91Ó°ÊÓ

Runners. In a study of exercise, a large group of male runners walk on a treadmill for six minutes. After this exercise, their heart rates vary with mean \(8.8\) beats per five seconds and standard deviation \(1.0\) beats per five seconds. The researcher records the number of heartbeats per five seconds for each runner over a period of time. This distribution takes only whole-number values, so it is certainly not Normal. a. Let \(x\) be the mean number of beats per five seconds after measuring heart rate for 24 five-second intervals (two minutes). What is the approximate distribution of \(x\) according to the central limit theorem? b. What is the approximate probability that \(x\) is less than 8 ? c. What is the approximate probability that the heart rate of a runner is less than 100 beats per minute? (Hint: Restate this event in terms of \(x .)\)

Step by step solution

01

Understand the Problem

We have a non-Normal distribution of heart rates after running, and we need to apply the Central Limit Theorem (CLT). We'll calculate probabilities related to mean values taken over several intervals.

02

Apply the Central Limit Theorem to the Mean

According to the CLT, when sampling distribution involves a large enough sample, the distribution of the sample mean is approximately Normal. Here, the sample mean, \( \bar{x} \), will have a mean \( \mu_x = \mu = 8.8 \) and standard deviation \( \sigma_x = \frac{\sigma}{\sqrt{n}} = \frac{1.0}{\sqrt{24}} \approx 0.204 \).

03

Find the Probability \(x < 8\)

First, calculate the z-score: \( z = \frac{8 - 8.8}{0.204} \approx -3.92 \). Use the standard normal distribution table or calculator to find the probability that \( z < -3.92 \), which is nearly 0 since it is a very extreme point.

04

Translate Heart Rate Condition into \(x\) Terms

A heart rate of less than 100 beats per minute corresponds to \( \frac{100}{12} \approx 8.33\) beats per five seconds (since 1 minute is 12 five-second intervals). We translate this condition: \( x \leq 8.33 \).

05

Calculate Probability \(x < 8.33\)

Calculate the z-score for \( x = 8.33 \): \( z = \frac{8.33 - 8.8}{0.204} \approx -2.31 \). Using the standard normal distribution, we find \( P(z < -2.31) \approx 0.0104 \). This is the probability that the heart rate of a runner is less than 100 beats per minute.

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Key Concepts

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

Probability

Probability is a fundamental concept in statistics, representing the likelihood of an event occurring. It measures how likely something is to happen, and is represented as a number between 0 and 1, where 0 means the event will definitely not occur, and 1 means it definitely will occur.

In the context of our heart rate study, probability helps us understand how likely it is for a runner's mean heart rate, measured in beats per five seconds, to fall below specific thresholds, like less than 8 beats or less than 8.33 beats. These probabilities require converting heart rates into standard scores, known as z-scores, which relate them to a normal distribution.

Using probability:

• We can determine the likelihood of events, such as a mean heart rate falling below a given threshold.
• It allows us to quantify uncertainty and variation in measurements, providing valuable insights into expected outcomes.

Probability becomes especially powerful when combined with the Central Limit Theorem, helping us make inferences about populations based on sample data.

Normal Distribution

The normal distribution, often referred to as the bell curve, plays a crucial role in the world of statistics. It describes a probability distribution that is symmetric around its mean, where most observations cluster around the central peak and probabilities for values tapering off equally in both directions. It is characterized by its mean and standard deviation.

In our study, although the heart rate measurements initially do not follow a normal distribution, we leverage the Central Limit Theorem (CLT) to approximate them as such when considering the mean of a sufficiently large sample.

Key features of normal distribution:

• It is symmetrical: Mean, median, and mode are all equal.
• The tails of the distribution extend indefinitely but contain minimal probability.
• Defined by the bell shape and the empirical rule: approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

The normal distribution simplifies the calculation of probabilities about sample means by allowing us to use z-scores and standard normal distribution tables.

Z-score

The z-score is a measure of how many standard deviations an element is from the mean. It is a key tool in the conversion of individual data points into contextually comparable units within a normal distribution.

In the heart rate study, z-scores enable us to determine how likely it is for any given mean heart rate to fall below a particular threshold, like 8 or 8.33 beats per five seconds.

The calculation of a z-score is straightforward:

• Formula: The z-score is calculated as \( z = \frac{x - \mu}{rac{ ext{standard deviation}}{ ext{sqrt n}}} \).
• This involves subtracting the population mean \( \mu \) from the observed value \( x \), and then dividing by the standard deviation of the sample mean.

Z-scores are particularly useful because they can be translated into probabilities by referring to standard normal distribution tables or using statistical software, illuminating the exact placement of data within the distribution curve.

Heart Rate Study

In our heart rate study, researchers are interested in understanding the effects of exercise on heart rates among male runners. The collected data involves the number of beats in five-second intervals post-exercise.

The core analysis revolves around the mean heart rate and its variation over time. Although the raw data might not be normally distributed (as it consists of whole numbers only), the Central Limit Theorem allows us to approximate the distribution of sample means as normal for a large sample size.

Significance of the study:

• Understanding Variability: It helps in understanding how heart rates stabilize post-exercise.
• Application of CLT: Offers practical insights into using the theorem to make predictions based on data samples.
• Real-world Context: Provides a physiological context for the application of statistical concepts, demonstrating how theoretical ideas apply to tangible, real-life data.

This study illustrates how researchers apply statistical ideas like the CLT and probability to interpret data and glean insights into human physiology.