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Chapter 12: Problem 8

The paper "Alcohol Consumption, Sleep, and Academic Performance Among College Students" (Journal of Studies on Alcohol and Drugs [2009]: 355-363) describes a study of \(n=236\) students that were randomly selected from a list of students enrolled at a liberal arts college in the northeastern region of the United States. Each student in the sample responded to a number of questions about their sleep patterns. For these 236 students, the sample mean additional time spent sleeping on weekend days compared to the other days of the week was reported to be 1.29 hours and the standard deviation was 1.09 hours. Suppose that you are interested in learning about the value of \(\mu,\) the mean additional time spent sleeping on weekend days for students at this college. The following table is similar to the table that appears in Example 12.4 . The "what you know" information has been provided. Complete the table by filling in the "how you know it" column.

Short Answer

Expert verified
The table is completed with the following information: 1. What you know: The population is the students at a liberal arts college in the northeastern region of the United States. - How you know it: Stated in the exercise. 2. What you know: The variable of interest is the mean additional time spent sleeping on weekend days. - How you know it: Implied in the exercise, as the study investigates sleep patterns and focuses on the mean additional time spent sleeping on weekend days. 3. What you know: The sample from the population has a total of \(n=236\) students. - How you know it: Stated in the exercise as the total number of students randomly selected. 4. What you know: The sample mean additional time spent sleeping on weekend days is \(\bar{x} = 1.29\) hours. - How you know it: Provided in the exercise as part of the results of the survey. 5. What you know: The sample standard deviation of additional time spent sleeping on weekend days is \(s = 1.09\) hours. - How you know it: Provided in the exercise as part of the results of the survey.

Step by step solution

01

Identify the Given Information

The sample consists of \(n=236\) students from a liberal arts college in the northeastern region of the United States. The sample mean additional time spent sleeping on weekend days compared to other days of the week was reported to be 1.29 hours and the standard deviation was 1.09 hours. We are interested in estimating the population mean (\(\mu\)) additional time spent sleeping on weekend days.
02

Make Use of the Given Information

In the table, we are asked about three things: "What you know", "Variable", and "How you know it". The given information is as follows: "What you know": \\ - The population is the students at a liberal arts college in the northeastern region of the United States. \\ - The variable of interest is the mean additional time spent sleeping on weekend days. \\ - The sample from the population has a total of \(n=236\) students. \\ - The sample mean additional time spent sleeping on weekend days is \(\bar{x} = 1.29\) hours. \\ - The sample standard deviation of additional time spent sleeping on weekend days is \(s = 1.09\) hours.
03

Complete the Table

Now let's fill in the remaining column "How you know it" based on the given information: 1. What you know: The population is the students at a liberal arts college in the northeastern region of the United States. - How you know it: This is stated in the exercise. 2. What you know: The variable of interest is the mean additional time spent sleeping on weekend days. - How you know it: It is implied in the exercise since the study investigates sleep patterns and focuses on the mean additional time spent sleeping on weekend days. 3. What you know: The sample from the population has a total of \(n=236\) students. - How you know it: Stated in the exercise as the total number of students randomly selected. 4. What you know: The sample mean additional time spent sleeping on weekend days is \(\bar{x} = 1.29\) hours. - How you know it: This information is provided in the exercise as part of the results of the survey. 5. What you know: The sample standard deviation of additional time spent sleeping on weekend days is \(s = 1.09\) hours. - How you know it: This information is also provided in the exercise as part of the results of the survey.

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Key Concepts

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

Population Mean Estimation
Understanding the concept of population mean estimation is crucial for interpreting data in studies and research. In the context of educational statistics, when we discuss population mean estimation, we are referring to the process of using sample data to predict the average outcome of an entire population. For example, a study looking at the sleep patterns of college students may not be able to survey every single individual due to time and resources constraints. Instead, a smaller group, known as a sample, is examined to estimate the sleep habits of all students.

Based on the sample data, the mean additional time spent sleeping on weekends, denoted as \(\bar{x}\), is used as the best estimate for the unknown population mean \(\mu\). With a sample size of \(n=236\) students, we assume this to be a representative subset from which the findings can reasonably infer about the larger group. Therefore, through statistical methods such as confidence intervals or hypothesis testing, we can estimate the mean additional sleep time for the entire population of students at that northeastern liberal arts college.
Sample Standard Deviation
The sample standard deviation is a metric that measures the amount of variation or dispersion from the mean within a dataset. To put it simply, it indicates how spread out the values in our sample are. In the given study, the sample standard deviation of additional sleep time on weekends is \(s = 1.09\) hours.

