Problem 8 - Basic Statistics for Business and Economics with Student

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 standard deviation of 2.44 words per essay. For her 10 a.m. section of 40 students, the mean number of misspelled words per essay was \(6.05 .\) Construct a 95 percent confidence interval for the mean number of misspelled words in the population of student essays.

Step-by-Step Solution

Answer

The 95% confidence interval for the mean is approximately (5.293, 6.807).

Step 1: Identify the Given Values

From the problem, we know that the standard deviation \( \sigma = 2.44 \), the sample mean \( \bar{x} = 6.05 \), and the sample size \( n = 40 \).

Step 2: Determine the Critical Value

For a 95% confidence interval, we need to determine the z-value that corresponds to the middle 95% of a standard normal distribution. This value is approximately \( z = 1.96 \).

Step 3: Calculate the Standard Error

The standard error (SE) of the mean is calculated as \( \frac{\sigma}{\sqrt{n}} \). Substituting the values, we have \( SE = \frac{2.44}{\sqrt{40}} \approx 0.386 \).

Step 4: Calculate the Margin of Error

The margin of error (ME) is determined by multiplying the z-value by the standard error: \( ME = z \times SE = 1.96 \times 0.386 \approx 0.757 \).

Step 5: Construct the Confidence Interval

The 95% confidence interval is constructed by adding and subtracting the margin of error from the sample mean: \( \bar{x} \pm ME = 6.05 \pm 0.757 \). So, the interval is approximately \( (5.293, 6.807) \).

Key Concepts

Normal Distribution

In statistics, the normal distribution is a common and useful distribution that describes how the values of a variable are dispersed in a symmetrical bell-shaped curve. This curve is centered around the mean, which represents the average of all values. Many naturally occurring phenomena tend to follow this pattern, which is why the normal distribution is often used in statistical analysis.
The key features of a normal distribution include:

• Symmetrical shape, meaning both sides of the curve are mirror images
• Mean, median, and mode are all at the center of the distribution
• The spread of the data is determined by the standard deviation
  

In the context of Dr. Patton's problem, the number of misspelled words per essay follows a normal distribution. This assumption allows us to use certain statistical techniques, such as constructing a confidence interval, which is only valid when the data follows a normal distribution.

Standard Deviation

Standard deviation is a measure of how much the values in a dataset deviate from the mean. It provides an idea of the variability or spread of the dataset. A smaller standard deviation means the data points are closer to the mean, whereas a larger one indicates a wider range of values.
Standard deviation is important because it helps to understand the dispersion and consistency within a dataset.
In Dr. Patton's class, the standard deviation of misspelled words per essay is 2.44, which means, on average, the number of misspelled words varies by 2.44 words from the mean. This variability is crucial when calculating the precise confidence interval for the average number of misspelled words.
Understanding the standard deviation helps in grasping how much the data extends from the mean and is essential in forming a reliable confidence interval.

Sample Mean

The sample mean is the average of all the values in a collected sample. It provides a snapshot of the average outcome in your sample set. While it may not be identical to the true population mean, it is calculated to be the best estimate of it. In our exercise, Dr. Patton found the sample mean to be 6.05 misspelled words per essay. This means that across the essays analyzed, this was the average number of spelling errors identified.
Having a sample mean allows researchers and statisticians to make estimations about a larger group's mean by using the sample data. This is an essential step in deriving further calculations, such as the margin of error and the confidence interval.

Margin of Error

The margin of error is a critical statistical calculation that represents the extent of possible error in your estimated sample mean when predicting the population mean. Essentially, it tells us how much we can expect the sample mean to vary from the true population mean.
In Dr. Patton's calculation, the margin of error was found to be approximately 0.757. This was determined using the critical z-value (1.96 for a 95% confidence interval) and the standard error derived from the standard deviation.

• The margin of error accounts for the uncertainty and variability inherent in any sample-based estimation
• Helps provide a range (confidence interval) within which the true population mean is expected to lie

It enables researchers to understand the possible range of variation and offers more meaningful insights than the sample mean alone. In this context, the actual mean number of misspelled words lies somewhere between 5.293 and 6.807. A well-calculated margin of error thus ensures confidence in these statistical predictions.