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Chapter 8: Problem 70

If \(X\) is a random variable, what is the difference between a sample mean of measurements of \(X\) and the expected value of \(X ?\) Illustrate by means of an example.

Short Answer

Expert verified
The sample mean is the average of observed outcomes, while the expected value is a weighted average of all possible values of a random variable. The main difference is that the sample mean is based on actual data from a sample, while the expected value is based on a theoretical calculation using all possible outcomes. For example, if we throw a die a finite number of times, the sample mean might not equal the expected value, but if we could throw the die infinitely, the sample mean would approach the expected value of 3.5.

Step by step solution

01

Defining Sample Mean and Expected Value

Sample mean is the average of the observed outcomes. It is practically computed as the sum of the observed outcomes divided by the number of outcomes. The expected value, also known as the population mean, is the long-term average in a random distribution. In other words, it is a weighted average of all possible values of a random variable, where each value is weighted by its probability of occurrence.
02

Demonstrating the Calculation of Sample Mean and Expected Value

Let鈥檚 assume a random variable \(X\) that represents the outcome when a die is thrown. There are 6 outcomes, each equally likely with a probability of \(1/6\). For sample mean, consider one case where we roll the die 5 times and the outcomes are 3, 4, 2, 6 and 1. The sample mean would be the sum of these outcomes divided by 5. i.e. \( (3+4+2+6+1)/5 = 3.2 \) For expected value, we consider all possible outcomes and their respective probabilities. Since each outcome from 1 to 6 has a probability of \(1/6\), the expected value is computed as \( E[X] = 1*(1/6) + 2*(1/6) + 3*(1/6) + 4*(1/6) + 5*(1/6) + 6*(1/6) = 3.5 \)
03

Understanding the Difference

The difference between sample mean and expected value is that sample mean is based on actual data from a sample, while the expected value is based on theoretical calculation considering all possible outcomes. In practice, the sample mean might vary from one sample to another and won't necessarily equal to the expected value, though they may get closer as the sample size increases. So, in our example, if we throw the die only a few times, the sample mean will not necessarily be close to 3.5 but if we could throw the die infinite times and calculate the mean, it will get closer and closer to 3.5, the expected value.

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Most popular questions from this chapter

Income Distribution up to \(\$ 100,000\) The following table shows the distribution of household incomes for a sample of 1,000 households in the United States with incomes up to \(\$ 100,000^{41}\) \begin{tabular}{|c|c|c|c|c|c|} \hline 2000 Income (thousands) & \(\$ 10\) & \(\$ 30\) & \(\$ 50\) & \(\$ 70\) & \(\$ 90\) \\ \hline Households & 270 & 280 & 200 & 150 & 100 \\ \hline \end{tabular} Compute the expected value \(\mu\) and the standard deviation \(\sigma\) of the associated random variable \(X\). If we define a "lower income" family as one whose income is more than one standard deviation below the mean, and a "higher income" family as one whose income is at least one standard deviation above the mean, what is the income gap between higher- and lower-income families in the United States? (Round your answers to the nearest \(\$ 1,000\).)

I \(\vee\) Supermarkets A survey of supermarkets in the United States yielded the following relative frequency table, where \(X\) is the number of checkout lanes at a randomly chosen supermarket: \(^{49}\) \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \(\boldsymbol{x}\) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \(\boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x})\) & \(.01\) & \(.04\) & \(.04\) & \(.08\) & \(.10\) & \(.15\) & \(.25\) & \(.20\) & \(.08\) & \(.05\) \\ \hline \end{tabular} a. Compute the mean, variance, and standard deviation (accurate to one decimal place). b. As financial planning manager at Express Lane Mart, you wish to install a number of checkout lanes that is in the range of at least \(75 \%\) of all supermarkets. What is this range according to Chebyshev's inequality? What is the least number of checkout lanes you should install so as to fall within this range?

Compute the (sample) variance and standard deviation of the given data sample. (You calculated the means in the Section 8.3 exercises. Round all answers to two decimal places.) 3,1,6,-3,0,5

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The following figures show the price of gold per ounce, in dollars, for the 10-business day period Feb. 2-Feb. 13, \(2009:^{.21}\) $$ 918,905,905,920,913,895,910,938,943,936 $$ Find the sample mean, median, and mode(s). What do your answers tell you about the price of gold?
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