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Chapter 6: Problem 26

The number of times \(x\) an adult human breathes per minute when at rest depends on the age of the human and varies greatly from person to person. Suppose the probability distribution for \(x\) is approximately normal, with the mean equal to 16 and the standard deviation equal to \(4 .\) If a person is selected at random and the number \(x\) of breaths per minute while at rest is recorded, what is the probability that \(x\) will exceed \(22 ?\)

Short Answer

Expert verified
Answer: The probability is approximately 0.0668, or 6.68%.

Step by step solution

01

Identify the parameters of the normal distribution

The given normal distribution has a mean(\(\mu\)) of 16 breaths per minute, and a standard deviation (\(\sigma\)) of 4 breaths per minute.
02

Calculate the Z-score for the given number of breaths per minute (22)

To find the z-score, we use the formula: \(z = \frac{x - \mu}{\sigma}\). In this case, \(x = 22\), \(\mu = 16\), and \(\sigma = 4\). Plugging in these values, we get: \(z = \frac{22 - 16}{4} = \frac{6}{4} = 1.5\)
03

Find the probability that the number of breaths per minute is greater than 22

To find the probability that the number of breaths per minute is greater than 22, we need to find the area to the right of the z-score (1.5) using the standard normal distribution table or a calculator. The standard normal distribution table shows the area to the left of a given z-score, so we'll need to calculate by finding the area to the left of 1.5 and subtracting it from 1 (since the total area under the curve is equal to 1). Using a z-score table or a calculator, we find that the area to the left of 1.5 is 0.9332. Now, we can find the area to the right of 1.5 (which represents the probability of having more than 22 breaths per minute) by subtracting the area to the left of 1.5 from 1: \(1 - 0.9332 = 0.0668\) So, the probability that a randomly selected person will breathe more than 22 times per minute while at rest is approximately 0.0668, or 6.68%.

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Key Concepts

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

Z-score
A Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is measured in terms of standard deviations from the mean. A Z-score tells you how many standard deviations a given data point is from the average. This can help you understand whether a value is typical for a given data set or whether it's unusual. For example, if a Z-score is 0, it means the value is exactly the same as the mean.
  • To calculate a Z-score, use the formula: \( z = \frac{x - \mu}{\sigma} \), where \( x \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
  • A positive Z-score indicates the data point is above the mean, whereas a negative Z-score signifies it's below the mean.
For instance, in the exercise, a Z-score of 1.5 was computed for a breathing rate of 22 breaths per minute. This means that 22 breaths per minute is 1.5 standard deviations higher than the average rate of 16 breaths per minute.
Probability
Probability is a measure of the likelihood that an event will occur. It is a core concept in statistics and is used to make inferences about a population based on a sample. In the context of normal distribution, probability helps us determine how likely it is for a random data point to fall within a particular range.
  • Probabilities range from 0 to 1, where 0 indicates an impossible event, and 1 signifies a certain event.
  • When working with a normal distribution, a probability is often found using Z-scores to determine how area under the curve corresponds to different intervals.
In the problem, we calculated the probability that the number of breaths per minute would exceed 22. By finding the area to the right of a Z-score of 1.5, we determined the chance that a person breathes more than 22 times per minute is roughly 6.68%.
Mean and Standard Deviation
The mean is the average of a set of numbers, providing a central value for the data. It's a key component in statistical analysis, helping to summarize the overall trend of the data. The standard deviation, on the other hand, measures how spread out the numbers are around the mean. It's a way to quantify the amount of variation or dispersion in a set of data.
  • The mean is calculated by summing all the values and then dividing by the number of values.
  • Standard deviation is computed as the square root of the variance, which is the average of the squared differences from the mean.
In the given exercise, the mean was 16 breaths per minute, and the standard deviation was 4 breaths per minute. This tells us that, on average, people's breathing rates at rest cluster around 16, with most rates falling within 4 breaths above or below this mean. Together, these metrics provide a clear picture of the distribution's central tendency and variability, which are crucial for further statistical analysis, such as calculating probabilities.

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