Problem 7 - Basic Statistics for Business & Economics

Question

Levinson's Department Store is having a special sale this weekend. Customers charging purchases of more than \(\$ 50\) to their store credit card will be given a special Levinson's Lottery card. The customer will scratch off the card, which will indicate the amount to be taken off the total amount of the purchase. Listed below are the amount of the prize and the percent of the time that amount will be deducted from the total amount of the purchase. $$ \begin{array}{|rc|} \hline \text { Prize Amount } & \text { Probability } \\ \hline \$ 10 & .50 \\ 25 & .40 \\ 50 & .08 \\ 100 & .02 \\ \hline \end{array} $$ a. What is the mean amount deducted from the total purchase amount? b. What is the standard deviation of the amount deducted from the total purchase?

Step-by-Step Solution

Answer

Mean = $21, Standard deviation ≈ $16.09.

Step 1: Understand the Problem

We have a lottery with prizes and their probabilities. We need to find the mean (average) and standard deviation of these prize amounts. This means calculating expected values using the given probabilities.

Step 2: Calculate the Mean (Expected Value)

The mean or expected value of a discrete random variable is computed by multiplying each outcome by its probability and then summing all these products. Evaluate this using the following formula:\[ \mu = \sum{(x_i \cdot p_i)} \]Here, \(x_i\) is the prize amount and \(p_i\) is the probability for that amount.\[ \mu = (10 \times 0.50) + (25 \times 0.40) + (50 \times 0.08) + (100 \times 0.02) \]Calculating each term:- \(10 \times 0.50 = 5\)- \(25 \times 0.40 = 10\)- \(50 \times 0.08 = 4\)- \(100 \times 0.02 = 2\)Sum these products:\[ \mu = 5 + 10 + 4 + 2 = 21 \]So, the mean amount deducted is \(\$21\).

Step 3: Calculate the Variance

Variance requires computing the expected value of the squared deviation from the mean. Use the formula:\[ \sigma^2 = \sum{((x_i - \mu)^2 \cdot p_i)} \]Calculate for each prize amount:- \((10 - 21)^2 \times 0.50 = 121 \times 0.50 = 60.5\)- \((25 - 21)^2 \times 0.40 = 16 \times 0.40 = 6.4\)- \((50 - 21)^2 \times 0.08 = 841 \times 0.08 = 67.28\)- \((100 - 21)^2 \times 0.02 = 6241 \times 0.02 = 124.82\)Sum these:\[\sigma^2 = 60.5 + 6.4 + 67.28 + 124.82 = 258.98\]

Step 4: Compute the Standard Deviation

The standard deviation is the square root of the variance:\[ \sigma = \sqrt{258.98} \approx 16.09 \]Thus, the standard deviation of the amount deducted is approximately \(16.09\).

Key Concepts

Expected Value

The expected value is essentially the average of all possible outcomes of a random variable, weighted by their probabilities. When it comes to probability distributions like our example with Levinson's Lottery, understanding the expected value allows us to determine the average prize amount customers might receive when they scratch the card.

To calculate the expected value, we multiply each potential prize amount by its probability and sum up these results. This gives us a comprehensive measure that highlights what can be anticipated over numerous trials.

For instance:

• Prizes range from \(10 to \)100.
• Probabilities indicate how frequently each prize might occur.
• We compute: \[ \mu = (10 \times 0.50) + (25 \times 0.40) + (50 \times 0.08) + (100 \times 0.02) = 21 \]

Thus, the expected value, or mean prize deduction, is $21.

Standard Deviation

Standard deviation is a statistic that measures the dispersion or spread of a set of values around their mean. In the context of our lottery card prize amounts, the standard deviation will show us how much the prize amounts typically vary from the average (mean) deduction of \(21.

A small standard deviation means the prize amounts are generally close to \)21. A larger standard deviation indicates that the prize amounts fluctuate more widely from the mean.

To calculate it, we first find the variance by determining the average squared differences from the mean, and then take the square root of this variance:

• Variance: \[ \sigma^2 = ((10 - 21)^2 \times 0.50) + ((25 - 21)^2 \times 0.40) + ((50 - 21)^2 \times 0.08) + ((100 - 21)^2 \times 0.02) = 258.98\]
• Standard Deviation:\[ \sigma = \sqrt{258.98} \approx 16.09 \]

This suggests a typical variation of about $16.09 from the mean prize amount.

Variance

Variance provides a numerical value that describes the spread of values in a probability distribution. Unlike the standard deviation, which is easy to interpret in the same units as the data, variance is expressed in squared units. However, variance is essential as a foundational step in calculating the standard deviation.

In the context of the lottery card, variance shows us how the prize amounts deviate, on average, from the expected value. Calculating variance involves more than just computing averages; it weights each squared deviation by the probability of occurrence.

To find variance:

• Subtract the mean from each prize amount, square it, and then multiply by the associated probability:\[ \sigma^2 = ((10 - 21)^2 \times 0.50) + ((25 - 21)^2 \times 0.40) + ((50 - 21)^2 \times 0.08) + ((100 - 21)^2 \times 0.02) \]
• Add up all these values to get the variance, which is 258.98.

While the variance itself might not be directly insightful without further computation, it provides the basis for calculating the more intuitive standard deviation.