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Chapter 7: Problem 60

In a population of 1000 subjects, 640 possess a certain characteristic. In a sample of 40 subjects selected from this population, 24 possess the same characteristic. What are the values of the population and sample proportions?

Short Answer

Expert verified
The population proportion is 0.64 and the sample proportion is 0.60.

Step by step solution

01

Identify the population size and quantity of interest

First identify the total number of subjects in the population, which is 1000, and the number of subjects with the characteristic, which is 640.
02

Calculate the population proportion

Population proportion is calculated as the number of subjects with the characteristic divided by the total number of subjects in the population. In mathematical form, it can be represented as \( p = \frac{x}{N} \), where \( p \) is the population proportion, \( x \) is the number of subjects with the characteristic and \( N \) is the total number of subjects. Substituting the given values, \( p = \frac{640}{1000} = 0.64. \) Thus, the population proportion is 0.64.
03

Identify the sample size and quantity of interest

Next, identify the total number of subjects in the sample, which is 40, and the number of subjects with the characteristic in the sample, which is 24.
04

Calculate the sample proportion

Sample proportion is calculated in a similar manner as the population proportion, as the number of subjects with the characteristic in the sample divided by the total number of subjects in the sample. It can be represented as \( \hat{p} = \frac{x_s}{n} \), where \( \hat{p} \) is the sample proportion, \( x_s \) is the number of subjects with the characteristic in the sample and \( n \) is the total number of subjects in the sample. Substituting the given values, \( \hat{p} = \frac{24}{40} = 0.60. \) Thus, the sample proportion is 0.60.

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Key Concepts

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

Population Proportion
Understanding the concept of population proportion is key in statistics. It refers to the fraction of the entire group that possesses a particular characteristic.
A population is the whole group that you are interested in studying, and in this example, it consists of 1000 subjects. Among them, 640 possess a certain characteristic.
This information allows us to calculate the population proportion, using the formula:
  • \( p = \frac{x}{N} \)
  • where \( x \) is the number of subjects with the characteristic (640 in this case), and \( N \) is the total population size (1000).
By substituting these values into the formula, we get the population proportion:
\( \frac{640}{1000} = 0.64 \).
So, 64% of the entire population possesses this characteristic.
Sample Proportion
The concept of sample proportion comes into play when examining a smaller group, drawn from a larger population. This smaller group is known as a "sample," which should ideally represent the whole population well.
In the given scenario, a sample of 40 subjects was selected from the population of 1000, and among them, 24 possessed the characteristic.
To find the sample proportion, we use a formula similar to that for the population proportion:
  • \( \hat{p} = \frac{x_s}{n} \)
  • where \( x_s \) is the number of subjects with the characteristic in the sample (24), and \( n \) is the sample size (40).
Substitute these values into the formula to get the sample proportion:
\( \frac{24}{40} = 0.60 \).
Therefore, 60% of the sample has the characteristic.
Statistical Sample
A statistical sample is a smaller set chosen from a larger population, intended to represent the whole group. It is crucial for data collection and analysis when surveying an entire population is impractical due to constraints like time and resources.
Sampling helps to estimate features of the entire population without examining every member. This technique saves time and effort while still allowing one to make reasonable inferences about the whole group. Sub-headline: The Importance of a Good Sample
  • The sample needs to accurately reflect the diversity of the population.
  • A well-chosen sample can lead to solid insights and enable valid predictions.
  • Random sampling is often employed to avoid biases and ensure the sample is representative.
Using samples correctly enables statisticians to draw conclusions that apply to entire populations based on detailed analysis of smaller groups. This forms the backbone of many statistical methods and practices.

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Most popular questions from this chapter

Consider a large population with \(p=.63\). Assuming \(n / N \leq .05\), find the mean and standard deviation of the sample proportion \(\hat{p}\) for a sample size of a. 100 b. 900

A city is planning to build a hydroelectric power plant. A local newspaper found that \(53 \%\) of the voters in this city favor the construction of this plant. Assume that this result holds true for the population of all voters in this city. a. What is the probability that more than \(50 \%\) of the voters in a random sample of 200 voters selected from this city will favor the construction of this plant? b. A politician would like to take a random sample of voters in which more than \(50 \%\) would favor the plant construction. How large a sample should be selected so that the politician is \(95 \%\) sure of this outcome?

Indicate in which of the following cases the central limit theorem will apply to describe the sampling distribution of the sample proportion. a. \(n=20\) and \(p=.45\) b. \(n=75\) and \(p=.22\) c. \(n=350\) and \(p=.01\) d. \(n=200\) and \(p=.022\)

In a Time Magazine/Aspen poll of American adults conducted by the strategic research firm Penn Schoen Berland, these adults were asked, "In your opinion, what is more important for the U.S. to focus on in the next decade?" Eighty- three percent of the adults polled said domestic issues (Time, July 11,2011 ). Assume that this percentage is true for the current population of American adults. Let \(\hat{p}\) be the proportion in a random sample of 1000 American adults who hold the above opinion. Find the mean and standard deviation of the sampling distribution of \(\hat{p}\) and describe its shape.

The delivery times for all food orders at a fast-food restaurant during the lunch hour are normally distributed with a mean of \(7.7\) minutes and a standard deviation of \(2.1\) minutes. Find the probability that the mean delivery time for a random sample of 16 such orders at this restaurant is a. between 7 and 8 minutes b. within 1 minute of the population mean c. less than the population mean by 1 minute or more
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