The alpha level (α) is the probability of rejecting the null hypothesis when it is actually true. It's also known as the significance level and complements the confidence level.
Mathematically, the alpha level can be found by subtracting the confidence level from 1. For a 99% confidence level, the alpha level would be \(\theta = 1 - 0.99 = 0.01\).ewline The alpha level is then divided by 2, because it is split equally between the two tails of the normal distribution. So in this case, \(\frac{\theta}{2} = 0.005\).
It's used to determine critical values in hypothesis tests; small alpha levels mean stricter criteria for rejecting the null hypothesis, reducing the risk of Type I errors (false positives).

Normal Distribution

The normal distribution, often called the bell curve, is a continuous probability distribution that is symmetrical around the mean.
Most data points are concentrated around the mean, and the probabilities for values taper off as you move further from the mean.
In this distribution, approximately 68% of the data lies within one standard deviation from the mean, 95% within two, and 99.7% within three, following the Empirical Rule.
In our exercise, we use the normal distribution to determine critical values by locating specific z-scores that correspond to defined tail probabilities, like 0.005 for each tail in a 99% confidence level scenario.
These critical values help us capture the middle percentage of the data, giving us the z-scores needed for constructing confidence intervals.