Problem 1 - Basic Statistics for Business & Economics

Question

A sample of 49 observations is taken from a normal population with a standard deviation of \(10 .\) The sample mean is \(55 .\) Determine the \(99 \%\) confidence interval for the population mean.

Step-by-Step Solution

Answer

The 99% confidence interval is (51.32, 58.68).

Step 1: Understanding the Problem

We need to find the 99% confidence interval for the population mean. We are given a sample of 49 observations with a sample mean of 55 and a known population standard deviation of 10.

Step 2: Identify the Formula

The formula for the confidence interval of the population mean when the population standard deviation is known is: \[ \bar{x} \pm z_{\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right)\]Where:- \(\bar{x}\) is the sample mean,- \(\sigma\) is the population standard deviation,- \(n\) is the sample size,- \(z_{\alpha/2}\) is the z-score corresponding to the confidence level.

Step 3: Determine the Z-score

For a 99% confidence interval, the significance level \(\alpha\) is 0.01. The z-score \(z_{\alpha/2}\) for 99% confidence is approximately 2.576.

Step 4: Calculate the Standard Error

The standard error of the mean is calculated as:\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{10}{\sqrt{49}} = \frac{10}{7} \approx 1.4286\]

Step 5: Compute the Margin of Error

The margin of error is calculated by multiplying the z-score by the standard error:\[ ME = z_{\alpha/2} \times SE = 2.576 \times 1.4286 \approx 3.6807\]

Step 6: Determine the Confidence Interval

The 99% confidence interval is given by:\[\bar{x} \pm ME = 55 \pm 3.6807\]Calculating the limits, we get:\[ (55 - 3.6807, 55 + 3.6807) = (51.3193, 58.6807)\]

Step 7: Conclusion

The 99% confidence interval for the population mean is approximately (51.32, 58.68).

Key Concepts

Sample Mean

The sample mean, often represented by \( \bar{x} \), is the average of a set of observations from a sample. In simple terms, it gives us a central value that summarizes all the data points in the sample. To calculate the sample mean, add up all the values in the sample and then divide by the number of observations. For instance, if you have a sample of 49 observations with each value adding up to 2,695, then the sample mean is \( \bar{x} = \frac{2695}{49} = 55 \).

• The sample mean is a key statistic used to estimate the population mean.
• It helps in making inferences about the entire population based on a smaller group.
• It's a foundational concept in creating confidence intervals and conducting hypothesis tests.

Understanding the sample mean in the context of a confidence interval gives you insight into how representative your sample might be of the larger population. It is important to remember that while the sample mean is an estimate, it's typically only accurate if the sample is random and the size is adequate.

Population Standard Deviation

Population standard deviation, denoted as \( \sigma \), is a measure of the variability or dispersion of a set of data points in a population. It's essential to grasp how spread out the values are from the population's mean. Unlike the sample standard deviation, population standard deviation considers every data point in the entire population.

• A small population standard deviation indicates that the data points are clustered closely around the mean.
• A large standard deviation shows that the data points are spread out over a wider range of values.
• It provides a way to understand how diverse the population's data is.

For example, in the exercise, a population standard deviation of 10 suggests moderate variability. This figure is crucial when calculating the confidence interval as it directly influences the standard error and ultimately the width of the interval.

Z-score

A z-score, symbolized by \( z \), is a statistical measurement that describes a data point's relation to the mean of a group of values. It's expressed in terms of standard deviations from the mean. For a standard normal distribution, the z-score helps determine how unusual a particular data point is. Specifically, it measures the number of standard deviations away from the mean a data point is.

• Z-scores are used extensively in statistics to assign probabilities.
• They allow us to compare scores from different data sets or different units of measure.
• In confidence intervals, specific z-scores correspond to different levels of confidence.

In our example problem, we wanted a 99% confidence interval. The z-score for a 99% confidence level is approximately 2.576, indicating that we are looking at values within 2.576 standard deviations from the mean to cover 99% of the population.

Normal Distribution

The normal distribution, sometimes known as the bell curve, is a probability distribution that is symmetric about the mean. Most of the observed data points are close to the mean. It is one of the most commonly used distributions in statistics due to its properties and the central limit theorem.

• It is defined by two parameters: the mean (\( \mu \)) and the standard deviation (\( \sigma \)).
• The area under the curve represents probabilities, with the total area equaling 1.
• A normal distribution can model many natural phenomena and measurement errors.

When data follows a normal distribution, tools like z-scores and confidence intervals become very powerful. In the exercise, knowing the sample is from a normal distribution allows us to confidently use the z-score method to estimate the population mean.