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Chapter 2: Problem 7

Give the relevant proportion using correct notation. A survey conducted of 1060 randomly selected US teens aged 13 to 17 found that 605 of them say they have made a new friend online. \(^{6}\)

Short Answer

Expert verified
The proportion of US teens aged 13 to 17, who reported having made a new friend online, is \(57.1\% \).

Step by step solution

01

Calculate the proportion

The proportion, denoted as \(p\), is calculated by dividing the number of 'successes' by the total number of attempts. In this case, the 'successes' are the teens who have made a new friend online. Thus, the proportion can be calculated using the formula \(p = \frac{x}{n}\), where \(x\) is the number of teens who have made a new friend online, and \(n\) is the total number of teens. So, \(p = \frac{605}{1060}\).
02

Simplify the proportion

To simplify the proportion, simply perform the division. So, \(p = \frac{605}{1060} = 0.57075\).
03

Express the proportion in the correct notation

In statistics, the proportion is usually expressed as a decimal, a fraction, or a percentage. In this case, it will be best to express the proportion as a percentage. To convert to percentage, multiply the decimal by 100. So, \(p = 0.57075 * 100 = 57.075\% \). The percentage should be rounded to the nearest tenth of a percent, thus the final proportion is \(57.1\% \).

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Key Concepts

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

Understanding Survey Analysis
Survey analysis is a cornerstone of statistical research. It's the method we use to make sense of data collected from surveys. In the problem provided, the idea was to figure out how many teens made friends online out of a larger group. This is typical in survey analysis, where we want to derive meaningful insights from small data sets.
Survey analysis involves:
  • Selecting a group of individuals to study, known as the sample.
  • Asking them questions or observing their behavior.
  • Compiling and interpreting the data received to make broader conclusions.
It鈥檚 important to ensure the sample accurately represents the larger population for the analysis to be valid. Here, 1060 teens were surveyed, and 605 reported making a new friend online. This result gives us insight into the social habits of teens in general, provided that the sample chosen was random and representative.
Decoding Statistical Notation
Statistical notation is like a language used to communicate complex statistical concepts concisely. When you first see terms like \( p \) or fractions like \( \frac{605}{1060} \), they might seem daunting. But understanding these symbols is essential for grasping statistics.
For instance:
  • \( p \): Represents the proportion or percentage of a subset within the data set (e.g., teens making new friends online).
  • \( x \): The count of successes or occurrences we're interested in (e.g., 605 teens).
  • \( n \): The total number of observations or the entire sample size (e.g., 1060 teens).
To communicate an analysis concisely, using notations like \( p = \frac{x}{n} \) provides a clear, standardized way to convey results that can be understood universally among statisticians.
Mastering Percentage Calculation
Converting a proportion to a percentage is a key skill in statistics, simplifying data interpretation. The percentage makes complex data more relatable and understandable.
To calculate a percentage:
  • First, find the proportion, which is the part divided by the whole \( \left(p = \frac{605}{1060}\right) \).
  • Simplify the fraction by performing the division, resulting in a decimal (e.g., \( 0.57075 \)).
  • Convert this to a percentage by multiplying by 100 (e.g., \( 0.57075 \times 100 = 57.075\% \)).
For clarity and convention, the result should be rounded to the nearest tenth of a percent, ending with an expression like \( 57.1\% \). This makes it easier to communicate findings to a wider audience without losing the essence of the data analysis.

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Most popular questions from this chapter

Ages of Husbands and Wives Suppose we record the husband's age and the wife's age for many randomly selected couples. (a) What would it mean about ages of couples if these two variables had a negative relationship? (b) What would it mean about ages of couples if these two variables had a positive relationship? (c) Which do you think is more likely, a negative or a positive relationship? (d) Do you expect a strong or a weak relationship in the data? Why? (e) Would a strong correlation imply there is an association between husband age and wife age?

