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Chapter 4: Problem 128

Use the following information to answer the next two exercises: The average number of times per eek that Mrs. Plum鈥檚 cats wake her up at night because they want to play is ten. We are interested in the number of times her cats wake her up each week. In words, the random variable \(X =\) _________________ a. the number of times Mrs. Plum鈥檚 cats wake her up each week. b. the number of times Mrs. Plum鈥檚 cats wake her up each hour. c. the number of times Mrs. Plum鈥檚 cats wake her up each night. d. the number of times Mrs. Plum鈥檚 cats wake her up.

Short Answer

Expert verified
The correct choice is a: the number of times Mrs. Plum鈥檚 cats wake her up each week.

Step by step solution

01

Determine the Focus of the Random Variable

Read the question carefully and identify the context in which the random variable is defined. Mrs. Plum鈥檚 cats waking her up is described as happening on a weekly basis.
02

Match the Context to the Answer Options

Look at each answer option and determine which one aligns with the weekly context. Option a mentions the number of times per week, which matches the context. Options b, c, and d mention different time frames (hour, night, and general without specificity).

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Key Concepts

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

Understanding Probability
Probability is the measure of the likelihood that a given event will occur. It is a fundamental concept in statistics and mathematics, assisting in quantifying uncertain outcomes. When dealing with random variables, such as the number of times Mrs. Plum's cats wake her up, probability helps us ascertain the chances of different occurrences over a specified period.

The probability of an event is expressed as a number between 0 and 1:
  • A probability of 0 means the event is impossible.
  • A probability of 1 signifies the event is certain to happen.
  • Values between 0 and 1 indicate various levels of likelihood.
In the context of Mrs. Plum, if the average is known to be ten times per week, one might want to look at how likely it is for the number of wake-ups to follow this average precisely, or deviate from it.
Purpose of Statistical Analysis
Statistical analysis involves collecting and examining data to identify patterns and trends. It is a vital tool for making informed decisions based on data insights. With Mrs. Plum and her cats, statistical analysis can help determine how often the average wake-up number varies from the weekly mean of ten.

This approach includes:
  • Descriptive statistics: which summarize the data to provide a clear understanding of what has occurred.
  • Inferential statistics: used to make predictions or inferences about a population based on a sample of data points.
For example, Mrs. Plum could use statistical analysis to project how frequently her sleep might be disturbed based on historical data. This could involve measuring the variance and standard deviation among her records.
Exploring Variable Context
Context is crucial when working with random variables. It defines the setting in which the data or the events are being considered. For variables like the ones describing Mrs. Plum's wake-up calls, context ensures the accuracy of interpretations and findings.

When identifying the random variable, pay attention to:
  • The time frame: Is it daily, hourly, weekly?
  • The specific occurrence: Is it focused on waking events particular to a certain period?
The original exercise specifically defines the interest in Mrs. Plum鈥檚 disturbances on a weekly basis, guiding us to select option 'a'. Recognizing the appropriate context confines our understanding to relevant data segments, enhancing precision in statistical endeavors.

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Most popular questions from this chapter

Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35% of the time, four events 25% of the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5% of the time. Define the random variable \(X\).

Use the following information to answer the next six exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly select freshman from the study until you find one who replies 鈥測es.鈥� You are interested in the number of freshmen you must ask. What values does the random variable \(X\) take on?

Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next, so that she can plan what classes to offer. Over the years, she has established the following probability distribution. \(\bullet\) Let \(X=\) the number of years a student will study ballet with the teacher. \(\bullet\) Let \(P(x)=\) the probability that a student will study ballet \(x\) years. On average, how many years would you expect a child to study ballet with this teacher?

Use the following information to answer the next four exercises: Ellen has music practice three days a week. She practices for all of the three days 85% of the time, two days 8% of the time, one day 4% of the time, and no days 3% of the time. One week is selected at random. Define the random variable \(X\).

Use the following information to answer the next two exercises: The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games. More than 96 percent of the very largest colleges and universities (more than 15,000 total enrollments) have some online offerings. Suppose you randomly pick 13 such institutions. We are interested in the number that offer distance learning courses. a. In words, define the random variable X. b. List the values that X may take on. c. Give the distribution of \(X . X \sim\) _____(___,___) d. On average, how many schools would you expect to offer such courses? e. Find the probability that at most ten offer such courses. f. Is it more likely that 12 or that 13 will offer such courses? Use numbers to justify your answer numerically and answer in a complete sentence.
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