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Chapter 10: Problem 3

The article "Facebook Use and Academic Performance Among College Students," Computers in Human Behavior [2015]: \(265-272\) ) estimated that \(70 \%\) of students at a large public university in California who are Facebook users log into their Facebook profiles at least six times a day. Suppose that you plan to select a random sample of 400 students at your college. You will ask each student in the sample if they are a Facebook user and if they log into their Facebook profile at least six times a day. You plan to use the resulting data to decide if there is evidence that the proportion for your college is different from the proportion reported in the article for the college in Califomia. What hypotheses should you test? (Hint: See Example \(10.3 .)\)

Short Answer

Expert verified
The hypotheses to be tested are: Null Hypothesis (\( H_0\)): \( p = 0.7 \) Alternative Hypothesis (\( H_a\)): \( p \ne 0.7 \)

Step by step solution

01

Define the proportions

Let \( p \) be the proportion of students at your college who log into their Facebook profiles at least six times a day and \( p_0 \) be the reported proportion at the college in California, which is \(0.7\).
02

State the null hypothesis

The null hypothesis is that the proportion of students at your college logging in at least six times a day is equal to the reported proportion of students at the college in California. Mathematically, the null hypothesis can be written as: \( H_0: p = p_0 \)
03

State the alternative hypothesis

The alternative hypothesis is that the proportion of students at your college logging in at least six times a day is not equal to the reported proportion of students at the college in California. Mathematically, the alternative hypothesis can be written as: \( H_a: p \ne p_0 \) So the hypotheses to be tested are: Null Hypothesis (\( H_0\)): \( p = 0.7 \) Alternative Hypothesis (\( H_a\)): \( p \ne 0.7 \)

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Key Concepts

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

Understanding the Null Hypothesis
When you're testing hypotheses in statistics, the null hypothesis ( H_0 ) is the starting assumption that there is no significant effect or difference. It's like saying, "Everything is as it seems, and nothing unusual is happening." In our exercise, the null hypothesis suggests that the proportion of students logging into Facebook six times a day at your college is the same as at the California university鈥�70%.
  • This hypothesis acts as a baseline. It's what you're initially testing against.
  • Null doesn't mean "false"! The null hypothesis is what you assume true until evidence suggests otherwise.
Keep this in mind: when conducting a test, you either reject the null hypothesis or fail to reject it. You never "accept" it. This is a common pitfall, and it's crucial to interpret it correctly.

In summary, the null hypothesis provides the "no difference" benchmark needed to measure against new data.
Exploring the Alternative Hypothesis
The alternative hypothesis ( H_a ) is the claim you're seeking evidence for; it represents a new effect or difference. In this scenario, the alternative hypothesis is that the proportion of students at your college who frequently log into Facebook is different from what was observed at the California university鈥攕omeone says, "Something's going on. It's not 70%!"
  • The alternative hypothesis usually represents what researchers suspect or hope to support through their study.
  • It's either two-tailed (like our exercise: not equal to 70%) or one-tailed (greater than or less than a certain value).
So, when data suggests the null hypothesis (no difference) is unlikely, we look for support in the alternative hypothesis. The goal is to gather enough statistical evidence through our sample to lean toward accepting this alternative viewpoint, showing there's something distinct with your college's Facebook habits. This helps researchers understand and interpret variations in their data.
Introduction to Proportions in Hypothesis Testing
Proportions are a way to express a fraction of a population with a certain characteristic. In our case, it's about quantifying how many students log into Facebook frequently at your college compared to the California university.
  • Proportions are often used because they're easy to understand and compare across different groups.
  • Mathematically, a proportion is calculated as the number of times an event occurs divided by the total number.
When using proportions in hypothesis testing:
- You're often comparing them to see if two groups differ significantly. - Proportions are particularly handy when dealing with large groups or samples, like in our 400-student scenario.

Understanding proportions is essential because it allows you to comprehend how widespread a characteristic is within a population. In hypothesis testing, it provides a meaningful metric to analyze and make informed decisions based on the data collected.

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Most popular questions from this chapter

For which of the following combinations of \(P\) -value and significance level would the null hypothesis be rejected? a. \(P\) -value \(=0.426 \quad \alpha=0.05\) b. \(P\) -value \(=0.033 \quad \alpha=0.01\) c. \(P\) -value \(=0.046 \quad \alpha=0.10\) d. \(P\) -value \(=0.026 \quad \alpha=0.05\) e. \(P\) -value \(=0.004 \quad \alpha=0.01\)

The report "Robot, You Can Drive My Car: Majority Prefer Driverless Technology" (Transportation Research Institute University of Michigan, www.umtri.umich.edu/what -were-doing/news/robot-you-can-drive-my-car-majority -prefer-driverless-technology, July \(22,2015,\) retrieved May 8 , 2017 ) describes a survey of 505 licensed drivers. Each driver in the sample was asked if they would prefer to keep complete control of the car while driving, to use a partially self-driving car that allowed partial driver control, or to turn full control over to a driverless car. Suppose that it is reasonable to regard this sample as a random sample of licensed drivers in the United States, and that you want to use the data from this survey to decide if there is evidence that fewer than half of all licensed drivers in the United States prefer to keep complete control of the car while driving. a. Describe the shape, center, and variability of the sampling distribution of \(\hat{p}\) for random samples of size 505 if the null hypothesis \(H_{0}: p=0.50\) is true. b. Would you be surprised to observe a sample proportion as small as \(\hat{p}=0.48\) for a sample of size 505 if the null hypothesis \(H_{0}: p=0.50\) were true? Explain why or why not. c. Would you be surprised to observe a sample proportion as small as \(\hat{p}=0.46\) for a sample of size 505 if the null hypothesis \(H_{0}: p=0.50\) were true? Explain why or why not, d. The actual sample proportion observed in the study was \(\hat{p}=0.44\). Based on this sample proportion, is there convincing evidence that fewer than \(50 \%\) of licensed drivers prefer to keep complete control of the car when driving, or is the sample proportion consistent with what you would expect to see when the null hypothesis is true? Support your answer with a probability calculation.

Public Policy Polling conducts an annual poll on sportsrelated issues. In 2015 , they found that in a sample of 1222 adult Americans, 794 said that they thought the designated hitter rule in professional baseball should be eliminated and that pitchers should be required to bat (www.publicpolicypolling.com/pdf/2015/PPP_Release_National_51216.pdf, retrieved December 1,2016 ). Suppose that this sample is representative of adult Americans. Based on the given information, is there convincing evidence that a majority of adult Americans think that the designated hitter rule should be eliminated and that pitchers should be required to bat?

A college has decided to introduce the use of plus and minus with letter grades, as long as there is convincing evidence that more than \(60 \%\) of the faculty favor the change. A random sample of faculty will be selected, and the resulting data will be used to test the relevant hypotheses. If \(p\) represents the proportion of all faculty who favor a change to plus-minus grading, which of the following pairs of hypotheses should be tested? $$ H_{0^{*}} p=0.6 \text { versus } H_{i}: p<0.6 $$ or $$ H_{0}: p=0.6 \text { versus } H_{e}: p>0.6 $$ Explain your choice.

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?
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