The calculation of standard deviation involves determining each data point's deviation from the mean, squaring it to negate any negative values, and averaging those squared deviations. A square root of this average yields the sample standard deviation. The smaller the standard deviation, the closer the individual points are to the mean. Conversely, a larger standard deviation indicates more variability among data points. Understanding the standard deviation helps researchers and educators to analyze the consistency of student behaviors, in this case, the amount of additional sleep students get on weekends.
Academic Performance and Sleep Patterns
Studies continuously reveal a link between academic performance and sleep patterns, making it an important area of investigation for educators and policy makers. The relationship between these two factors is grounded in the premise that adequate sleep is vital for cognitive functions such as memory, attention, and critical thinking鈥攁ll of which are necessary for learning and academic success.

When examining sleep patterns, researchers look at not just the quantity of sleep, but also its quality and consistency. For college students, additional hours of sleep on weekends may indicate an attempt to compensate for sleep deprivation during the week, which can have mixed implications for academic performance. Interestingly, oversleeping on weekends could also disrupt the body's natural rhythm, potentially resulting in poorer academic outcomes. By conducting statistical analyses on sleep data, such as in the referenced study, educators can better understand how to support their students' academic endeavors through healthier sleep habits.

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Most popular questions from this chapter

The dodo was a species of flightless bird that lived on the island of Mauritius in the Indian Ocean. The first record of human interaction with the dodo occurred in 1598 , and within 100 years the dodo was extinct due to hunting by humans and other newly introduced invasive species. After the extinction, the word "dodo" became synonymous with stupidity, implying that the birds lacked the intelligence to avoid or escape extinction. The closest existing relatives of the dodo are pigeons and doves. Researchers at the American Museum of Natural History used computed tomography (CT) scans to measure the brain size ("endocranial capacity") of one of the few existing preserved dodo birds, and to measure the brain sizes in samples of eight birds that are close relatives of dodos ("The First Endocast of the Extinct Dodo and an Anatomical Comparison Amongst Close Relatives," Zoological Journal of the Linnean Society [2016]: 950-953) The brain size for the dodo was \(4.17 \log \mathrm{mm}^{3}\). The following table contains the brain sizes for the sample of birds from related species (approximate values from a graph in the paper). a. Use the output at the bottom of the page from the Shiny App "Randomization Test for One Mean" to help you to carry out a randomization test of the hypothesis that the population mean brain size for birds that are relatives of the dodo differs from the established dodo brain size of 4.17 . b. What does the result of your test indicate about the brain size of the dodo?

The authors of the paper "Changesin Quantity, Spending, and Nutritional Characteristics of Adult, Adolescent and Child Urban Corner Store Purchases After an Environmental Intervention"(Preventative Medicine [2015]: \(81-85\) ) wondered if increasing the availability of healthy food options would also increase the amount people spend at the corner store. They collected data from a representative sample of 5949 purchases at corner stores in Philadelphia after the stores increased their healthy food options. The sample mean amount spent for this sample of purchases was \(\$ 2.86\) and the sample standard deviation was \(\$ 5.40\). a. Notice that for this sample, the sample standard deviation is greater than the sample mean. What does this tell you about the distribution of purchase amounts? b. Before the stores increased the availability of healthy foods, the population mean total amount spent per purchase was thought to be about \(\$ 2.80\). Do the data from this study provide convincing evidence that the population mean amount spent per purchase is greater after the change to increased healthy food options? Carry out a hypothesis test with a significance level of 0.05

How much money do people spend on graduation gifts? In \(2016,\) the National Retail Federation (www.nrf.com) surveyed 2511 consumers who reported that they bought one or more graduation gifts in 2016 . The sample was selected to be representative of adult Americans who purchased graduation gifts in 2016 . For this sample, the mean amount spent per gift was \(\$ 53.73 .\) Suppose that the sample standard deviation was \(\$ 20 .\) Construct and interpret a \(98 \%\) confidence interval for the mean amount of money spent per graduation gift in 2016 .

Give as much information as you can about the \(P\) -value of a \(t\) test in each of the following situations: a. Two-tailed test, \(n=16, t=1.6\) b. Upper-tailed test, \(n=14, t=3.2\) c. Lower-tailed test, \(n=20, t=-5.1\) d. Two-tailed test, \(n=16, t=6.3\)

An airplane with room for 100 passengers has a total baggage limit of 6000 pounds. Suppose that the weight of baggage checked by an individual passenger, \(x\), has a mean of 50 pounds and a standard deviation of 20 pounds. If 100 passengers will board a flight, what is the approximate probability that the total weight of their baggage will exceed the limit? (Hint: With \(n=100\), the total weight exceeds the limit when the mean weight \(\bar{x}\) exceeds \(6000 / 100 .\) )
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