Use data on college students collected from the American College Health Association-National College Health Assessment survey \(^{18}\) conducted in Fall 2011 . The survey was administered at 44 colleges and universities representing a broad assortment of types of schools and representing all major regions of the country. At each school, the survey was administered to either all students or a random sample of students, and more than 27,000 students participated in the survey. Binge Drinking Students in the ACHANCHA survey were asked, "Within the last two weeks, how many times have you had five or more drinks of alcohol at a sitting?" The results are given in Table \(2.13 .\) Table 2.13 In the last two weeks, how many times have you had five or more drinks of alcohol? $$\begin{array}{l|rr|r}\hline & \text { Male } & \text { Female } & \text { Total } \\\\\hline 0 & 5402 & 13,310 & 18,712 \\\1-2 & 2147 & 3678 & 5825 \\\3-4 & 912 & 966 & 1878 \\\5+ & 495 & 358 & 853 \\\\\hline \text { Total } & 8956 & 18,312 & 27,268 \\\\\hline\end{array}$$ (a) What percent of all respondents answered zero? (b) Of the students who answered five or more days, what percent are male? (c) What percent of males report having five or more drinks at a sitting on three or more days in the last two weeks? (d) What percent of females report having five or more drinks at a sitting on three or more days in the last two weeks?

Find and interpret the z-score for the data value given. The value 8.1 in a dataset with mean 5 and standard deviation 2

2Does Sexual Frustration Increase the Desire for Alcohol? Apparently, sexual frustration increases the desire for alcohol, at least in fruit flies. Scientists \(^{35}\) randomly put 24 fruit flies into one of two situations. The 12 fruit flies in the "mating" group were allowed to mate freely with many available females eager to mate. The 12 in the "rejected" group were put with females that had already mated and thus rejected any courtship advances. After four days of either freely mating or constant rejection, the fruit flies spent three days with unlimited access to both normal fruit fly food and the same food soaked in alcohol. The percent of time each fly chose the alcoholic food was measured. The fruit flies that had freely mated chose the two types of food about equally often, choosing the alcohol variety on average \(47 \%\) of the time. The rejected males, however, showed a strong preference for the food soaked in alcohol, selecting it on average \(73 \%\) of the time. (The study was designed to study a chemical in the brain called neuropeptide that might play a role in addiction.) (a) Is this an experiment or an observational study? (b) What are the cases in this study? What are the variables? Which is the explanatory variable and which is the response variable? (c) We are interested in the difference in means, where the means measure the average percent preference for alcohol \((0.47\) and 0.73 in this case). Find the difference in means and give the correct notation for your answer, using the correct notation for a mean, subscripts to identify groups, and a minus sign. (d) Can we conclude that rejection increases a male fruit fly's desire for alcohol? Explain.

Use data on college students collected from the American College Health Association-National College Health Assessment survey \(^{18}\) conducted in Fall 2011 . The survey was administered at 44 colleges and universities representing a broad assortment of types of schools and representing all major regions of the country. At each school, the survey was administered to either all students or a random sample of students, and more than 27,000 students participated in the survey. Students in the ACHA-NCHA survey were asked, "Within the last 12 months, have you been in a relationship (meaning an intimate/coupled/partnered relationship) that was emotionally abusive?" The results are given in Table 2.12 . (a) What percent of all respondents have been in an emotionally abusive relationship? (b) What percent of the people who have been in an emotionally abusive relationship are male? (c) What percent of males have been in an emotionally abusive relationship? (d) What percent of females have been in an emotionally abusive relationship? Table 2.12 Have you been in an emotionally abusive relationship? $$\begin{array}{l|rr|r} \hline & \text { Male } & \text { Female } & \text { Total } \\ \hline \text { No } & 8352 & 16,276 & 24,628 \\ \text { Yes } & 593 & 2034 & 2627 \\ \hline \text { Total } & 8945 & 18,310 & 27,255 \\ \hline\end{array}$$
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