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Chapter 6: Problem 38

Croissant Bakery, Inc. offers special decorated cakes for birthdays, weddings, and other occasions. They also have regular cakes available in their 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. $$\begin{array}{|cc|}\hline \text { Number of Cakes } & \\\\\text { Sold in a Day } & \text { Probability } \\\\\hline 12 & .25 \\\13 & .40 \\\14 & .25 \\\15 & .10 \\\\\hline\end{array}$$

Short Answer

Expert verified
Mean: 13.2, Variance: 0.86, Standard Deviation: 0.927

Step by step solution

01

Understand the Given Table

We are provided with a probability distribution table for the number of cakes sold in a day. It includes the possible number of cakes and the probability of each outcome. Our task is to calculate the mean, variance, and standard deviation of this distribution.
02

Calculate the Mean

The mean of a probability distribution is given by the sum of the product of each outcome and its probability. So, compute as follows: \( \mu = (12 \times 0.25) + (13 \times 0.40) + (14 \times 0.25) + (15 \times 0.10) \). Calculate each product and sum them up.
03

Compute Each Product for the Mean

Calculate each product: \( 12 \times 0.25 = 3.0 \), \( 13 \times 0.40 = 5.2 \), \( 14 \times 0.25 = 3.5 \), and \( 15 \times 0.10 = 1.5 \).
04

Sum the Products for the Mean

Add the products calculated in Step 3: \( 3.0 + 5.2 + 3.5 + 1.5 = 13.2 \). Therefore, the mean \( \mu \) is 13.2.
05

Calculate the Variance

The variance is given by \( \sigma^2 = \sum (x_i - \mu)^2 P(x_i) \). Subtract the mean from each number of cakes sold, square the result, multiply by the probability, and sum for all possible outcomes.
06

Calculate Each Squared Deviation Product

Compute: \((12 - 13.2)^2 \times 0.25 = 0.36 \), \((13 - 13.2)^2 \times 0.40 = 0.016 \), \((14 - 13.2)^2 \times 0.25 = 0.16\), \((15 - 13.2)^2 \times 0.10 = 0.324 \).
07

Sum Squared Deviation Products

Add the squared deviation products: \(0.36 + 0.016 + 0.16 + 0.324 = 0.86 \). The variance \( \sigma^2 \) is 0.86.
08

Calculate the Standard Deviation

The standard deviation is the square root of the variance. Therefore, \( \sigma = \sqrt{0.86} \approx 0.927 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation
The mean, also known as the expected value, is a measure of the center of a probability distribution. Think of it as the average number of cakes the bakery expects to sell in a day. To calculate the mean, you need to multiply each possible number of cakes sold by its respective probability. Then you add all these products together.

For the bakery, the calculations are as follows:
  • Multiply: 12 cakes by 0.25 = 3.0
  • Multiply: 13 cakes by 0.40 = 5.2
  • Multiply: 14 cakes by 0.25 = 3.5
  • Multiply: 15 cakes by 0.10 = 1.5

The next step is to sum these products:
3.0 + 5.2 + 3.5 + 1.5 = 13.2.
This sum gives you the mean, which is 13.2 in this case. This suggests that, on average, about 13 cakes are sold per day.
Variance Calculation
Variance helps us understand how much the number of cakes sold varies from the mean on average. It's a measure of the distribution's spread or dispersion. To compute variance, you subtract the mean from each possible outcome, square that result, multiply by the probability, and then sum these products.

Let's break down the steps:
  • Compare each cake quantity to the mean (13.2) and square the result:
    • For 12 cakes: (12 - 13.2)^2 = 1.44
    • For 13 cakes: (13 - 13.2)^2 = 0.04
    • For 14 cakes: (14 - 13.2)^2 = 0.64
    • For 15 cakes: (15 - 13.2)^2 = 3.24
  • Multiply each squared difference by its probability:
    • 12 cakes: 1.44 脳 0.25 = 0.36
    • 13 cakes: 0.04 脳 0.40 = 0.016
    • 14 cakes: 0.64 脳 0.25 = 0.16
    • 15 cakes: 3.24 脳 0.10 = 0.324
  • Add these products together: 0.36 + 0.016 + 0.16 + 0.324 = 0.86.
The variance, therefore, is 0.86. This means there is relatively small variability in the number of cakes sold each day from the average.
Standard Deviation Calculation
Standard deviation provides insight into the variation of values relative to the mean. It is calculated as the square root of the variance, offering a measure of the spread of a data set. This value is crucial because it translates the variance into the same units as the mean, making it easier to interpret.

In this example, the variance is 0.86, so the standard deviation is calculated by taking its square root:
  • Find the square root: \( \sigma = \sqrt{0.86} \approx 0.927 \)

This tells us that most daily cake sales will deviate from the mean of 13.2 by about 0.927 cakes. In simple terms, it means that if one day the sale is less or more than the average, it's typically by nearly one cake.

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Most popular questions from this chapter

A federal study reported that 7.5 percent of the U.S. workforce has a drug problem. A drug enforcement official for the State of Indiana wished to investigate this statement. In his sample of 20 employed workers: a. How many would you expect to have a drug problem? What is the standard deviation? b. What is the likelihood that none of the workers sampled has a drug problem? c. What is the likelihood at least one has a drug problem?

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Suppose 1.5 percent of the antennas on new Nokia cell phones are defective. For a random sample of 200 antennas, find the probability that: a. None of the antennas is defective. b. Three or more of the antennas are defective.
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