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Chapter 3: Problem 8

Give the correct notation for the quantity described and give its value. Proportion of US adults who own a cell phone. In a survey of 2252 US adults, \(82 \%\) said they had a cell phone.

Short Answer

Expert verified
The correct notation for the given quantity is 'p' and its value is approximately 0.82 or 82%. The estimated number of US adults in the survey who own a cell phone is around 1846.

Step by step solution

01

Identify the notation for proportion

In statistical notation, the word 'proportion' is commonly symbolized by the letter 'p'. So in this context, we consider 'p' to represent the proportion of US adults who own a cell phone.
02

Convert the given percentage into a proportion

The exercise states that 82% of the respondents own a cell phone. We convert this percentage into a decimal by dividing by 100, so \( p = \frac{82}{100} = 0.82 \)
03

Calculate the number of US adults surveyed who own a cell phone

We multiply the total number of US adults surveyed by the proportion who own a cell phone to obtain the number of adults who own a cell phone. So, the number of adults with a cell phone = \( p \times \text{total number of adults} = 0.82 \times 2252 \approx 1846 \)

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Key Concepts

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

Statistical Notation
Understanding statistical notation is crucial when addressing problems in statistics as it offers a universal language to represent complex data succinctly. One fundamental term is the 'proportion', often represented by a lowercase 'p'. A proportion is a type of ratio that expresses the fraction of the total sample that exhibits a particular trait.
For example, when a survey indicates that '82%' of a population has a certain characteristic鈥攍ike owning a cell phone鈥攖he relative frequency of this occurrence is described using the proportion 'p'. In statistical notation, this would be represented as 'p' for proportion, followed by the actual proportion value after calculations are done. This notation plays a significant role in the accurate communication and analysis of statistical data.
Converting Percentages
Percentages are ubiquitous in everyday life and in the field of statistics. They are used to represent how a part relates to a whole, and they can be converted into proportions or decimals for various calculations. To convert a percentage to a proportion, you simply divide by 100.
For instance, if a survey report states that '82%' of individuals possess a certain item, this percentage is converted into a decimal (which represents the proportion) by dividing by 100, resulting in 0.82. This is important in statistics as it transforms the percent value into a form suitable for further statistical operations, such as finding averages, probabilities, and more. Converting percentages is a fundamental skill, as it transitions the data from a general public's understanding to a more analytical, statistical perspective.
Sampling in Surveys
Sampling is a pivotal concept in the field of statistics, especially in conducting surveys. It refers to the process of selecting a subset of a population to represent the whole. Sampling is used because it is often impractical or impossible to collect data from every member of a population.
There are various sampling methods, such as random sampling, stratified sampling, and cluster sampling, each with its own advantages and limitations. In the context of our example, a survey of '2252 U.S. adults' implies that these people were chosen as a sample from the broader population of U.S. adults, aiming to approximate the behavior or characteristics of the entire group. The quality of the sample鈥攁nd therefore the validity of the survey results鈥攄epends heavily on how well it represents the overall population. Understanding sampling methods is essential for interpreting survey results and for designing surveys that can produce accurate and reliable data.

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Most popular questions from this chapter

Student Misinterpretations Suppose that a student is working on a statistics project using data on pulse rates collected from a random sample of 100 students from her college. She finds a \(95 \%\) confidence interval for mean pulse rate to be \((65.5,\) 71.8). Discuss how each of the statements below would indicate an improper interpretation of this interval. (a) I am \(95 \%\) sure that all students will have pulse rates between 65.5 and 71.8 beats per minute. (b) I am \(95 \%\) sure that the mean pulse rate for this sample of students will fall between 65.5 and 71.8 beats per minute. (c) I am \(95 \%\) sure that the confidence interval for the average pulse rate of all students at this college goes from 65.5 to 71.8 beats per minute. (d) I am sure that \(95 \%\) of all students at this college will have pulse rates between 65.5 and 71.8 beats per minute. (e) I am \(95 \%\) sure that the mean pulse rate for all US college students is between 65.5 and 71.8 beats per minute. (f) Of the mean pulse rates for students at this college, \(95 \%\) will fall between 65.5 and 71.8 beats per minute. (g) Of random samples of this size taken from students at this college, \(95 \%\) will have mean pulse rates between 65.5 and 71.8 beats per minute.

Topical Painkiller Ointment The use of topical painkiller ointment or gel rather than pills for pain relief was approved just within the last few years in the US for prescription use only. \(^{12}\) Insurance records show that the average copayment for a month's supply of topical painkiller ointment for regular users is \(\$ 30 .\) A sample of \(n=75\) regular users found a sample mean copayment of \(\$ 27.90\). (a) Identify each of 30 and 27.90 as a parameter or a statistic and give the appropriate notation for each. (b) If we take 1000 samples of size \(n=75\) from the population of all copayments for a month's supply of topical painkiller ointment for regularusers and plot the sample means on a dotplot, describe the shape you would expect to see in the plot and where it would be centered. (c) How many dots will be on the dotplot you described in part (b)? What will each dot represent?

Predicting Election Results Throughout the US presidential election of \(2012,\) polls gave regular updates on the sample proportion supporting each candidate and the margin of error for the estimates. This attempt to predict the outcome of an election is a common use of polls. In each case below, the proportion of voters who intend to vote for each candidate is given as well as a margin of error for the estimates. Indicate whether we can be relatively confident that candidate A would win if the election were held at the time of the poll. (Assume the candidate who gets more than \(50 \%\) of the vote wins.) (a) Candidate A: 54\% Candidate B: \(46 \%\) Margin of error: \(\pm 5 \%\) (b) Candidate A:52\% Candidate B: \(48 \%\) Margin of error: \(\pm 1 \%\) \(\begin{array}{llll}\text { (c) Candidate A: 53\% } & \text { Candidate B: } 47 \% & \text { Margin }\end{array}\) of error: \(\pm 2 \%\) (d) Candidate A: 58\% Candidate B: 42\% Margin of error: \(\pm 10 \%\)

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 3.90 to 3.95 , 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: \(\begin{array}{lll}A .66 \text { to } 74 & B .67 \text { to } 73 & \text { C. } 67.5 \text { to } 72.5\end{array}\) Using the data to find a \(99 \%\) confidence interval.

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 3.90 to 3.95 , 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: \(\begin{array}{lll}A .66 \text { to } 74 & B .67 \text { to } 73 & \text { C. } 67.5 \text { to } 72.5\end{array}\) Using an original sample of size \(n=45\)
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