Question

Find the z-values needed to calculate large-sample confidence intervals for the confidence levels given. Confidence coefficient \(1-\alpha=90\)

Short answer

Answer: The critical z-value for a 90% confidence level is approximately 1.645.

Step-by-step solution

Calculate the Value of \(\alpha\)#

The confidence coefficient is given as \(1-\alpha=90\%\). First, we'll express this percentage as a decimal. \(90\%=0.9\). Now let's find \(\alpha\). Using the equation \(1-\alpha=0.9\), we can solve for \(\alpha\): $$\alpha = 1-0.9$$ $$\alpha = 0.1$$ Now, we have the value of \(\alpha=0.1\).

Find Z-score for the Confidence Level#

To find the z-score that corresponds to the \(90\%\) confidence level, we'll look at the area in two tails, since it's a two-tailed test. We'll divide our \(\alpha\) value by 2 and distribute it evenly between both tails. $$\frac{\alpha}{2} = \frac{0.1}{2}=0.05$$ So we have \(0.05\) in each tail. This means we are looking for the Z-score that has an area of \(0.95\) to its left (since \(0.9+0.05=0.95\)). We can use a standard normal distribution table or calculator to find this z-score. The z-score corresponding to the \(0.95\) area is approximately \(1.645\).

Z-values for the Confidence Interval#

We have found the critical z-value, \(1.645\), for the \(90\%\) confidence level. With this z-value, we can calculate the large-sample confidence intervals as follows: $$\text{Confidence Interval} = \bar{x} \pm 1.645 \cdot \frac{\sigma}{\sqrt{n}}$$ where \(\bar{x}\) is the sample mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Key concepts

Z-Value

Understanding the z-value is pivotal when determining confidence intervals in statistics. The z-value, also referred to as the z-score, quantifies the number of standard deviations a data point is from the mean in a standard normal distribution. To calculate a z-value, we use the formula:
\( z = \frac{x - \mu}{\sigma} \),
where \( x \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. In the context of confidence intervals, the z-value acts as a critical value that indicates the number of standard deviations required to encapsulate a certain proportion of the data within the interval, given the confidence level specified.

For example, in the exercise, we seek to calculate large-sample confidence intervals for a confidence level of 90%. The critical z-value is identified by finding the standard normal distribution value that leaves 5% of the distribution in the tails (because of the two-tailed test). Thus, we find the z-value using a z-table or calculator, which corresponds to a cumulative area of 0.95 to the left, and it is approximately 1.645.

Confidence Level

The confidence level expresses the degree of certainty we want to have in our statistical inferences. It is denoted as \(1 - \alpha\) and presented as a percentage, typically such as 90%, 95%, or 99%. When we speak of a 90% confidence level, as seen in the given exercise, this represents that we can be 90% certain that our confidence interval contains the true population parameter. Mathematical interpretation of this concept involves identifying the probability that the true parameter lies within the interval includes \(1 - \alpha\) of the time. Thus, a 90% confidence level means there's a 10% chance (\(\alpha = 0.1\)) that the interval does not contain the true parameter.

To create a confidence interval with a particular confidence level, one must adjust the width of the interval by using the appropriate critical z-value, which scales with the standardized data distribution according to the selected confidence level.

Standard Normal Distribution

The standard normal distribution, a key concept in statistics, is a special case of the normal distribution with a mean (\( \mu \)) of 0 and a standard deviation (\( \sigma \)) of 1. It's graphically represented as a bell curve where most data points lie near the mean, declining in frequency as they move away towards either tail.

Using the standard normal distribution, statisticians can make inferences about populations based on sample data. For example, using z-values determined from this distribution allows for the creation of confidence intervals. When a z-value is identified for a given confidence level, that z-score corresponds to a specific point on the curve, which indicates the cumulative area under the curve to that point. The area under the curve to the left of a positive z-value, like the 1.645 in the exercise, signifies the proportion of data within the confidence interval to the left of the corresponding value. It's critical for students to understand this concept clearly, as it is fundamental to various statistical methods, including hypothesis testing and confidence interval estimations.