Problem 7 - Basic Statistics for Business & Economics

Question

Bob Nale is the owner of Nale's Quick Fill. Bob would like to estimate the mean number of gallons of gasoline sold to his customers. Assume the number of gallons sold follows the normal distribution with a population standard deviation of 2.30 gallons. From his records, he selects a random sample of 60 sales and finds the mean number of gallons sold is \(8.60 .\) a. What is the point estimate of the population mean? b. Develop a \(99 \%\) confidence interval for the population mean. c. Interpret the meaning of part (b).

Step-by-Step Solution

Answer

a) The point estimate is 8.60 gallons. b) The 99% confidence interval is (7.836, 9.364) gallons. c) It means we are 99% confident the true mean is between 7.836 and 9.364 gallons.

Step 1: Point Estimate of the Population Mean

The point estimate of the population mean is simply the sample mean. Given that the mean number of gallons sold in the sample is \(8.60\), this will also be our point estimate for the population mean.

Step 2: Determine the Critical Value for Confidence Interval

Since we are calculating a \(99\%\) confidence interval with a known standard deviation and normal distribution, we will use the standard normal distribution (Z-distribution). The critical value (Z) for a \(99\%\) confidence interval is approximately \(2.576\).

Step 3: Calculate the Standard Error of the Sample Mean

The standard error (SE) of the sample mean is computed using the formula \(SE = \frac{\sigma}{\sqrt{n}}\), where \(\sigma = 2.30\) is the population standard deviation, and \(n = 60\) is the sample size. Thus, \(SE = \frac{2.30}{\sqrt{60}} \approx 0.2965\).

Step 4: Calculate the Margin of Error

The margin of error (ME) is determined by multiplying the standard error by the critical value. So, \(ME = 2.576 \times 0.2965 \approx 0.7638\).

Step 5: Construct the Confidence Interval

The confidence interval is calculated by adding and subtracting the margin of error from the sample mean. Therefore, the \(99\%\) confidence interval is \(8.60 \pm 0.7638\), resulting in an interval of approximately \( (7.836, 9.364) \).

Step 6: Interpret the Confidence Interval

We are \(99\%\) confident that the true population mean number of gallons sold per customer is between \(7.836\) and \(9.364\) gallons. This means that if we were to take numerous samples and construct a confidence interval in the same way, \(99\%\) of those intervals would include the true mean.

Key Concepts

Normal Distribution

The concept of a normal distribution is crucial in statistics. This is especially true when analyzing data such as the gallons of gasoline sold in a store like Nale's Quick Fill. The normal distribution is a continuous probability distribution.
It is often illustrated as a symmetrical bell-shaped curve. This shape signifies that most observations cluster around the central peak, which is the mean.

Key characteristics of a normal distribution include:

• The mean, median, and mode are all equal.
• It is symmetrical about the mean.
• The tails approach the horizontal axis asymptotically, meaning they never touch it.
• Empirically, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two, and 99.7% within three.

Understanding that gasoline sales follow a normal distribution allows Bob to use valuable tools like the Z-distribution for estimating population parameters.

Population Mean

In the context of statistics, the population mean refers to the average of a set of values for the entire population.
In Bob's case, this is the average number of gallons of gasoline sold to all his customers. However, calculating the population mean directly is often impractical. This is because it would require data from every transaction, which can be massive.

Instead, statisticians like Bob use sample data to estimate the population mean by using tools such as confidence intervals. The point estimate, or best guess, of the population mean is often the sample mean from a collected sample. This approach is both economical and efficient.

Sample Mean

The sample mean is a crucial concept in understanding statistics and data analysis. In Bob's situation, the sample mean is 8.60 gallons, which he obtained by examining 60 sales records.
This serves as a point estimate for the entire population mean. In simpler terms, it's the calculated average from a subset of the data collected.

Here’s why sample mean is beneficial:

• It's a practical way to make statistical inferences without having to investigate the entire population.
• It provides a snapshot of the overall trends and ensures decision-making with informed evidence.

By using the sample mean, Bob has a reliable basis for constructing a confidence interval to estimate the population mean more accurately.

Standard Error

The standard error is an essential statistical measure that aids in understanding how accurately a sample represents a population.
It quantifies the variability or 'spread' of the sample mean estimates when multiple samples are taken from the same population. Specifically, the standard error decreases as the sample size increases.

The formula for calculating the standard error is:\[ SE = \frac{\sigma}{\sqrt{n}} \]where \( \sigma \) is the population standard deviation, and \( n \) is the sample size. For Bob's Quick Fill, it was found to be approximately 0.2965 using \( \sigma = 2.30 \) gallons and \( n = 60 \) sales records.

Standard error plays a role in determining the margins of error and confidence intervals, shedding light on the precision of the estimated population mean from the sample mean. This is paramount for Bob to feel confident in his understanding of how much gasoline, on average, his customers use.