/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 285 Exercise 4.86 on page 263 introd... [FREE SOLUTION] | 91Ó°ÊÓ

91Ó°ÊÓ

Exercise 4.86 on page 263 introduces a matched pairs study in which 47 participants had cell phones put on their ears and then had their brain glucose metabolism (a measure of brain activity) measured under two conditions: with one cell phone turned on for 50 minutes (the "on" condition) and with both cell phones off (the "off" condition). Brain glucose metabolism is measured in \(\mu \mathrm{mol} / 100 \mathrm{~g}\) per minute, and the differences of the metabolism rate in the on condition minus the metabolism rate in the off condition were computed for all participants. The mean of the differences was 2.4 with a standard deviation of \(6.3 .\) Find and interpret a \(95 \%\) confidence interval for the effect size of the cell phone waves on mean brain metabolism rate.

Short Answer

Expert verified
The 95% confidence interval for the effect size of cell phone waves on mean brain metabolism rate can be calculated as \( 2.4 \pm 1.96 * \frac{6.3}{\sqrt{47}} \), which gives the indicated range. The interpretation of these results is that we can be 95% confident that the true mean difference in brain glucose metabolism rate for the population lies within this range.

Step by step solution

01

Identify the Sample Size, Sample Mean, and Standard Deviation

From the problem, we know that the sample size (\(n\)) is 47, the sample mean (labelled as \(\bar{X}\)) is 2.4, and the standard deviation (labelled as \(s\)) is 6.3.
02

Calculate Standard Error

The Standard Error(SE) measures the accuracy with which a sample represents a population. SE is calculated as the standard deviation divided by the square root of the sample size. In this case, it is \( SE = \frac{s}{\sqrt{n}} = \frac{6.3}{\sqrt{47}} \).
03

Determine the Confidence Level and Critical Value

From the problem, we know that the confidence level is 95%. Since our data should be normally distributed, we use the z-score for a 95% confidence level. The critical value (labelled as \(Z_{\alpha/2}\)) is approximately 1.96.
04

Calculate the Confidence Interval

The formula for a 95% confidence interval is \(\bar{X} \pm Z_{\alpha/2} * SE\). Plugging in our calculated values gives us the confidence interval as follows: \(2.4 \pm 1.96 * SE \).
05

Interpret the Output

The final step involves interpreting the resulting confidence interval. It can be presumed that, with 95% certainty, the true population mean brain glucose metabolism difference between 'on' and 'off' states of the cell phone lies within the calculated interval.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Confidence Interval
A confidence interval provides a range of values that is likely to contain the true population parameter we are interested in, such as a mean difference, with a certain level of confidence, usually expressed as a percentage. In our matched pairs study, the problem is asking for a 95% confidence interval. This means that we expect the true difference in brain metabolism rates—attributable to cell phone use—to lie within this interval 95 times out of 100, if the experiment were repeated many times.
Confidence intervals are calculated using a formula, which incorporates the sample mean, the standard error, and the critical value derived from the chosen confidence level. Here, our sample mean difference is 2.4, referring to the "on" condition minus the "off" condition. The critical value for a 95% interval, using a normal distribution, is 1.96, representing a z-score.
  • First, compute the standard error (SE), which is the standard deviation divided by the square root of the sample size.
  • The critical value is then multiplied by the SE, and the result is added to and subtracted from the sample mean to get the lower and upper bounds of our interval.
So, in practice, a 95% confidence interval gives us a reliable statistical "range" reflecting where the true mean difference could lie, considering our sample data.
Brain Glucose Metabolism
Brain glucose metabolism is a measure of how the brain uses glucose, often seen as an indicator of brain activity. This is crucial in neuroscience because the brain needs a constant supply of glucose for energy, especially during stressful or demanding tasks. In our study, we explore how cell phone usage might affect this metabolism.
When your brain is more active, it tends to consume more glucose. Thus, measuring glucose metabolism can reveal changes or impacts on brain function. The study mentioned utilizes this metric to assess the biological effect of cell phone exposure, by comparing metabolism rates with the phones either turned on or off.
  • The study involves a paired design, meaning that each participant acts as their own control. This reduces the variability that might come from individual differences in metabolism.
  • Such a direct comparison allows researchers to isolate the effects of cell phone exposure on each participant's brain activity.
Understanding how and if our brain's glucose metabolism changes with environmental factors like cell phone waves can lead us to comprehend implications on cognitive tasks or potential long-term effects.
Effect Size
Effect size is a statistical measure that describes the strength or magnitude of a phenomenon. In research, assessing the effect size is essential for understanding the practical significance of a study's findings, above and beyond statistical significance. In our case involving brain glucose metabolism, the effect size helps us determine how much cell phone exposure could potentially alter brain activity.
Calculating effect size can help quantify the difference we observe concerning brain glucose levels when cell phones are "on" compared to "off." Without assessing effect size, a significant statistical result might be misleading, due to either very large sample sizes driving significance, or the observed effect being practically insignificant.
  • Notably, when reported alongside confidence intervals, effect size equips researchers and readers to understand both the potential range and magnitude of an effect.
  • This provides a clearer picture of the true impact of cell phone usage on brain function.
Such dimensions of the study give insights beyond mere numbers—informing public health guidelines or future technological assessments.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises 6.1 to \(6.6,\) if random samples of the given size are drawn from a population with the given proportion: (a) Find the mean and standard error of the distribution of sample proportions. (b) If the sample size is large enough for the Central Limit Theorem to apply, draw a curve showing the shape of the sampling distribution. Include at least three values on the horizontal axis. Samples of size 100 from a population with proportion 0.41

6.8 Percent over 600 on Math SAT In the class of \(2010,25 \%\) of students taking the Mathematics portion of the SAT (Scholastic Aptitude Test) \(^{3}\) scored over a 600 . If we take random samples of 100 members of the class of 2010 and compute the proportion who got over a 600 on the Math SAT for each sample, what will be the mean and standard deviation of the distribution of sample proportions?

We see in the AllCountries dataset that the percent of the population that is elderly (over 65 years old) is 17.0 in Austria and 15.9 in Denmark. Suppose we take random samples of size 200 from each of these countries and compute the difference in sample proportions \(\hat{p}_{A}-\hat{p}_{D}\), where \(\hat{p}_{A}\) represents the proportion of the sample that is elderly in Austria and \(\hat{p}_{D}\) represents the proportion of the sample that is elderly in Denmark. Find the mean and standard deviation of the differences in sample proportions.

Standard Error from a Formula and Simulation In Exercises 6.15 to \(6.18,\) find the mean and standard error of the sample proportions two ways: (a) Use StatKey or other technology to simulate at least 1000 sample proportions. Give the mean and standard error and comment on whether the distribution appears to be normal. (b) Use the formulas in the Central Limit Theorem to compute the mean and standard error. Are the results similar to those found in part (a)? Sample proportions of sample size \(n=10\) from a population with \(p=0.2\)

Use StatKey or other technology to generate a bootstrap distribution of sample means and find the standard error for that distribution. Compare the result to the standard error given by the Central Limit Theorem, using the sample standard deviation as an estimate of the population standard deviation. Mean commute time in Atlanta, in minutes, using the data in CommuteAtlanta with \(n=500\), \(\bar{x}=29.11,\) and \(s=20.72\)

See all solutions

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.