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Chapter 8: Problem 79

Use the following information to answer the next 16 exercises: The Ice Chalet offers dozens of different beginning ice-skating classes. All of the class names are put into a bucket. The 5 P.M., Monday night, ages 8 to 12, beginning ice-skating class was picked. In that class were 64 girls and 16 boys. Suppose that we are interested in the true proportion of girls, ages 8 to 12, in all beginning ice-skating classes at the Ice Chalet. Assume that the children in the selected class are a random sample of the population. What is being counted?

Short Answer

Expert verified
The number of girls in the class is being counted.

Step by step solution

01

Identify the Groups

In the given scenario, two groups are mentioned: girls and boys in the ice skating class. We are interested in these groups for the purpose of calculating proportions.
02

Determine What is Being Counted

The problem specifically refers to 'the true proportion of girls' in the class. This implies that the count of interest is the number of girls in the sample of beginning ice-skating classes, which is 64 girls.

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Key Concepts

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

Understanding Proportion in Statistics
In statistics, a proportion is one way of expressing a part of a whole. It tells us how large one part is relative to the total, often expressed as a percentage or fraction. For instance, in the context of the Ice Chalet exercise, we have a total class of 80 children, consisting of 64 girls and 16 boys.
  • To compute the proportion of girls in the class, divide the number of girls by the total number of children.
  • The formula for calculating proportion is: \( \text{Proportion} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \)
  • In this case, \( \text{Proportion of Girls} = \frac{64}{80} = 0.8 \) or 80%.
Proportions are crucial for inferring information about larger groups (or populations) from small samples, providing insights into trends or characteristics of different groups. When analyzing any dataset, understanding proportions can help in comparing various segments within the data effectively.
What is a Random Sample?
A random sample is a subset of individuals taken from a population, where each individual has an equal chance of being selected. This concept ensures that the sample is representative of the population, minimizing bias and providing a reliable basis for making statistical inferences.
  • Random sampling is vital because it allows researchers to draw accurate conclusions about the larger population without examining every member.
  • In the Ice Chalet example, the selected class is assumed to be a random sample of all beginning ice-skating classes. This means every class had an equal opportunity to be chosen.
  • This randomness ensures that the characteristics of the sample approximate those of the overall population, thereby making any findings more applicable to the entire group.
Using random samples helps maintain the integrity and validity of the statistical analysis by ensuring results are not skewed by preexisting biases or anomalies in the data.
Defining Population in Statistical Terms
In statistics, 'population' refers to the entire group of individuals or items that we want to understand or make inferences about. It is the "bigger picture," representing everyone or everything under consideration for a particular study.
  • The population can be very large or small depending on the context. For Ice Chalet, the population is all children aged 8 to 12 taking beginning ice-skating classes.
  • Understanding the population is crucial because it defines the scope of your study and the applicability of your results.
  • When we gather data from a sample (a subset of the population), the goal is to make patterns and insights applicable to the entire population. Thus, choosing a representative sample is critical.
Statistical studies must accurately define their target population as it guides both the methodology and the inferences that can be legitimately drawn from the sample data.

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Most popular questions from this chapter

Use the following information to answer the next three exercises: According to a Field Poll, 79% of California adults (actual results are 400 out of 506 surveyed) feel that education and our schools is one of the top issues facing California. We wish to construct a 90% confidence interval for the true proportion of California adults who feel that education and the schools is one of the top issues facing California. The error bound is approximately _____. a. 1.581 b. 0.791 c. 0.059 d. 0.030

Use the following information to answer the next seven exercises: The U.S. Census Bureau conducts a study to determine the time needed to complete the short form. The Bureau surveys 200 people. The sample mean is 8.2 minutes. There is a known standard deviation of 2.2 minutes. The population distribution is assumed to be normal. If the Census did another survey, kept the error bound the same, and surveyed only 50 people instead of 200, what would happen to the level of confidence? Why?

Use the following information to answer the next ten exercises: A sample of 20 heads of lettuce was selected. Assume that the population distribution of head weight is normal. The weight of each head of lettuce was then recorded. The mean weight was 2.2 pounds with a standard deviation of 0.1 pounds. The population standard deviation is known to be 0.2 pounds. What would happen if 40 heads of lettuce were sampled instead of 20, and the confidence level remained the same?

Use the following information to answer the next 14 exercises: The mean age for all Foothill College students for a recent Fall term was 33.2. The population standard deviation has been pretty consistent at 15. Suppose that twenty-five Winter students were randomly selected. The mean age for the sample was 30.4. We are interested in the true mean age for Winter Foothill College students. Let X = the age of a Winter Foothill College student. What is \(\overline{x}\) estimating?

Among various ethnic groups, the standard deviation of heights is known to be approximately three inches. We wish to construct a 95% confidence interval for the mean height of male Swedes. Forty-eight male Swedes are surveyed. The sample mean is 71 inches. The sample standard deviation is 2.8 inches. a. i. \(\overline{x}=\) _____ ii. \(\sigma=\) _____ iii. \(n=\) _____ b. In words, define the random variables \(X\) and \(\overline{X}\) . c. Which distribution should you use for this problem? Explain your choice. d. Construct a 95\(\%\) confidence interval for the population mean height of male Swedes. i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound. e. What will happen to the level of confidence obtained if \(1,000\) male Swedes are surveyed instead of 48\(?\) Why?
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