Problem 12 - Basic Statistics for Business & Economics

Question

The U.S. Dairy Industry wants to estimate the mean yearly milk consumption. A sample of 16 people reveals the mean yearly consumption to be 45 gallons with a standard deviation of 20 gallons. Assume the population distribution is normal. 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 90\% confidence interval, what is the value of \(t ?\) d. Develop the \(90 \%\) confidence interval for the population mean. e. Would it be reasonable to conclude that the population mean is 48 gallons?

Step-by-Step Solution

Answer

Best estimate of mean is 45 gallons; use t-distribution due to small sample. The 90% confidence interval is (36.235, 53.765). Yes, 48 is reasonable.

Step 1: Determine the Population Mean and Best Estimate

The problem does not provide the true population mean. The best estimate for the population mean in this context is the sample mean. Therefore, our best estimate of the population mean is 45 gallons.

Step 2: Determine Why to Use t-Distribution

We use the t-distribution because the sample size is small (n=16, which is less than 30), and the population standard deviation is unknown. This requires assuming that the population follows a normal distribution.

Step 3: Identify t-Value for 90% Confidence Interval

For a 90% confidence interval and 15 degrees of freedom (n-1), we look up the t-table to find the critical t-value. The t-value is approximately 1.753.

Step 4: Calculate the Confidence Interval

Use the formula for the confidence interval: \( \bar{x} \pm t \times \frac{s}{\sqrt{n}} \). Here, \(\bar{x} = 45\), \(s = 20\), and \(n = 16\). Calculate: \[ 45 \pm 1.753 \times \frac{20}{\sqrt{16}} = 45 \pm 8.765 \]This results in the interval (36.235, 53.765).

Step 5: Evaluate 48 Gallons Within Confidence Interval

Since 48 gallons lies within the 90% confidence interval (36.235, 53.765), it is reasonable to conclude that the true population mean could be 48 gallons.

Key Concepts

t-distribution

The t-distribution is a type of probability distribution that is often used in statistics when the sample size is small and the population standard deviation is not known. In such cases, the normal distribution cannot be applied directly due to its reliance on the known population standard deviation. Instead, the t-distribution becomes the tool of choice.

One key feature of the t-distribution is that it is similar to the normal distribution but has heavier tails. These heavier tails make it more accommodating for the uncertainty introduced by small sample sizes. As the sample size increases, the t-distribution approaches the normal distribution.

When using the t-distribution, the critical value is found from a t-table, which requires knowing the degrees of freedom. The degrees of freedom usually equal the sample size minus one (n-1). This adjustment helps in making more accurate estimates of the population mean from a small sample. By using the t-distribution, we can construct confidence intervals that more accurately reflect the variability present in small samples.

sample mean

The sample mean is a crucial component in statistics because it serves as an estimate of the population mean, especially when the population mean is unknown. In many practical scenarios, like the one with the U.S. Dairy Industry, researchers are limited to a sample and must make inferences about the population based on it.

To calculate the sample mean, simply sum up all observed values in the sample and divide by the number of observations. For example, if we have 16 people and their yearly milk consumption averages to 45 gallons, then 45 gallons is the sample mean.

• It provides a single value representation of the central tendency of data from a sample.
• The sample mean helps determine the best estimate for the population mean when the actual population mean is not available.

This estimation assumes that the sample accurately represents the population. Thus, while never exact, the sample mean becomes a strong approximation tool for statistical analysis.

normal distribution

The normal distribution is often referred to as the "bell curve" due to its symmetric shape. It's a continuous probability distribution that is fundamental to many statistical processes, including calculating confidence intervals, because many variables in nature and research follow this pattern.

One important characteristic of the normal distribution is that it is defined by its mean and standard deviation. The mean determines the center of the curve, while the standard deviation measures the spread of the data around the mean.

• Approximately 68% of the data falls within one standard deviation of the mean.
• About 95% falls within two standard deviations, and around 99.7% within three.

In the given exercise, the assumption that the sample is drawn from a normally distributed population justifies the use of statistical methods like the t-distribution for estimating confidence intervals. If the population weren't assumed normal, different methods might be necessary.

population mean

The population mean is the average of a set of characteristics in an entire population. While it's the true value we're often interested in, finding it can be challenging, especially in large populations.

Since measuring every member of a population is typically infeasible, researchers use the sample mean to estimate the population mean. In this context, knowing the population mean would provide insight into general behaviors or characteristics, like average milk consumption in the U.S.

• It is the goal of many statistical analyses to make inferences about this value.
• The accuracy of these inferences depends heavily on how well the sample represents the population.

By developing confidence intervals for the population mean, statisticians can state, with a given level of certainty, that the population mean falls within a specified range. This allows conclusions to be made even when the exact population mean is not observed directly.