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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

Gender in the Rock and Roll Hall of Fame From its founding through \(2015,\) the Rock and Roll Hall of Fame has inducted 303 groups or individuals. Forty-seven of the inductees have been female or have included female members. \(^{20}\) The full dataset is available in RockandRoll. (a) What proportion of inductees have been female or have included female members? Use the correct notation with your answer. (b) If we took many samples of size 50 from the population of all inductees and recorded the proportion female or with female members for each sample, what shape do we expect the distribution of sample proportions to have? Where do we expect it to be centered?

Use data from a study designed to examine the effect of doing synchronized movements (such as marching in step or doing synchronized dance steps) and the effect of exertion on many different variables, such as pain tolerance and attitudes toward others. In the study, 264 high school students in Brazil were randomly assigned to one of four groups reflecting whether or not movements were synchronized (Synch= yes or no) and level of activity (Exertion= high or low). \(^{49}\) Participants rated how close they felt to others in their group both before (CloseBefore) and after (CloseAfter) the activity, using a 7-point scale (1=least close to \(7=\) most close ). Participants also had their pain tolerance measured using pressure from a blood pressure cuff, by indicating when the pressure became too uncomfortable (up to a maximum pressure of \(300 \mathrm{mmHg}\) ). Higher numbers for this Pain Tolerance measure indicate higher pain tolerance. The full dataset is available in SynchronizedMovement. For each of the following problems: (a) Give notation for the quantity we are estimating, and define any relevant parameters. (b) Use StatKey or other technology to find the value of the sample statistic. Give the correct notation with your answer. (c) Use StatKey or other technology to find the standard error for the estimate. (d) Use the standard error to give a \(95 \%\) confidence interval for the quantity we are estimating. (e) Interpret the confidence interval in context. Does Synchronization Increase Feelings of Closeness? Use the closeness ratings given after the activity (CloseAfter) to estimate the difference in mean rating of closeness between those who have just done a synchronized activity and those who do a non-synchronized activity.

Exercises 3.112 to 3.115 give information about the proportion of a sample that agree with a certain statement. Use StatKey or other technology to find a confidence interval at the given confidence level for the proportion of the population to agree, using percentiles from a bootstrap distribution. StatKey tip: Use "CI for Single Proportion" and then "Edit Data" to enter the sample information. Find a \(99 \%\) confidence interval if, in a random sample of 1000 people, 382 agree, 578 disagree, and 40 can't decide.

Investigating the Width of a Confidence Interval Comparing Exercise 3.120 to Exercise \(3.121,\) you should have found that the confidence interval when utilizing the paired structure of the data was narrower than the confidence interval ignoring this structure (this will generally be the case, and is the primary reason for pairing). How else could we change the width of the confidence interval? More specifically, for each of the following changes, would the width of the confidence interval likely increase, decrease, or remain the same? (a) Increase the sample size. (b) Simulate more bootstrap samples. (c) Decrease the confidence level from \(99 \%\) to \(95 \%\).

Number of Text Messages a Day A random sample of \(n=755\) US cell phone users age 18 and older in May 2011 found that the average number of text messages sent or received per day is 41.5 messages, 32 with standard error about \(6.1 .\) (a) State the population and parameter of interest. Use the information from the sample to give the best estimate of the population parameter. (b) Find and interpret a \(95 \%\) confidence interval for the mean number of text messages.
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