Find α \alpha α for a 90% confidence interval.

α = 0.10 \alpha=0.10 α = 0.10

α = 0.90 \alpha=0.90 α = 0.90

α = 0.05 \alpha=0.05 α = 0.05

α = 0.01 \alpha=0.01 α = 0.01

7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals: Videos & Practice Problems

Video Lessons Worksheet

Topic summary

Confidence intervals provide a range of values likely to contain a population parameter, enhancing the precision of point estimates like $x ̄$ . The confidence level, denoted as $c$ , indicates the probability that the interval includes the parameter, with $α =1− c$ representing the area outside the interval. The margin of error, $e$ , defines the distance from the point estimate to the interval's endpoints, calculated using critical z-values for specific confidence levels.

concept

Introduction to Confidence Intervals

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Introduction to Confidence Intervals Video Summary

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 point estimates rarely provide the exact value of the parameter, it is more beneficial to create a range of values that the parameter is likely to fall within. This range is known as a confidence interval.

A confidence interval is constructed based on a specified confidence level, which represents the probability that the interval contains the true parameter value. For example, a 95% confidence interval indicates that there is a 95% probability that the interval encompasses the parameter being estimated. This confidence level is often denoted as C, where C = 1 - α. 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 1 - 0.95, resulting in α = 0.05. This α value is split between the two tails of the distribution, leading to an area of α/2 = 0.025 in each tail.

To construct a confidence interval, one must also understand the concept of margin of error (E). The margin of error is the maximum expected difference between the point estimate and the true parameter value. It is the distance from the point estimate to the endpoints of the confidence interval. The endpoints can be calculated by taking the point estimate (ŷ) and adding or subtracting the margin of error: ŷ - E and ŷ + E.

For instance, if the point estimate ŷ is 4 and the margin of error E is 2, the endpoints of the confidence interval would be calculated as follows: 4 - 2 = 2 and 4 + 2 = 6. Therefore, the confidence interval is [2, 6]. This can also be expressed in compact form as ŷ ± E, which in this case would be 4 ± 2.

When interpreting the confidence interval, one would state that there is a 95% confidence that the parameter y falls between the two calculated endpoints, 2 and 6. Understanding these concepts is crucial for accurately constructing and interpreting confidence intervals, which will be further explored in subsequent problems and examples.

Problem

Find $\alpha$ for a 90% confidence interval.

$\alpha=0.90$

$\alpha=0.10$

$\alpha=0.05$

$\alpha=0.01$

Problem

A market analyst is estimating the average monthly spending on online subscriptions among young adults. The point estimate y for the average monthly spending is $dollar sign 45.60$ , with a margin of error $E$ of $dollar sign 3.20$ at a $90 percent sign$ confidence level. Write and interpret a $90 percent sign$ Confidence Interval for the average monthly spending.

($44.00, $47.2); We are 90% confident that ($44.00, $47.2) contains the true average monthly spending.

($42.72, $48.48); We are 90% confident that ($42.72, $48.48) contains the true average monthly spending.

($45.28, $45.92); We are 90% confident that ($45.28, $45.92) contains the true average monthly spending.

($42.40, $48.40); We are 90% confident that ($42.40, $48.40) contains the true average monthly spending.

Problem

A company is interested in estimating lunch costs in their cafeteria, so they survey employees asking for their monthly lunch expenses. The staff calculated a 95% confidence interval for the mean amount of money spent on lunch to be ($580, $720). What are the point estimate and margin of error for this sample data?

$y hat equals dollar sign 600$ , $E equals dollar sign 20$

$y hat equals dollar sign 650$ , $E equals dollar sign 70$

$ŷ=\$600$ , $E equals dollar sign 40$

$ŷ=\$650$ , $E equals dollar sign 140$

concept

Critical Values: z Scores

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Critical Values: z Scores Video Summary

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 $ 0.05 $. Since this area is split between the two tails, the area for one tail is $ \alpha/2 = 0.025 $. This value indicates the probability that the parameter lies in one of the tails.

To find the critical z value, you can either use a z-table or a calculator. For the negative z-score, you would use the area of $ 0.025 $, while for the positive z-score, you would add this area to the confidence level (0.95), resulting in an area of $ 0.975 $. This leads to the critical z value of $ z_{\alpha/2} = 1.96 $ for a 95% confidence interval.

It is important to note that the critical z score remains consistent for a given confidence level. For example, for a 90% confidence interval, the calculation yields $ \alpha/2 = 0.05 $, resulting in a critical z value of $ z_{\alpha/2} = 1.645 $. Similarly, for a 99% confidence interval, $ \alpha/2 = 0.005 $ gives a critical z value of $ z_{\alpha/2} = 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.

Problem

Find the critical value, $z_{\frac{\alpha}{2}}$ , for an 80% confidence interval.

0.10

1.282

0.40

0.90

Here’s what students ask on this topic:

What is a confidence interval and why is it important in statistics?

A confidence interval is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter. It provides more information than a single point estimate by indicating the reliability of the estimate. The confidence level, such as 95%, represents the probability that the interval contains the parameter. This is crucial in statistics as it allows researchers to make inferences about a population with a known level of certainty, accounting for sampling variability. Confidence intervals are widely used in hypothesis testing, quality control, and decision-making processes, providing a measure of precision and reliability in statistical conclusions.

How do you calculate the margin of error for a confidence interval?

The margin of error for a confidence interval is calculated using the critical value and the standard error of the sample statistic. The formula is: $e = z$ _α/2×SE, where $z$ _α/2 is the critical z-value corresponding to the desired confidence level, and $SE$ is the standard error of the sample statistic. The critical z-value can be found using a z-table or calculator. The margin of error indicates the maximum expected difference between the sample statistic and the population parameter, providing a measure of the interval's precision.

What is the relationship between confidence level and alpha in a confidence interval?

The confidence level and alpha are complementary in a confidence interval. The confidence level, denoted as $c$ , represents the probability that the interval contains the true population parameter. Alpha, denoted as $α$ , is the probability that the parameter lies outside the interval. The relationship is expressed as $c = 1 - α$ . For example, a 95% confidence level corresponds to an alpha of 0.05, meaning there is a 5% chance that the parameter is not within the interval. Understanding this relationship helps in determining the reliability and precision of the interval estimate.

How do you interpret a 95% confidence interval?

Interpreting a 95% confidence interval involves understanding that it provides a range of values within which the true population parameter is likely to fall, with 95% certainty. For example, if a 95% confidence interval for a population mean is (2, 6), it means we are 95% confident that the true mean lies between 2 and 6. This does not imply that there is a 95% probability that the parameter is within this range for a specific sample; rather, it means that if we were to take many samples and construct intervals in the same way, 95% of those intervals would contain the true parameter. This interpretation helps in making informed decisions based on statistical data.

What are critical z-values and how are they used in confidence intervals?

Critical z-values are specific points on the standard normal distribution that correspond to the desired confidence level of a confidence interval. They are used to determine the margin of error and the endpoints of the interval. For a given confidence level, the critical z-value, denoted as $z$ _α/2, is found by dividing alpha by two and locating the corresponding z-score. Common critical z-values include 1.96 for a 95% confidence level, 1.645 for 90%, and 2.576 for 99%. These values are essential in constructing confidence intervals, as they help quantify the uncertainty associated with the sample estimate, providing a range that likely contains the true population parameter.