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Chapter 3: Problem 84

In a random sample of 1000 people, 382 people agree, 578 disagree, and 40 are undecided.

Short Answer

Expert verified
The proportion of people who agree is 0.382, the proportion of people who disagree is 0.578, and the proportion of people who are undecided is 0.04.

Step by step solution

01

Calculate the proportion of people who agree

To calculate the proportion of people who agree, divide the number of people who agree by the total number of people. This can be achieved with the following formula: \(\frac{Number of people who agree}{Total number of people}\). Substitute the given numbers into the formula: \(\frac{382}{1000}\).
02

Calculate the proportion of people who disagree

To calculate the proportion of people who disagree, divide the number of people who disagree by the total number of people. This is achieved with the following formula: \(\frac{Number of people who disagree}{Total number of people}\). Substitute the given numbers into the formula: \(\frac{578}{1000}\).
03

Calculate the proportion of people who are undecided

To calculate the proportion of people who are undecided, divide the number of people who are undecided by the total number of people. This is done with the formula: \(\frac{Number of people who are undecided}{Total number of people}\). Substitute the numbers into the formula: \(\frac{40}{1000}\).

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Key Concepts

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

Understanding Random Samples
When we use a random sample, it means we are selecting a subset of a larger group or population in such a way that every member of this population has an equal chance of being chosen. This method is crucial because it helps us produce unbiased and representative results. This is why in our exercise, a random sample of 1000 people is used; it allows us to fairly assess the general opinions of a much larger group.
  • Ensures fairness: Each person has an equal chance of selection.
  • Reduces bias: Less likely to over-represent any particular group.
  • Makes data reliable: More representative of the whole population.
It's important to note that the randomness in selection does not mean the results are random, but rather they are reflective of the entire population. Always ensure your sample size is large enough to be meaningful.
Basics of Data Analysis
Data analysis involves carefully examining the data collected, often to uncover patterns or to answer specific questions. In this context, data analysis supports better decision-making by interpreting data correctly. From our sample, analyzing the data involves calculating proportions, which provides a clear picture of the distribution of opinions among the group sampled.
  • Calculating proportions: Helps determine the percentage of people with specific opinions.
  • Interpretation: What does it mean when more people disagree than agree?
  • Informed decisions: Provides a basis for understanding trends or patterns.
As you analyze such data, make sure to look for consistency and relevance to ensure your data-driven conclusions are valid and reliable. Each step like calculating the proportion of agreement, disagreement, or indecision is a part of understanding the entire data set.
Statistical Formulas in Practice
Statistical formulas help us to bring structure and meaning to data. In our example, we use straightforward proportion calculations. These are fundamental to many statistical approaches and involve some simple mathematics.
  • Proportion formula: \[\text{Proportion} = \frac{\text{Number in category}}{\text{Total number}} \]
  • Breakdown: This tells what fraction of the entire sample possesses a certain quality (like agreeing).
  • Application: Used in various fields such as psychology, market research, and political surveys.
These formulas are pivotal in deducing clear and precise results from raw data, helping to communicate findings understandably. Make sure to substitute the correct values as shown in the calculations to avoid errors, and remember to express your final answer as a percentage for clarity.

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Most popular questions from this chapter

Daily Tip Revenue for a Waitress Data 2.12 on page 123 describes information from a sample of 157 restaurant bills collected at the First Crush bistro. The data is available in RestaurantTips. Two intervals are given below for the average tip left at a restaurant; one is a \(90 \%\) confidence interval and one is a \(99 \%\) confidence interval. Interval A: 3.55 to 4.15 Interval B: 3.35 to 4.35 (a) Which one is the \(90 \%\) confidence interval? Which one is the \(99 \%\) confidence interval? (b) One waitress generally waits on 20 tables in an average shift. Give a range for her expected daily tip revenue, using both \(90 \%\) and \(99 \%\) confidence. Interpret your results.

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In estimating the mean score on a fitness exam, we use an original sample of size \(n=30\) and a bootstrap distribution containing 5000 bootstrap samples to obtain a \(95 \%\) confidence interval of 67 to \(73 .\) In Exercises 3.106 to 3.111 , a change in this process is described. If all else stays the same, which of the following confidence intervals \((A, B,\) or \(C)\) is the most likely result after the change: \(\begin{array}{ll}A .66 \text { to } 74 & B .67 \text { to } 73\end{array}\) C. 67.5 to 72.5 Using 1000 bootstrap samples for the distribution.

Bisphenol A in Your Soup Cans Bisphenol A (BPA) is in the lining of most canned goods, and recent studies have shown a positive association between BPA exposure and behavior and health problems. How much does canned soup consumption increase urinary BPA concentration? That was the question addressed in a recent study \(^{34}\) in which consumption of canned soup over five days was associated with a more than \(1000 \%\) increase in urinary BPA. In the study, 75 participants ate either canned soup or fresh soup for lunch for five days. On the fifth day, urinary BPA levels were measured. After a two-day break, the participants switched groups and repeated the process. The difference in BPA levels between the two treatments was measured for each participant. The study reports that a \(95 \%\) confidence interval for the difference in means (canned minus fresh) is 19.6 to \(25.5 \mu \mathrm{g} / \mathrm{L}\). (a) Is this a randomized comparative experiment or a matched pairs experiment? Why might this type of experiment have been used? (b) What parameter are we estimating? (c) Interpret the confidence interval in terms of BPA concentrations. (d) If the study had included 500 participants instead of \(75,\) would you expect the confidence interval to be wider or narrower?

Adolescent Brains Are Different Researchers continue to find evidence that brains of adolescents behave quite differently than either brains of adults or brains of children. In particular, adolescents seem to hold on more strongly to fear associations than either children or adults, suggesting that frightening connections made during the teen years are particularly hard to unlearn. In one study, \({ }^{25}\) participants first learned to associate fear with a particular sound. In the second part of the study, participants heard the sound without the fear-causing mechanism, and their ability to "unlearn" the connection was measured. A physiological measure of fear was used, and larger numbers indicate less fear. We are estimating the difference in mean response between adults and teenagers. The mean response for adults in the study was 0.225 and the mean response for teenagers in the study was \(0.059 .\) We are told that the standard error of the estimate is 0.091 . (a) Give notation for the quantity being estimated. (b) Give notation for the quantity that gives the best estimate, and give its value. (c) Give a \(95 \%\) confidence interval for the quantity being estimated. (d) Is this an experiment or an observational study?
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