Problem 3 - Basic Statistics for Business and Economics with Student

Question

A sample of 10 observations is selected from a normal population for which the population standard deviation is known to be 5 . 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 percent confidence interval even though the sample is less than \(30 .\) c. Determine the 95 percent confidence interval for the population mean.

Step-by-Step Solution

Answer

a. SEM is 1.5811. b. The sample is from a normal population. c. 95% CI is [16.901, 23.099].

Step 1: Calculate the standard error of the mean

The standard error of the mean (SEM) is determined using the formula: \( SEM = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the population standard deviation, and \( n \) is the sample size.Given \( \sigma = 5 \) and \( n = 10 \), the SEM is:\[SEM = \frac{5}{\sqrt{10}} = \frac{5}{3.162} \approx 1.5811\]

Step 2: Reason for using the formula for small sample size

We can use the formula (9-1) for determining the 95 percent confidence interval because the population is known to be normally distributed, and we have the population standard deviation. These conditions allow the use of the normal distribution to construct confidence intervals even though the sample size is less than 30.

Step 3: Calculate the 95% Confidence Interval

For a 95% confidence interval with known population standard deviation, the formula is: \( \bar{x} \pm Z \times SEM \). Here, \( \bar{x} = 20 \) is the sample mean, and \( Z \) is the critical value of the standard normal distribution for a 95% confidence level (1.96).Substituting the values we have:\[CI = 20 \pm 1.96 \times 1.5811\]This yields:\[CI = 20 \pm 3.099 \]Hence, the 95% confidence interval is approximately \([16.901, 23.099]\).

Key Concepts

Standard Error

The standard error is a crucial concept when dealing with sample statistics. It provides an estimate of the variation of a sample mean from the true population mean.
In simpler terms, it tells us how much the sample mean might fluctuate if you repeated your study many times. The standard error helps quantify how much sampling variability exists.
For example, if you have a large standard error, it means there is a lot of variability from sample to sample.

• Smaller standard error indicates sample mean is closer to the population mean.
• Larger standard error suggests wider spread of sample means.

For a normal distribution, the formula to calculate the standard error of the mean (SEM) is:\[SEM = \frac{\sigma}{\sqrt{n}}\]where \( \sigma \) is the population standard deviation, and \( n \) is the sample size. This formula shows that as your sample size increases, the standard error decreases.
That means larger samples give more reliable estimates of the population mean.

Normal Distribution

The normal distribution is a fundamental principle in statistics, often referred to as the bell curve due to its shape. It is symmetric, with most of the observations clustering around the central peak. The mean, median, and mode of a normal distribution are all equal.
Normal distribution is used widely in constructing confidence intervals, especially when the population standard deviation is known, as in this case.

• Useful in predicting probabilities and behaviors of the data.
• Empowers us to make predictions about the data within certain intervals.

The bell-shaped curve follows the empirical rule:

• 68% of the data falls within one standard deviation from the mean.
• 95% falls within two standard deviations.
• 99.7% falls within three standard deviations.

For smaller sample sizes, like less than 30, we can still use the normal distribution given a known population standard deviation. That’s because, under these conditions, the Central Limit Theorem assures us the distribution of the sample means follows a normal distribution.

Sample Mean

The sample mean is a statistical estimate of the central tendency of a sample dataset. It is calculated by summing up all the sample data points and then dividing by the number of data points. It serves as an estimate of the true population mean.
The sample mean is important because it provides a single value that describes the center of the sample data. In our exercise, the sample mean is 20, which indicates that the average of the selected sample observations hovers around this number.
When trying to gauge the population mean, the sample mean acts as a representative measure, especially when the sample size is sufficient.

• It forms the basis for many statistical tests and confidence intervals.
• Critical in determining various statistical analyses like ANOVA and t-tests.

When we use the sample mean in the formula for confidence intervals, it provides the midpoint for our interval estimation. This allows us to make incisive predictions about the population mean with a specified level of confidence.