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Chapter 3: Problem 79

Product codes of two, three, four, or five letters are equally likely. What are the mean and standard deviation of the number of letters in the codes?

Short Answer

Expert verified
The mean is 3.5 letters, and the standard deviation is approximately 1.118 letters.

Step by step solution

01

Determine Possible Outcomes

The possible number of letters in the product codes are 2, 3, 4, and 5. These represent the different outcomes of our experiment.
02

Assign Probabilities

Since each outcome is equally likely, the probability for each number of letters is the same. There are four possible outcomes (2, 3, 4, 5), so each has a probability of \( \frac{1}{4} \).
03

Calculate the Mean (Expected Value)

The mean of the number of letters is calculated by: \( \mu = (2 \times \frac{1}{4}) + (3 \times \frac{1}{4}) + (4 \times \frac{1}{4}) + (5 \times \frac{1}{4}) \). Calculating this gives us: \[ \mu = \frac{2}{4} + \frac{3}{4} + \frac{4}{4} + \frac{5}{4} = \frac{14}{4} = 3.5 \].
04

Calculate Each Outcome's Squared Deviation from the Mean

For each number of letters (2, 3, 4, 5), calculate the squared deviation from the mean: - For 2: \((2 - 3.5)^2 = 2.25\)- For 3: \((3 - 3.5)^2 = 0.25\)- For 4: \((4 - 3.5)^2 = 0.25\)- For 5: \((5 - 3.5)^2 = 2.25\).
05

Calculate Variance

Using the squared deviations calculated, find the variance: \[ \sigma^2 = (2.25 \times \frac{1}{4}) + (0.25 \times \frac{1}{4}) + (0.25 \times \frac{1}{4}) + (2.25 \times \frac{1}{4}) \]. Calculate this to get: \[ \sigma^2 = \frac{2.25}{4} + \frac{0.25}{4} + \frac{0.25}{4} + \frac{2.25}{4} = 1.25 \].
06

Calculate Standard Deviation

The standard deviation is the square root of the variance: \( \sigma = \sqrt{1.25} \approx 1.118 \).

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Key Concepts

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

Mean Calculation
The mean, often called the "average," is a way of describing the central value of a dataset. Imagine you have a set of numbers, like the number of letters in different product codes, which can be 2, 3, 4, or 5.
  • To find the mean, you assign a probability to each possible outcome. Here, since there are 4 different outcomes and they are equally likely, the probability for each is \( \frac{1}{4} \).
  • You then multiply each number by its probability and add up all those products.
For the product codes, the mean would be:\[ \mu = (2 \times \frac{1}{4}) + (3 \times \frac{1}{4}) + (4 \times \frac{1}{4}) + (5 \times \frac{1}{4}) = 3.5 \]Thus, the mean number of letters in the codes is 3.5, indicating that if you averaged out all possible codes, you'd expect each to have about 3.5 letters.
Standard Deviation
The standard deviation is a statistic that tells us how much variation or dispersion exists from the mean. In simpler terms, it's a measure of how "spread out" the numbers in your dataset are around the average.
  • First, you find the squared deviation for each outcome. For example, if the mean number of letters is 3.5, compute \((2 - 3.5)^2, (3 - 3.5)^2, ...\) for each possible number of letters.
  • Average these squared deviations to get the variance. The variance in our case is calculated as \[ \sigma^2 = 1.25 \].
  • The standard deviation is the square root of the variance. So, \( \sigma = \sqrt{1.25} \approx 1.118 \).
This means most of the product code lengths will vary roughly within 1.118 letters from the mean.
Probability Distribution
A probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range.
  • For the product code example, the random variable is the number of letters, with possible values of 2, 3, 4, and 5.
  • Each of these outcomes has a probability of \( \frac{1}{4} \). This makes it a uniform probability distribution, as all outcomes are equally probable.
Understanding the distribution helps in predicting which number of letters is more likely to occur. Since this is uniform, any of the number of letters is no more likely than the other.
Expected Value
The expected value is incredibly important in probability and statistics because it essentially provides a measure of the center of a probability distribution. It tells us the average result we can expect if we were to repeat an experiment many times.
  • The calculation for expected value is similar to the mean calculation, where each outcome is multiplied by its probability and then added together.
  • For the product codes, the expected value, or average number of letters, is also 3.5.
This value doesn't need to be a possible outcome itself but is the average we'd expect over many trials. It's a foundational concept in understanding and predicting future events based on past data.

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Most popular questions from this chapter

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