Question

A simple random sample of 50 items from a population with \(\sigma=6\) resulted in a sample mean of 32 a. Provide a \(90 \%\) confidence interval for the population mean. b. Provide a \(95 \%\) confidence interval for the population mean. c. Provide a \(99 \%\) confidence interval for the population mean.

Short answer

The 90%, 95%, and 99% confidence intervals for the population mean are \((30.573, 33.427)\), \((30.342, 33.658)\), and \((29.860, 34.140)\), respectively.

Step-by-step solution

Identify the desired confidence intervals

The exercise asks for 90%, 95%, and 99% confidence intervals for the population mean. We will find the confidence intervals for each of these values separately. ##Step 2: Determine the z-scores for the desired confidence levels##

Determine the z-scores for the desired confidence levels

For the given confidence levels, we have: - 90% level: \(z_{\frac{\alpha}{2}} = 1.645\) - 95% level: \(z_{\frac{\alpha}{2}} = 1.960\) - 99% level: \(z_{\frac{\alpha}{2}} = 2.576\) These z-scores can be found using a z-table or statistical software. ##Step 3: Calculate the confidence intervals for each confidence level##

Calculate the confidence intervals for each confidence level

We'll now calculate the confidence intervals for each level, using the formula we mentioned earlier. a. 90% confidence interval: \[ \bar{x} \pm z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}} = 32 \pm 1.645\frac{6}{\sqrt{50}} = (30.573, 33.427) \] b. 95% confidence interval: \[ \bar{x} \pm z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}} = 32 \pm 1.960\frac{6}{\sqrt{50}} = (30.342, 33.658) \] c. 99% confidence interval: \[ \bar{x} \pm z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}} = 32 \pm 2.576\frac{6}{\sqrt{50}} = (29.860, 34.140) \] Therefore, we have the following confidence intervals for the population mean: - 90% confidence interval: \((30.573, 33.427)\) - 95% confidence interval: \((30.342, 33.658)\) - 99% confidence interval: \((29.860, 34.140)\)

Key concepts

Sample Mean

The **sample mean** is a fundamental concept in statistics, serving as a simple representation of data gathered from a sample. When you take a random sample from a population, the sample mean is essentially the average value of the data points in that sample. This value is crucial because it provides an estimate of the population mean, which is often unknown and is what we're ultimately interested in. Calculating the sample mean is straightforward. You sum up all the values in the sample and then divide by the number of values. For example, if you have a sample of 50 items and you've calculated a sample mean of 32, it means the average of these 50 values is 32. This helps us to make inferences about the larger population. Understanding the sample mean is essential for constructing confidence intervals, as it is the central point around which these intervals are built. Confidence intervals give us a range of values, centered around the sample mean, which are believed to cover the population mean with a certain level of confidence. With larger sample sizes, the sample mean is typically a better estimate of the population mean.

Z-scores

**Z-scores** are a statistical measure that describes a value's relationship to the mean of a group of values. They are expressed as the number of standard deviations away from the mean a particular value lies. In confidence intervals, z-scores are crucial for determining the width of the interval. Confidence intervals are calculated by subtracting and adding a certain number of standard errors to the sample mean. That ‘certain number’ is represented by the z-score, which is chosen based on the desired confidence level. For instance:

• A 90% confidence level adopts a z-score of approximately 1.645.
• A 95% confidence level uses a z-score of about 1.960.
• A 99% confidence level employs a z-score of roughly 2.576.

These z-scores are derived from the standard normal distribution (often found in a z-table) and help control how broad or narrow the confidence interval will be, balancing precision with certainty.

Population Mean

The **population mean** is a crucial theoretical concept in statistics that represents the average of a whole population. Unlike the sample mean, which is calculated from a sample of the population, the population mean is typically unknown and is what statisticians are interested in estimating. Confidence intervals are a powerful tool to make estimations about the population mean. They provide a range within which the true population mean is expected to fall, with a specific probability. In the exercise, different confidence intervals were calculated around a sample mean of 32. These intervals suggest that, while we cannot say exactly what the population mean is, we can estimate it with varying levels of confidence. The width of a confidence interval depends on several factors including the sample size, the standard deviation of the population, and the z-score chosen based on the confidence level. Bigger samples tend to result in narrower confidence intervals, making the estimates closer to the actual population mean. This is a cornerstone of inferential statistics, allowing researchers to make informed decisions or predictions based on their data.