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Chapter 10: Problem 57

Refer to the instructions given prior to this exercise. The paper "College Students' Social Networking Experiences on Facebook" (Journal of Applied Developmental Psychology [2009]: \(227-238\) ) summarized a study in which 92 students at a private university were asked whether they used Facebook just to pass the time. Twenty-three responded "yes" to this question. The researchers were interested in estimating the proportion of students at this college who use Facebook just to pass the time.

Short Answer

Expert verified
The sample proportion is calculated using the formula \(\hat{p} = \frac{x}{n}\), where \(x\) is the number of positive responses (23 students) and \(n\) is the sample size (92 students). In this case, \(\hat{p} = \frac{23}{92} = 0.25\), which means that 25% of the sample of 92 students use Facebook just to pass the time. The sample proportion can be used as an estimate for the population proportion, so we can estimate that approximately 25% of the entire student population at this college uses Facebook just to pass the time.

Step by step solution

01

Identify the sample size and number of positive responses

First, we need to identify our sample size (n) and the number of positive responses, or "yes" replies in our study. According to the information given, the sample size is 92 students, and there are 23 students who responded "yes" to using Facebook just to pass the time.
02

Calculate the sample proportion

Now we need to calculate the sample proportion (often denoted as \(\hat{p}\)). The sample proportion is the fraction of the students in the sample who responded "yes" to the question. We can calculate the sample proportion using the formula: \(\hat{p} = \frac{x}{n}\) Where \(x\) is the number of positive responses (23 students) and \(n\) is the sample size (92 students).
03

Compute the sample proportion

Using the formula from Step 2 and the provided values, we can compute the sample proportion: \(\hat{p} = \frac{23}{92} = 0.25\) So, the sample proportion is 0.25, or 25%. This means that, in the sample of 92 students, 25% of them use Facebook just to pass the time.
04

Use the sample proportion to estimate the population proportion

The sample proportion (\(\hat{p}\)) can be used as an estimate for the population proportion (often denoted as \(p\)). This is because we assume that the sample is representative of the whole population, and therefore the proportions should be similar. So, based on our sample, we can estimate that approximately 25% of the entire student population at this college uses Facebook just to pass the time. Keep in mind that this is only an estimate, and there may be some sampling error or variation among different samples.

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Key Concepts

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

Sample Proportion
Understanding the sample proportion is crucial when conducting statistical research. It represents the fraction or percentage of our sample that exhibits a particular characteristic. In educational contexts, such as studies on social networking experiences, knowing the sample proportion allows researchers to make inferences about the larger student population.

For example, in the study mentioned, the sample proportion, denoted as \(\hat{p}\), was calculated by dividing the number of students who use Facebook to pass the time (23) by the total number of students sampled (92), resulting in \(\hat{p} = \frac{23}{92} = 0.25\). This figure (25%) serves as an estimate of the proportion we might expect in the larger population. However, it is important to remember that the sample proportion may vary from the actual population proportion due to factors such as sampling error.
Sample Size
The sample size in a study significantly affects the accuracy of the statistical estimates. A larger sample size generally means a more precise estimate of the population proportion. In our case, 92 students were sampled for the Facebook usage study.

A key point for students to understand is that the reliability of the sample proportion as an estimate for the population proportion improves with a larger sample size. However, there are diminishing returns; after a certain point, increasing the sample size yields only slight improvements. Statistical formulas and software can help determine the ideal sample size needed for given confidence levels and margins of error.
Statistics Education
In statistics education, the focus is not merely on how to compute values but also on understanding what these values represent in real-world contexts. It's important to teach students not just the process of calculating something like a sample proportion, but also its implications for real-life scenarios such as social media usage trends.

Real-world examples, such as estimating the proportion of students using social networking sites, make abstract statistical concepts more tangible. This approach to statistics education encourages critical thinking and data literacy, which are essential skills in today鈥檚 data-driven society.
Social Networking Experiences
When students engage with studies related to social networking experiences, they often find the content directly relevant to their lives, which can enhance learning and comprehension. This study on Facebook usage, for instance, not only sheds light on social media trends among college students but also provides an opportunity to discuss privacy, mental health, and personal productivity.

