Problem 35 - Basic Statistics for Business & Economics

Question

Croissant Bakery Inc. offers special decorated cakes for birthdays, weddings, and other occasions. It also has regular cakes available in its bakery. The following table gives the total number of cakes sold per day and the corresponding probability. Compute the mean, variance, and standard deviation of the number of cakes sold per day. $$ \begin{array}{cc} \hline \text { Number of Cakes } & \\ \text { Sold in a Day } & \text { Probability } \\ \hline 12 & .25 \\ 13 & .40 \\ 14 & 25 \\ 15 & .10 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Mean: 13.2, Variance: 0.86, Standard Deviation: 0.927.

Step 1: Identify the variables

First, let's identify the random variable and its possible values and probabilities. The random variable, X, is the number of cakes sold in a day. It can take on the values 12, 13, 14, and 15 with probabilities 0.25, 0.40, 0.25, and 0.10 respectively.

Step 2: Calculate the Mean (Expected Value)

The mean of a discrete random variable X is calculated using the formula: \( E(X) = \sum x_i P(x_i) \), where \( x_i \) are the values of the random variable and \( P(x_i) \) are the probabilities. Here, we compute:\[ E(X) = 12 \times 0.25 + 13 \times 0.40 + 14 \times 0.25 + 15 \times 0.10 = 3 + 5.2 + 3.5 + 1.5 = 13.2 \]

Step 3: Calculate the Variance

The variance of a discrete random variable is given by \( Var(X) = \sum (x_i - E(X))^2 P(x_i) \). Using the previous mean value (13.2), we calculate:\[ Var(X) = (12-13.2)^2 \times 0.25 + (13-13.2)^2 \times 0.40 + (14-13.2)^2 \times 0.25 + (15-13.2)^2 \times 0.10 \]\[ = 1.44 \times 0.25 + 0.04 \times 0.40 + 0.64 \times 0.25 + 3.24 \times 0.10 \]\[ = 0.36 + 0.016 + 0.16 + 0.324 = 0.86 \]

Step 4: Calculate the Standard Deviation

The standard deviation is the square root of the variance. Therefore, we compute:\[ SD(X) = \sqrt{Var(X)} = \sqrt{0.86} \approx 0.927 \]

Key Concepts

Mean Calculation

In probability distributions, the mean—or expected value—represents the average outcome you can expect if you perform an experiment numerous times. It is a crucial measure that gives us an idea of the central tendency of a probability distribution. To calculate the mean for a discrete random variable, you multiply each outcome by its probability and then sum up all those products. For example, if you have the number of cakes sold per day and their corresponding probabilities like Croissant Bakery Inc., you use the formula:\[ E(X) = \sum x_i P(x_i) \]Where \( x_i \) are the possible outcomes (number of cakes), and \( P(x_i) \) is the probability of each outcome. In our example:

• 12 cakes sold with probability 0.25
• 13 cakes sold with probability 0.40
• 14 cakes sold with probability 0.25
• 15 cakes sold with probability 0.10

The calculation becomes \( 12 \times 0.25 + 13 \times 0.40 + 14 \times 0.25 + 15 \times 0.10 = 13.2 \). Hence, the mean number of cakes sold per day is 13.2. This tells us on average the bakery sells approximately 13 cakes a day.

Variance Calculation

Variance is a key measure of the spread or variability of a probability distribution. It tells us how much the outcomes differ from the mean. To calculate variance for a discrete random variable, we use the following formula:\[ Var(X) = \sum (x_i - E(X))^2 P(x_i) \]This involves finding the squared difference between each possible outcome and the mean, multiplying each squared difference by its probability, and then summing up these values. For Croissant Bakery Inc., where the mean is 13.2, we calculate:

• For 12 cakes: \((12 - 13.2)^2 \times 0.25\)
• For 13 cakes: \((13 - 13.2)^2 \times 0.40\)
• For 14 cakes: \((14 - 13.2)^2 \times 0.25\)
• For 15 cakes: \((15 - 13.2)^2 \times 0.10\)

This gives us a variance of 0.86. This value quantifies how much the number of cakes sold daily fluctuates around the mean. A higher variance would indicate more variability in the number sold each day.

Standard Deviation

Standard deviation is another fundamental concept in statistics that provides insight into the dispersion of a dataset relative to its mean. It is simply the square root of the variance, making it easier to interpret since it is in the same unit as the data. For Croissant Bakery Inc., with a calculated variance of 0.86, the standard deviation is computed as:\[ SD(X) = \sqrt{Var(X)} = \sqrt{0.86} \approx 0.927 \]This means that, on average, the number of cakes sold each day will vary by about 0.927 cakes from the mean value of 13.2 cakes. Standard deviation effectively communicates how diverse the daily cake sales are in comparison to the mean, making it a valuable measure for understanding data distributions. Remember, a smaller standard deviation indicates that the data points tend to be close to the mean, while a larger value implies more spread.