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Chapter 3: Problem 26

Daily activity. It appears that people who are mildly obese are less active than leaner people. One study looked at the average number of minutes per day that people spend standing or walking. \({ }^{10}\) Among mildly obese people, minutes of activity varied according to the \(N(373,67)\) distribution. Minutes of activity for lean people had the \(N(526,107)\) distribution. Within what limits do the active minutes for about \(95 \%\) of the people in each group fall? Use the 68-95-99.7 rule.

Short Answer

Expert verified
For mildly obese: 239 to 507 minutes; for lean: 312 to 740 minutes.

Step by step solution

01

Understand the 68-95-99.7 Rule

The 68-95-99.7 rule (also known as the empirical rule) is a statistical rule which states that for a normal distribution, almost all data will fall within three standard deviations of the mean. Specifically, about 68% of the data falls within one standard deviation, 95% within two, and 99.7% within three.
02

Identify Mean and Standard Deviation

For mildly obese people, the mean (\(\mu\)) is 373 minutes and the standard deviation (\(\sigma\)) is 67 minutes. For lean people, the mean is 526 minutes and the standard deviation is 107 minutes.
03

Calculate Range for Mildly Obese Group

Using the 68-95-99.7 rule, to find the range for 95% of the data, calculate \(\mu \pm 2\sigma\). For mildly obese people: \(373 \pm 2\times67 = 373 \pm 134\). Therefore, the range is from 239 to 507 minutes.
04

Calculate Range for Lean Group

Similarly, for lean people: calculate \(\mu \pm 2\sigma\). So, \(526 \pm 2\times107 = 526 \pm 214\). Thus, the range is from 312 to 740 minutes.

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Key Concepts

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

Normal Distribution
The concept of normal distribution is fundamental in statistics. It describes how data points are spread over a range, and is often represented by a symmetrical bell-shaped curve. This distribution is widely found in natural settings and indicates that most data points are concentrated around the mean, with fewer appearing as you move away from the center.
In the study mentioned, we look at a normal distribution where minutes of activity for two groups follow a predictable pattern:
  • For mildly obese individuals, activity minutes are distributed according to a normal distribution with a mean of 373 and a standard deviation of 67.
  • Lean individuals have a different normal distribution with a mean of 526 and a standard deviation of 107.
Understanding the normal distribution allows statisticians to make predictions and calculate probabilities about the data at hand.
68-95-99.7 Rule
The 68-95-99.7 rule, also known as the empirical rule, is a critical shortcut used in statistics for analyzing normal distributions. This rule simplifies how we view data by stating:
  • 68% of the data falls within one standard deviation from the mean.
  • 95% of the data is within two standard deviations.
  • 99.7% of data points lie within three standard deviations.
Using this rule allows quick insight into data spread without complex calculations. In our activity study, it guides us to find where 95% of activity times for mildly obese and lean individuals will fall. You simply need to calculate the mean plus or minus two times the standard deviation to find this range.
Mean and Standard Deviation
The mean and standard deviation are two key statistics used to describe a data set's characteristics in a normal distribution. The mean is the average value of the data set, representing the center point or "typical" value.
  • For mildly obese people, the mean is 373 minutes of activity.
  • For lean individuals, it's 526 minutes.
The standard deviation measures how spread out the values in a data set are around the mean.
  • A larger standard deviation, like the 107 minutes for lean people, indicates more variability in activity minutes.
  • A smaller one, such as the 67 minutes for mildly obese people, suggests less variability.
Together, these statistics help define the shape and spread of the normal distribution curve.
Statistical Analysis
Statistical analysis involves collecting and examining data to draw meaningful conclusions. It's essential in understanding complex data through the lens of methods like the 68-95-99.7 rule and tools like mean and standard deviation.
  • In this context, statistical analysis provided insights into how activity levels differ between mildly obese and lean individuals.
  • By using mean, standard deviation, and the rule, researchers can determine the expected range of daily activity minutes for the majority (95%) of each group.
This analysis aids in making informed decisions and identifying patterns, differences, or trends within data. Moreover, it informs interventions or programs aimed at increasing physical activity levels in these populations.

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Most popular questions from this chapter

Monsoon Rains. The summer monsoon rains in India follow approximately a Normal distribution with mean 852 millimeters \((\mathrm{mm})\) of rainfall and standard deviation \(82 \mathrm{~mm}\). (a) In the drought year \(1987,697 \mathrm{~mm}\) of rain fell. In what percent of all years will India have \(697 \mathrm{~mm}\) or less of monsoon rain? (b) "Normal rainfall" means within \(20 \%\) of the long-term average, or between \(682 \mathrm{~mm}\) and \(1022 \mathrm{~mm}\). In what percent of all years is the rainfall normal?

Body mass index. Your body mass index (BMI) is your weight in kilograms divided by the square of your height in meters. Many online BMI calculators allow you to enter weight in pounds and height in inches. High BMI is a common but controversial indicator of overweight or obesity. A study by the National Center for Health Statistics found that the BMI of American young men (ages 20-29) is approximately Normal with mean \(26.8\) and standard deviation 5.2. 12 (a) People with BMI less than \(18.5\) are often classified as "underweight." What percent of men aged 20-29 are underweight by this criterion? (b) People with BMI over 30 are often classified as "obese." What percent of men aged \(20-29\) are obese by this criterion?

Where are the quartiles? How many standard deviations above and below the mean do the quartiles of any Normal distribution lie? (Use the standard Normal distribution to answer this question.)

The Medical College Admissions Test. Almost all medical schools in the United States require students to take the Medical College Admission Test (MCAT). \({ }^{8}\) A new version of the exam was introduced in spring 2015 and is intended to shift the focus from what applicants know to how well they can use what they know. One result of the change is that the scale on which the exam is graded has been modified, with the total score of the four sections on the test ranging from 472 to 528 . In spring 2015 , the mean score was \(500.0\) with a standard deviation of \(10.6\). (a) What proportion of students taking the MCAT had a score over 510 ? (b) What proportion had scores between 505 and 515 ?

Grading managers. In Exercise 3.44, we saw that Ford Motor Company once graded its managers in such a way that the top \(10 \%\) received an A grade, the bottom \(10 \%\) a C, and the middle \(80 \%\) a B. Let's suppose that performance scores follow a Normal distribution. How many standard deviations above and below the mean do the \(\mathrm{A} / \mathrm{B}\) and \(\mathrm{B} / \mathrm{C}\) cutoffs lie? (Use the standard Normal distribution to answer this question.)
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