Problem 11 - Basic Statistics for Business & Economics

Question

The owner of Britten's Egg Farm wants to estimate the mean number of eggs produced per chicken. A sample of 20 chickens shows they produced an average of 20 eggs per month with a standard deviation of 2 eggs per month. a. What is the value of the population mean? What is the best estimate of this value? b. Explain why we need to use the \(t\) distribution. What assumption do you need to make? c. For a \(95 \%\) confidence interval, what is the value of \(t ?\) d. Develop the \(95 \%\) confidence interval for the population mean. e. Would it be reasonable to conclude that the population mean is 21 eggs? What about 25 eggs?

Step-by-Step Solution

Answer

The best estimate of the population mean is 20 eggs. We use the t-distribution due to a small sample size and unknown population standard deviation. The 95% confidence interval is 19.065 to 20.935 eggs, ruling out a mean of 21 or 25 eggs.

Step 1: Estimate the Population Mean

Since we are dealing with a sample mean and have no information about the population mean in the problem, the best estimate for the population mean (denoted as \( \mu \)) is the sample mean. Therefore, the value of the sample mean, which is 20 eggs, serves as the best estimate for the population mean.

Step 2: Justify the Use of the t-Distribution

The \( t \)-distribution is used instead of the normal distribution when the sample size is small (typically \( n < 30\)) and when the population standard deviation is unknown. In this problem, we have a sample size of 20 and no information about the population standard deviation. We assume the underlying population distribution is approximately normal.

Step 3: Determine the t-Value for 95% Confidence Interval

For a \(95\%\) confidence interval with 19 degrees of freedom (which is \( n - 1 \) or \( 20 - 1 \)), we use a \( t \)-table to find the critical \( t \)-value. This value is approximately 2.093.

Step 4: Calculate the 95% Confidence Interval

To develop the confidence interval, use the formula:\[ \bar{x} \pm t \times \frac{s}{\sqrt{n}} \]where \( \bar{x} = 20 \), \( t = 2.093 \), \( s = 2 \), and \( n = 20 \).Calculate:\[ 20 \pm 2.093 \times \frac{2}{\sqrt{20}} \]\[ = 20 \pm 0.935 \]So the 95% confidence interval is approximately \( 19.065 \) to \( 20.935 \).

Step 5: Evaluate Population Mean Claims

For the population mean to be reasonably considered as 21 eggs, 21 should fall within the confidence interval \( 19.065 \) to \( 20.935 \). Since it does not, it is unreasonable to conclude the mean is 21. Likewise, 25 is even beyond this range, making it also unreasonable.

Key Concepts

t-Distribution

When dealing with small sample sizes, such as in our case where we have 20 chickens, the t-distribution becomes very useful. The t-distribution is often used in statistics when the sample size is small and the population's standard deviation is unknown.
In simple terms, the t-distribution is like a cousin to the normal distribution but has thicker tails. This means it is more prone to producing values that fall far from the mean compared to the normal distribution.
This thicker tail property makes the t-distribution ideal for handling smaller datasets where variance could cause mean estimates to vary more significantly. Since we do not know the population standard deviation and only have a sample of 20 chickens, using the t-distribution allows for more accurate confidence interval estimates.
We also assume the data follows an approximate normal distribution, which is an important condition for the t-distribution to work effectively. Hence, in our context, we utilize the t-distribution with care to ensure we have a valid estimate for the confidence interval.

Sample Mean

The sample mean is fundamental as it serves as the best estimate of the population mean when we only have sample data. In our example, the sample of 20 chickens produced an average of 20 eggs per month, and this average represents the sample mean.
The sample mean is calculated by adding all the values for the number of eggs produced and dividing by the number of chickens in the sample (in this case, 20).
In mathematical terms, if we let\[ \bar{x} \]represent the sample mean, then \[ \bar{x} = \frac{\sum x}{n} \], where \( \sum x \) is the total number of eggs produced by all chickens in the sample, and \( n \) is the number of chickens.
Thus for our data, the value of 20 eggs helps set the stage for analyzing and making inferences about the larger population of chickens based on this sample's performance.

Population Mean

The population mean represents the average value in the entire population, which in this case is the hypothetical mean number of eggs all chickens at Britten's Egg Farm produce.
In many real-world scenarios, capturing the entire population's mean directly is impractical; hence, we rely on the sample mean to provide an estimation. This is why the sample mean of 20 eggs is our best estimate for the population mean, denoted as \( \mu \).
While we can only estimate \( \mu \) via sample data, constructing a confidence interval helps provide a range within which the population mean is likely to fall. This aids in quantitative assessments and decisions, providing insight and supporting conclusions that can guide business or scientific conclusions.
To determine a more precise range for \( \mu \), we develop a confidence interval using statistical methods, as demonstrated in the solution.

Degrees of Freedom

Degrees of freedom are an essential aspect when working with statistical estimations, especially with the t-distribution. They provide a level of flexibility and vary based on the size of the data set.
Conceptually, degrees of freedom refer to the number of independent values or quantities that can be assigned to a statistical distribution, free from any constraints.
For our sample, calculating the degrees of freedom involves subtracting one from our sample size, that is, \( n - 1 \). Therefore, with our 20 chickens, we have 19 degrees of freedom.

• This influences the shape and properties of the t-distribution we use.
• It determines which t-value to access from the t-table we need to construct our confidence interval.

In simpler terms, degrees of freedom help to correct for the fact that we're using sample data to estimate the mean of a larger population, allowing us more flexibility and accuracy in our inference.