Problem 3 - Basic Statistics for Business & Economics

Question

A sample of 250 observations is selected from a normal population with a population standard deviation of \(25 .\) The sample mean is 20 . a. Determine the standard error of the mean. b. Explain why we can use formula \((9-1)\) to determine the \(95 \%\) confidence interval. c. Determine the \(95 \%\) confidence interval for the population mean.

Step-by-Step Solution

Answer

Standard Error of the Mean is 1.58. We can use formula (9-1) due to large sample size and known population standard deviation. The 95% confidence interval is (16.90, 23.10).

Step 1: Calculate the Standard Error of the Mean

The standard error of the mean (SEM) is calculated using the formula for SEM: \( \text{SEM} = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the population standard deviation, and \( n \) is the sample size. Given that \( \sigma = 25 \) and \( n = 250 \), we can substitute these values into the formula:\[ \text{SEM} = \frac{25}{\sqrt{250}} \approx \frac{25}{15.81} \approx 1.58 \].

Step 2: Explanation of Formula (9-1) for Confidence Interval

Formula (9-1) is used to compute confidence intervals when the sample size is large or when the population standard deviation is known. In this case, both conditions are met: the sample size is relatively large (n = 250), allowing us to invoke the Central Limit Theorem, ensuring that the sampling distribution of the sample mean is approximately normal. Additionally, the population standard deviation is known (25). Hence, using the Z-distribution for confidence intervals is appropriate.

Step 3: Calculate the 95% Confidence Interval

The 95% confidence interval is calculated using the formula: \( \text{CI} = \bar{x} \pm Z \times \text{SEM} \), where \( \bar{x} \) is the sample mean, and Z is the Z-score for a 95% confidence level (approximately 1.96).Given that \( \bar{x} = 20 \) and \( \, \text{SEM} = 1.58 \), the confidence interval is calculated as follows:\[ \text{CI} = 20 \pm 1.96 \times 1.58 \]\[ \text{CI} = 20 \pm 3.10 \]Thus, the 95% confidence interval is approximately (16.90, 23.10).

Key Concepts

Standard Error of the Mean

The Standard Error of the Mean (SEM) is a measure that helps us understand how much the sample mean is expected to vary due to random chance. It's essentially the standard deviation of the sample mean distribution. Calculating the SEM involves knowing both the population standard deviation and the sample size. The formula is given by\[\text{SEM} = \frac{\sigma}{\sqrt{n}}\]where \(\sigma\) represents the population standard deviation, and \(n\) denotes the sample size. For example, if \(\sigma = 25\) and \(n = 250\), the calculation becomes \(\text{SEM} = \frac{25}{\sqrt{250}}\), giving approximately 1.58. A smaller SEM means the sample mean is more likely to be close to the population mean.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental concept in statistics that assures us that when we take a large enough sample from a population, the distribution of the sample mean will tend to be normal. This holds true even if the population itself is not normally distributed, provided the sample size is large enough—commonly, a sample size of 30 or more is considered sufficient. In our case, with a sample size of 250, we can confidently apply the CLT. This means that the values we calculate, like the sample mean, will form a normal distribution, allowing us to utilize normal probability theory for further calculations like confidence intervals.

Z-distribution

The Z-distribution, or standard normal distribution, is a bell-shaped curve characterized by a mean of 0 and a standard deviation of 1. It's a special form of the normal distribution used when dealing with large samples or when the population standard deviation is known. Using the Z-distribution in confidence interval calculations enables us to determine how far off our sample mean likely is from the true population mean. When we apply the Z-distribution to the 95% confidence interval, we use a Z-score of approximately 1.96, which encapsulates 95% of the data under a normal curve. This process helps us indicate the range where we expect the true population mean to lie 95% of the time.

Population Standard Deviation

The population standard deviation is a measure of the amount of variation or dispersion in a set of values within an entire population. Unlike the sample standard deviation, which estimates this spread based on a smaller group, the population standard deviation focuses on the complete group. Knowing the population standard deviation is crucial for certain statistical analyses, such as calculating the SEM and confidence intervals when using the Z-distribution. In the context of our example, having the population standard deviation \(\sigma = 25\) ensures the accuracy and applicability of our calculations by directly feeding into the SEM calculation and enabling the use of the Z-distribution for reliable confidence interval estimation.