91Ó°ÊÓ

A \(90 \%\) confidence interval for the number of hours that full-time college students sleep during a weekday is lower bound: 7.8 hours and upper bound: 8.8 hours. Which of the following represents a reasonable interpretation of the result? For those that are not reasonable, explain the flaw. (a) \(90 \%\) of full-time college students sleep between 7.8 hours and 8.8 hours. (b) We are \(90 \%\) confident that the mean number of hours of sleep that full- time college students get any day of the week is between 7.8 hours and 8.8 hours. (c) There is a \(90 \%\) probability that the mean hours of sleep that full-time college students get during a weekday is between 7.8 hours and 8.8 hours. (d) We are \(90 \%\) confident that the mean hours of sleep that fulltime college students get during a weekday is between 7.8 hours and 8.8 hours.

Step by step solution

01

Analyzing Statement (a)

Statement (a) implies that 90% of individual full-time college students sleep between 7.8 and 8.8 hours. However, confidence intervals pertain to the population mean, not individual values. This interpretation is incorrect because the confidence interval is about the average sleep time, not the range for 90% of individuals.

02

Analyzing Statement (b)

Statement (b) suggests we are 90% confident that the mean number of hours of sleep full-time college students get any day of the week falls between 7.8 and 8.8 hours. This is incorrect because the interval pertains to sleep during a weekday, not any day of the week.

03

Analyzing Statement (c)

Statement (c) implies a 90% probability that the mean sleep hours during a weekday falls within 7.8 to 8.8 hours. This is incorrect as confidence intervals do not represent a probability for the parameter but a method that, in the long run, will cover the true parameter value in 90% of repeated samples.

04

Analyzing Statement (d)

Statement (d) correctly states we are 90% confident that the mean hours of sleep full-time college students get during a weekday lies between 7.8 and 8.8 hours. This is the appropriate interpretation of a confidence interval as it refers to the population mean within the specified days and confidence level.

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Key Concepts

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

Mean Hours of Sleep

The term 'mean hours of sleep' refers to the average number of hours that full-time college students sleep during a specific period, in this case, a weekday.
For example, if you have the sleep data for five students: 7.5, 8, 8.5, 7.8, and 8.2 hours, the mean can be calculated by summing these values and then dividing by the number of students.
Mathematically: \(\text{Mean} = \frac{7.5 + 8 + 8.5 + 7.8 + 8.2}{5} = 8\text{ hours} \).
The mean provides a central value around which individual sleep hours vary. Understanding the mean helps identify general sleep patterns in a population rather than focusing on outliers.

Confidence Level

A confidence level indicates the probability that a calculated confidence interval actually contains the true population parameter, which in this case is the mean hours of sleep.
In the given exercise, we have a 90% confidence level, meaning there is a 90% chance the interval from 7.8 to 8.8 hours includes the true mean sleep duration.
When we say we're 90% confident, it's based on the idea that if we were to take 100 different samples and compute confidence intervals for each, we'd expect about 90 of those intervals to contain the true mean.
Rather than indicating a probability about a specific parameter, it tells us about the reliability of the estimation method over multiple trials.

Full-Time College Students

The term 'full-time college students' refers to students enrolled for enough credit hours to be considered full-time by their educational institution, typically around 12-15 credit hours per semester.
It's important to distinguish these students as their lifestyle, which includes attending classes, studying, and possibly working part-time, can significantly impact their sleep patterns.
In research, narrowing down our study group to full-time students helps to control for variables such as course load and activity levels that might affect sleep, making our confidence interval more accurate for this specific population.

Weekday Sleep Patterns

Weekday sleep patterns may differ significantly from weekend patterns due to structured schedules like classes, study sessions, and extracurricular activities.
The consistency of a college student's weekday schedule can help in drawing more reliable conclusions about their sleep habits during the week.
In contrast to weekends where sleep patterns might be irregular due to social activities or catching up on rest, weekdays provide a more controlled environment to study sleep duration.
Thus, the confidence interval provided is particularly relevant for understanding the sleep behavior of full-time college students from Monday through Friday.