Introduction to Confidence Intervals: Videos & Practice Problems

When estimating a parameter, such as the mean (denoted as μ), a single point estimate like the sample mean (x̄) is often used. However, since this point estimate may not accurately reflect the true parameter, it is beneficial to create a range of values, known as a confidence interval, which is likely to contain the parameter. A confidence interval provides a more comprehensive understanding of the parameter's possible values.

The confidence level, typically expressed as a percentage (e.g., 95%), indicates the probability that the confidence interval encompasses the true parameter. In this case, a 95% confidence level means there is a 95% chance that the interval contains the parameter. This confidence level is denoted by the symbol C, which is calculated as:

$C=1-\alpha$

Here, α represents the significance level, which is the probability that the parameter falls outside the confidence interval. For a 95% confidence level, α is calculated as:

$\alpha =1-0.95=0.05$

This α value represents the combined area of the tails outside the confidence interval, meaning each tail has an area of α/2, which in this case is 0.025.

To construct a confidence interval, we also need to understand the margin of error (E), which is the maximum likely amount of error in our estimate. It represents the distance from the point estimate to the endpoints of the confidence interval. The endpoints can be calculated using the formula:

$y=\mathrm{ŷ}\pm E$

Where ŷ is the point estimate. For example, if we have a point estimate of 4 and a margin of error of 2, the endpoints of the confidence interval would be:

$4-2=2$

and

$4+2=6$

Thus, the confidence interval is (2, 6). This means we can interpret the result by stating that we are 95% confident that the parameter y lies within this interval. Confidence intervals can also be expressed in different notations, such as interval notation (2, 6) or compact form (4 ± 2).

Understanding these concepts is crucial as we will encounter various problems requiring the calculation of confidence intervals and margins of error in future exercises.

To construct a confidence interval, it is essential to determine the margin of error, which is often not provided directly. Instead, we can calculate it using a critical value, specifically the z-score, denoted as \( z_{\alpha/2} \). This z-score is derived from the standard normal distribution and represents the endpoints of the confidence interval, effectively separating the tail ends on either side.

When working with confidence intervals, the total area of the two tails combined is represented by \( \alpha \), which is calculated as \( 1 - c \), where \( c \) is the confidence level. For instance, with a 95% confidence interval, \( \alpha \) equals \( 1 - 0.95 = 0.05 \). Since this area is split between the two tails, the area for one tail is \( \alpha/2 = 0.025 \).

To find the critical z value, we can either use a z-table or a calculator. For the negative z-score, we use the area of one tail, \( 0.025 \). For the positive z-score, we need to add the area of the confidence interval to this tail area, resulting in \( 0.025 + 0.95 = 0.975 \). Using these areas, we find that \( z_{\alpha/2} \) for a 95% confidence interval is approximately \( 1.96 \).

This critical z value remains consistent for any given confidence level. For example, for a 90% confidence interval, \( \alpha \) is \( 1 - 0.90 = 0.10 \), leading to \( \alpha/2 = 0.05 \) and a critical z value of \( 1.645 \). Similarly, for a 99% confidence interval, \( \alpha \) is \( 1 - 0.99 = 0.01 \), resulting in \( \alpha/2 = 0.005 \) and a critical z value of \( 2.576 \).

Understanding how to find these critical z scores is crucial for constructing confidence intervals and calculating the corresponding margin of error. By applying these concepts, you can effectively analyze data and make informed statistical inferences.