Problem 7 - Basic Statistics for Business and Economics with Student

Question

Bob Nale is the owner of Nale's Texaco GasTown. 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 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 percent confidence interval for the population mean. c. Interpret the meaning of part (b).

Step-by-Step Solution

Answer

a. Point estimate is 8.60 gallons. b. 99% CI: (7.8351, 9.3649). c. True mean is likely between 7.8351 and 9.3649 gallons.

Step 1: Identify the Point Estimate

The point estimate of the population mean is simply the sample mean, as it provides our best single estimate of the true population mean value. In this case, the sample mean is given as 8.60 gallons. Thus, the point estimate of the population mean is 8.60 gallons.

Step 2: Determine the Standard Error

The standard error (SE) is calculated using the formula \( SE = \frac{\sigma}{\sqrt{n}} \), where \( \sigma = 2.30 \) (standard deviation) and \( n = 60 \) (sample size). Substitute the given values:\[ SE = \frac{2.30}{\sqrt{60}} \approx 0.2968 \]

Step 3: Find the Z-score for 99% Confidence Level

For a 99% confidence interval, we need the Z-score corresponding to the two-tailed 1% level. From standard Z-tables, the critical Z-score is approximately 2.576, as it encompasses 99% of the normal distribution.

Step 4: Calculate the Margin of Error

The Margin of Error (ME) is calculated by multiplying the Z-score by the standard error: \[ ME = Z \times SE = 2.576 \times 0.2968 \approx 0.7649 \]

Step 5: Calculate the Confidence Interval

The confidence interval is found by adding and subtracting the margin of error from the sample mean:\[CI = \bar{x} \pm ME = 8.60 \pm 0.7649\]Hence, the 99% confidence interval is \[ (8.60 - 0.7649, 8.60 + 0.7649) = (7.8351, 9.3649) \]

Step 6: Interpret the Confidence Interval

The 99% confidence interval means that we are 99% confident that the true population mean number of gallons sold lies between 7.8351 and 9.3649. This interval estimation accounts for the sample variability over repeated sampling.

Key Concepts

Point Estimate

When we talk about a point estimate in statistics, it is a single value that serves as the best guess of an unknown population parameter. In simpler terms, it is the sample mean which provides the direct estimate of the population mean. In the case of Nale's Texaco GasTown, the point estimate for the mean number of gallons sold is the average found in the sample, which is 8.60 gallons.

This sample mean is used as a direct approximation because it pulls information from the sample data collected and assumes that this sample is representative of the entire population. Although it provides a useful figure, it's always good to remember that it is based on sample data, not the entire population, which means there is always a margin of error.

Standard Deviation

The concept of standard deviation is key for understanding how data values are distributed around the mean. It measures the spread of the data points, showing us how much variation there is from the average.

For example, if the standard deviation is small, the data points tend to be close to the mean, showing less variability. In the case of Bob Nale's gasoline sales, the standard deviation is given as 2.30 gallons.

• This tells us how much individual sales diverge from the mean.
• It provides crucial information about the distribution of gasoline sales over time.

By knowing the standard deviation, we have a clearer picture of how consistent or varied the variables are, which is crucial for further calculations, such as the standard error.

Standard Error

The standard error (SE) indicates the precision of the sample mean as an estimate of the population mean. It helps to understand how much the sample mean would vary from the true population mean if we took multiple samples.

To calculate the SE, you use the formula: \[ SE = \frac{\sigma}{\sqrt{n}} \] where \( \sigma \) is the standard deviation of the population, and \( n \) is the sample size. For Bob's data, with a \( \sigma \) of 2.30 and a sample size of 60, we compute \[ SE \approx 0.2968 \]

• This shows the average deviation that a sample mean will have from the actual population mean.
• The larger the sample size, the smaller the SE, indicating more reliable estimates.

The SE forms the basis for constructing the confidence interval, giving insight into the reliability of the point estimate.

Margin of Error

The margin of error (ME) provides the range within which we expect the population mean to fall, given the sample data. It helps quantify the uncertainty associated with the sample estimate.

To find the ME, you multiply the standard error by the critical value from the Z-distribution (specifically for the desired confidence level). For Bob’s exercise, using a 99% confidence level with a Z-score of 2.576, \[ ME = 2.576 \times 0.2968 \approx 0.7649 \]

• This means the estimate of 8.60 gallons can reasonably fall 0.7649 gallons above or below.
• The larger the ME, the wider the confidence interval, signaling less precision.

The margin of error is essential for determining confidence intervals, making it possible to state how much faith we can place in our estimate based on sample data.