Bringing in studies based on social networking experiences into the classroom adds an element of interest and engagement that can boost students' motivation to learn statistics. It also provides a platform to discuss how data is collected and utilized in research, highlighting ethical considerations and the importance of critical evaluation of sources and methodologies.

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Most popular questions from this chapter

A press release about a paper that appeared in The Journal of Youth and Adolescence (www.springer.com/ about1springer/media/springertselect?SGWID50-11001-6 \(-1433942-0,\) August \(26,2013,\) retrieved May 8,2017\()\) was titled "Video Games Do Not Make Vulnerable Teens More Violent." The press release includes the following statement about the study described in the paper: "Study finds no evidence that violent video games increase antisocial behavior in youths with pre- existing psychological conditions." In the context of a hypothesis test with the null hypothesis being that video games do not increase antisocial behavior, explain why the title of the press release is misleading.

The article "Public Acceptability in the UK and the USA of Nudging to Reduce Obesity: The Example of Reducing Sugar-Sweetened Beverages" (PLOS One, June 8,2016 ) describes a survey in which each person in a representative sample of 1082 adult Americans was asked about whether they would find different types of interventions acceptable in an effort to reduce consumption of sugary beverages. When asked about a tax on sugary beverages, 459 of the people in the sample said they thought that this would be an acceptable intervention. These data were used to test \(H_{0}: p=0.5\) versus \(H_{a^{*}}: p<0.5\) and the null hypothesis was rejected. a. Based on the hypothesis test, what can you conclude about the proportion of adult Americans who think that taxing sugary beverages is an acceptable intervention in an effort to reduce consumption of sugary beverages? b. Is it reasonable to say that the data provide strong support for the alternative hypothesis? c. Is it reasonable to say that the data provide strong evidence against the null hypothesis?

According to a large national survey conducted by the Pew Research Center ("What Americans Think About NSA Surveillance, National Security and Privacy," May \(2,2015,\) wwW.pewresearch.org, retrieved December 1,2016 ), \(54 \%\) of adult Americans disapprove of the National Security Agency collecting records of phone and Internet data. Suppose that this estimate was based on a random sample of 1000 adult Americans. a. Is there convincing evidence that a majority of adult Americans feel this way? Test the relevant hypotheses using a 0.05 significance level. b. The actual sample size was much larger than 1000 . If you had used the actual sample size when doing the calculations for the test in Part (a), would the \(P\) -value have been larger than, the same as, or smaller than the \(P\) -value you obtained in Part (a)? Provide a justification for your answer.

Researchers at the University of Washington and Harvard University analyzed records of breast cancer screening and diagnostic evaluations ("Mammogram Cancer Scares More Frequent Than Thought," USA TODAY, April 16,1998 ). Discussing the benefits and downsides of the screening process, the article states that although the rate of falsepositives is higher than previously thought, if radiologists were less aggressive in following up on suspicious tests, the rate of false-positives would fall, but the rate of missed cancers would rise. Suppose that such a screening test is used to decide between a null hypothesis of \(H_{0}:\) no cancer is present and an alternative hypothesis of \(H_{a}:\) cancer is present. (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) a. Would a false-positive (thinking that cancer is present when in fact it is not) be a Type I error or a Type II error? b. Describe a Type I error in the context of this problem, and discuss the consequences of making a Type I error. c. Describe a Type II error in the context of this problem, and discuss the consequences of making a Type II error. d. Recall the statement in the article that if radiologists were less aggressive in following up on suspicious tests, the rate of false-positives would fall but the rate of missed cancers would rise. What aspect of the relationship between the probability of a Type I error and the probability of a Type II error is being described here?

A number of initiatives on the topic of legalized gambling have appeared on state ballots. A political candidate has decided to support legalization of casino gambling if he is convinced that more than two-thirds of American adults approve of casino gambling. Suppose that 1035 of the people in a random sample of 1523 American adults said they approved of casino gambling. Is there convincing evidence that more than two-thirds approve?
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