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Chapter 8: Problem 66

Explain how you can use a sample to estimate an expected value.

Short Answer

Expert verified
To estimate an expected value using a sample, first collect a representative sample of individuals or items from the population of interest. Next, calculate the sample mean by summing up all the values in the sample and dividing by the number of observations in the sample. This sample mean can be used as an estimate of the expected value (or population mean). It is important to remember that there might be some error in this estimation due to the limited sample size and potential biases in the sampling method. In our example with high school students' weekly homework hours, we calculated a sample mean of 3.8 hours, which we used as an estimate for the expected value of the population.

Step by step solution

01

Understanding sample, population, and expected value

A sample is a subset of individuals or items selected from a larger population. The population refers to the entire group of individuals or items of interest. The expected value represents the long-term average of a random variable based on a large number of repeated trials. In this case, we want to estimate the expected value of a population using a sample.
02

The concept of sample mean

The sample mean is a statistic that we can use to estimate the expected value or population mean. It is calculated by adding up all the values of the sample and dividing this sum by the number of observations in the sample. Since the sample is a representative part of the population, the sample mean should approximate the population mean (expected value).
03

Example problem statement

Suppose you want to estimate the average number of hours high school students spend on homework every week. You have collected data on weekly homework hours from a random sample of 10 students. Sample data: \[3, 4, 2, 6, 4, 5, 1, 3, 4, 6\] Using this sample, let's estimate the expected value of the average number of hours high school students spend on homework every week.
04

Calculate the sample mean

To calculate the sample mean, follow these steps: 1. Add up all the values in the sample: \(3 + 4 + 2 + 6 + 4 + 5 + 1 + 3 + 4 + 6 = 38\) 2. Divide the sum by the number of observations in the sample: \(\frac{38}{10} = 3.8\) The sample mean is 3.8 hours.
05

Use the sample mean as an estimate for the expected value

Using the sample mean, we can estimate that the expected value of the average number of hours high school students spend on homework each week is approximately 3.8 hours. This estimation is based on the assumption that the sample is representative of the entire population of high school students; however, it is important to remember that there might be some error in our estimation due to the limited sample size and potential biases in our sampling method.

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Most popular questions from this chapter

According to a study, \({ }^{31}\) the probability that a randomly selected teenager shopped at a mall at least once during a week was .63. How many teenagers in a randomly selected group of 40 would you expect to shop at a mall during the next week? HINT [See Example 5.]

The mean batting average in major league baseball is about \(0.250\). Supposing that batting averages are normally distributed, that the standard deviation in the averages is \(0.05\), and that there are 250 batters, what is the expected number of batters with an average of at least \(0.400 ?^{54}\)

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Compute the (sample) variance and standard deviation of the data samples given in Exercises \(1-8 .\) You calculated the means in the last exercise set. Round all answers to two decimal nlaces. $$ 2.5,-5.4,4.1,-0.1,-0.1 $$
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