Problem 56 - Basic Statistics for Business and Economics with Student

Question

The following graph portrays the distribution of the number of Biggie-sized soft drinks sold at a nearby Wendy's for the last 141 days. The mean number of drinks sold per day is 91.9 and the standard deviation is 4.67 If we use the Empirical Rule, sales will be between what two values on 68 percent of the days? Sales will be between what two values on 95 percent of the days?

Step-by-Step Solution

Answer

68% of sales are between 87.23 and 96.57; 95% of sales are between 82.56 and 101.24.

Step 1: Understand the Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 Rule, states that for a normal distribution: - approximately 68% of data falls within one standard deviation of the mean, - about 95% falls within two standard deviations, - and about 99.7% falls within three standard deviations.

Step 2: Calculate the Range for 68% of the Days

To find the range where 68% of the data falls, we calculate one standard deviation above and below the mean. Given the mean (\(ar{x}\)) is 91.9 and the standard deviation (\( ext{sd}\)) is 4.67:- Lower bound = Mean - SD = 91.9 - 4.67 = 87.23- Upper bound = Mean + SD = 91.9 + 4.67 = 96.57. Thus, 68% of the sales fall between 87.23 and 96.57 drinks.

Step 3: Calculate the Range for 95% of the Days

For 95% of the data, we consider two standard deviations: - Compute the bounds by subtracting and adding twice the standard deviation from the mean. - Lower bound = Mean - 2*SD = 91.9 - 2*4.67 = 82.56 - Upper bound = Mean + 2*SD = 91.9 + 2*4.67 = 101.24. Therefore, 95% of the sales are between 82.56 and 101.24 drinks.

Key Concepts

Normal Distribution

The concept of a normal distribution is a key element in statistics and data analysis. It refers to a probability distribution that is symmetric around the mean, depicting that data near the mean are more frequent in occurrence than data far from it. This results in the characteristic "bell curve" shape when plotted on a graph. Normal distributions are important because they often resemble the real-world distribution of many types of data, including heights, test scores, and measurement errors.
In the context of the exercise, the distribution of Biggie-sized soft drinks sold at Wendy's over the last 141 days is assumed to be normal. This assumption allows us to apply statistical rules, such as the Empirical Rule, to make predictions about sales data.

• The mean, median, and mode of a normal distribution are equal and located at the center of the distribution.
• The shape of the curve is determined by the mean and standard deviation. The mean indicates where the center of the curve is, while the standard deviation indicates the distance from the center to the point where the curve begins to flatten.

Standard Deviation

Standard deviation is a measure that describes how spread out the numbers in a data set are around the mean. A small standard deviation indicates that the data points are close to the mean, while a large standard deviation shows that the data points are spread out over a wider range.
In our exercise, the standard deviation of the number of drinks sold is 4.67. This tells us how much the number of drinks sold each day varies from the average of 91.9 drinks.

• Standard deviation is crucial for understanding the variability of data. In practical terms, it helps businesses make informed decisions by showing the typical difference from average sales.
• Combined with the mean, standard deviation can give insights into the likelihood of certain sales outcomes, as per the Empirical Rule.

Mean

The mean, often referred to as the average, is a single value representing the center of a data set. It is calculated by summing all the values and then dividing by the number of values.
In the problem provided, the mean number of Biggie-sized soft drinks sold daily is 91.9. This number provides a central point around which the daily sales data are distributed.

• The mean is a useful summary figure that shows the typical, or expected, value in a data set.
• When the data distribution is normal, the mean coincides with the median and mode, which can simplify analysis and interpretation.
• Understanding the mean helps to put other aspects of data distribution, like variance and standard deviation, into context.