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Chapter 7: Problem 62

In a population of 18,700 subjects, \(30 \%\) possess a certain characteristic. In a sample of \(2.50\) subjects selected from this population, \(25 \%\) possess the same characteristic. How many subjects in the population and sample, respectively, possess this characteristic?

Short Answer

Expert verified
In the population, there are \(18,700 \times 0.3 = 5610\) subjects with the characteristic. In the sample, there are \(2.5 \times 0.25 = 0.625\) subjects with the characteristic.

Step by step solution

01

Calculation for Population

First, find the number of subjects with the characteristic in the whole population. This can be calculated by multiplying the total population of 18700 subjects by 0.30 (which represents 30% in decimal form). Therefore, the calculation is \(18,700 \times 0.30\).
02

Calculation for Sample

Next, calculate the number of subjects with the characteristic in the sample. This can be calculated by multiplying the sample size of 2.50 subjects by 0.25 (which represents 25% in decimal form). So, the calculation is \(2.50 \times 0.25\).
03

Final Count

By performing the calculations from step 1 and step 2, you find the number of subjects with the characteristic, both in the larger population and in the smaller sample.

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Key Concepts

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

Population
In statistics, the concept of a "population" refers to the complete set of all possible observations or outcomes for a particular study or survey. This could be anything from all adults in a country to all cars manufactured by a particular company.
Populations provide the context for statistical analysis, aiming to understand large groups by studying selected individuals.
  • Huge numbers: Often involves a large number of subjects like our scenario with 18,700 people.
  • Desirable for complete analysis: Ideally, you'd study every member to gather data.
  • Limitations: In reality, accessing every member can be costly or impractical.
Understanding a population is key when you want to generalize your findings to an entire group.
Sample
Since studying an entire population might be impractical, we often use a "sample". A sample is a smaller, manageable number of subjects drawn from the population. Sampling allows us to collect data efficiently and cost-effectively while still drawing conclusions about the whole group.
  • Selection: Typical samples include a few hundred or thousand subjects, but our example illustrates a simple sample of 2.50 subjects.
  • Representation: Samples should reflect the characteristics of the population well.
  • Avoids bias: Proper sampling methods help avoid sampling bias and lead to more reliable conclusions.
By analyzing samples, statisticians can infer what might be true for the entire population.
Percentage
Percentages are a universal way to express how large one quantity is relative to another. They are an essential part of understanding data in statistics because they simplify comparisons between different sets of numbers.
  • Understanding: Expressed as a fraction of 100, percentages make it easy to communicate proportional relationships.
  • Population Example: When we say 30% of a population has a characteristic, it breaks down to a simple proportion like 0.30.
  • Sample Calculation: Similarly, 25% of a sample translates to a proportion of 0.25.
Percentages allow us to quickly grasp how widespread a characteristic is within different groups.
Characteristic
In statistics, a "characteristic" refers to a specific attribute or trait that researchers are interested in studying within a population or sample. Characteristics help categorize and analyze groups to derive meaningful insights.
  • Defining Traits: Could be a health condition, preference, or any quantifiable trait.
  • Assessment: Researchers identify and measure these traits within their subjects.
  • Example Case: The exercise mentions a characteristic whose presence is quantified both in the population and the sample.
Understanding these characteristics helps researchers make informed conclusions about their study groups.

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Most popular questions from this chapter

In a population of 5000 subjects, 600 possess a certain characteristic. \(A\) sample of 120 subjects selected from this population contains 18 subjects who possess the same characteristic. What are the values of the population and sample proportions?

Let \(\hat{p}\) be the proportion of elements in a sample that possess a characteristic. \(\mathbf{a}_{\mathbf{+}}\) What is the mean of \(\hat{p}\) ? b. What is the standard deviation of \(\hat{p}\) ? Assume \(n / N \leq .05\). c. What condition(s) must hold true for the sampling distribution of \(\hat{p}\) to be approximately normal?

Refer to Exercise \(7.93 .\) The print on the package of 100 -watt General Electric soft-white light-bulbs says that these bulbs have an average life of 750 hours. Assume that the lives of all such bulbs have a normal distribution with a mean of 750 hours and a standard deviation of 55 hours. Find the probability that the mean life of a random sample of 25 such bulbs will be a. greater than 735 hours b. between 725 and 740 hours c. within 15 hours of the population mean d. less than the population mean by 20 hours or more

Refer to Exercise 7.94. On average, Britons spend 225 minutes per day communicating electronically. Assume that currently such communication times for all Britons are normally distributed with a mean of 225 minutes per day and a standard deviation of 62 minutes per day. Find the probability that the mean time spent communicating electronically per day by a random sample of 20 Britons will be a. less than 200 minutes b. between 230 and 240 minutes c. within 20 minutes of the population mean d. more than 260 minutes

The print on the package of 100 -watt General Electric soft-white lightbulbs claims that these bulbs have an average life of 750 hours. Assume that the lives of all such bulbs have a normal distribution with a mean of 750 hours and a standard deviation of 55 hours, I.ct \(x\) be the mean life of a random sample of 25 such bulbs. Find the mean and standard deviation of \(\bar{x}\), and describe the shape of its sampling distribution.
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