Problem 8 - Basic Statistics for Business & Economics

Question

Dr. Patton is a professor of English. Recently she counted the number of misspelled words in a group of student essays. She noted the distribution of misspelled words per essay followed the normal distribution with a population standard deviation of 2.44 words per essay. For her 10 a.m. section of 40 students, the mean number of misspelled words was \(6.05 .\) Construct a \(95 \%\) confidence interval for the mean number of misspelled words in the population of student essays.

Step-by-Step Solution

Answer

The 95% confidence interval is (5.295, 6.805).

Step 1: Identify the known values

We are given the following values:- Population standard deviation (\( \sigma \)) = 2.44- Sample mean (\( \bar{x} \)) = 6.05- Sample size (\( n \)) = 40- Confidence level = 95%

Step 2: Find the z-score for the confidence level

For a 95% confidence interval, the z-score that corresponds to the critical value is 1.96. This is because 95% of the data lies within 1.96 standard deviations from the mean in a standard normal distribution.

Step 3: Calculate the standard error

The standard error of the mean is calculated using the formula:\[ SE = \frac{\sigma}{\sqrt{n}} \]Substituting the known values, we get:\[ SE = \frac{2.44}{\sqrt{40}} \approx 0.3856 \]

Step 4: Calculate the margin of error

The margin of error is calculated using the formula:\[ ME = z \times SE \]Substituting the values, we get:\[ ME = 1.96 \times 0.3856 \approx 0.755 \]

Step 5: Determine the confidence interval

Finally, construct the confidence interval using the sample mean and the margin of error:\[CI = \bar{x} \pm ME \]Substituting the values, we have:\[CI = 6.05 \pm 0.755 \]This results in the interval:\[(6.05 - 0.755, 6.05 + 0.755) = (5.295, 6.805)\]

Key Concepts

Normal Distribution

The normal distribution is a key concept in statistics and is often referred to as the bell curve due to its distinct shape. It represents a distribution where most occurrences take place around the mean, and probabilities for values decrease symmetrically as you move away from the mean. Understanding this distribution is vital for interpreting data in fields from social sciences to economics.

The importance of the normal distribution lies in its natural occurrence in various datasets, such as heights, test scores, or errors in measurements. In the context of Dr. Patton's exercise, the number of misspelled words per essay follows a normal distribution, which allows us to use certain statistical techniques to make predictions about the population.

• The normal distribution simplifies the process of determining probabilities and confidence intervals, as it provides a basis for finding the z-scores.
• Central Limit Theorem supports using the normal distribution for sample means regardless of the population distribution if the sample size is large enough.

Population Standard Deviation

Population standard deviation is a statistical measure that quantifies the amount of variation or dispersion within a set of values in a population. It is denoted by the Greek letter sigma (\( \sigma \)), and in simple terms, it helps to assess how spread out the data points are from the mean.

In Dr. Patton's problem, the population standard deviation is given as \( 2.44 \) misspelled words per essay, providing essential information about the variability of the data across the entire group of students.

• A higher population standard deviation indicates more spread out data points, while a lower one suggests that the data points are closer to the mean.
• Knowing the standard deviation helps in constructing a confidence interval by calculating the standard error, which leads to understanding how much we can trust the mean of the sample to represent the population mean.

Sample Mean

The sample mean, often denoted by \( \bar{x} \), is the average value obtained from a sample, providing an estimate of the population mean. It is calculated by summing up all data points within the sample and dividing by the number of data points.

For Dr. Patton's students, the sample mean is \( 6.05 \) misspelled words per essay. This figure serves as a central value to construct the confidence interval.

• The sample mean acts as an unbiased estimator of the population mean, especially when the sample is randomly selected and sufficiently large.
• While individual samples might vary, the sample mean gives a useful indication of where the true population mean might lie, informing our confidence interval calculations.

Standard Error

Standard error (SE) is a measure of the statistical accuracy of an estimate, indicating how much the sample mean is expected to fluctuate from the true population mean. It is derived by dividing the population standard deviation by the square root of the sample size.

In this exercise, the standard error is calculated as \( SE = \frac{2.44}{\sqrt{40}} \approx 0.3856 \). This value indicates the level of precision we have in the sample mean of \( 6.05 \).

• The smaller the standard error, the more precise the estimate of the population mean.
• It plays a crucial role in determining the margin of error for the confidence interval, which ultimately gauges the reliability of the estimated mean.

Understanding standard error aids in making informed decisions based on sample data.