Question

Breathing rates for humans can be as low as 4 breaths per minute or as high as 70 or 75 for a person doing strenuous exercise. Suppose that the resting breathing rates for college-age students have a distribution that is mound- shaped, with a mean of 12 and a standard deviation of 2.3 breaths per minute. What fraction of all students have breathing rates in the following intervals? a. 9.7 to 14.3 breaths per minute b. 7.4 to 16.6 breaths per minute c. More than 18.9 or less than 5.1 breaths per minute

Short answer

In a college, the breathing rates of students follow a mound-shaped distribution with a mean of 12 breaths per minute and a standard deviation of 2.3 breaths per minute. What are the approximate fractions of students with breathing rates in the following intervals: a. 9.7 to 14.3 breaths per minute b. 7.4 to 16.6 breaths per minute c. More than 18.9 or less than 5.1 breaths per minute Results: a. Approximately 68.26% of students have breathing rates between 9.7 and 14.3 breaths per minute. b. Approximately 95.44% of students have breathing rates between 7.4 and 16.6 breaths per minute. c. Approximately 0.26% of students have a breathing rate more than 18.9 or less than 5.1 breaths per minute.

Step-by-step solution

The exercise states that the mean (μ) is 12 breaths per minute and the standard deviation (σ) is 2.3 breaths per minute. #Step 2: Calculate the z-scores for each interval#

To find the z-scores, we use the formula: z = (X - μ)/σ, where X is the value of interest. For each part of the exercise, we will find the z-scores corresponding to the interval limits: a. 9.7 to 14.3 breaths per minute: - z1 = (9.7 - 12)/2.3 = -1 - z2 = (14.3 - 12)/2.3 = 1 b. 7.4 to 16.6 breaths per minute: - z1 = (7.4 - 12)/2.3 = -2 - z2 = (16.6 - 12)/2.3 = 2 c. More than 18.9 or less than 5.1 breaths per minute: - z1 = (18.9 - 12)/2.3 = 3 - z2 = (5.1 - 12)/2.3 = -3 #Step 3: Find the fractions corresponding to z-scores using a standard normal table#

Refer to a standard normal table (also called a z-table) to find the fractions of students with breathing rates in the specified intervals: a. 9.7 to 14.3 breaths per minute: - P(-1 < z < 1) = 0.8413 - 0.1587 = 0.6826 b. 7.4 to 16.6 breaths per minute: - P(-2 < z < 2) = 0.9772 - 0.0228 = 0.9544 c. More than 18.9 or less than 5.1 breaths per minute: - P(z > 3) = 1 - 0.9987 = 0.0013 - P(z < -3) = 0.0013 - P(z > 3 or z < -3) = 0.0013 + 0.0013 = 0.0026 #Step 4: State the results#

In conclusion, the fractions of students with breathing rates in the specified intervals are: a. 9.7 to 14.3 breaths per minute: approximately 68.26% of students b. 7.4 to 16.6 breaths per minute: approximately 95.44% of students c. More than 18.9 or less than 5.1 breaths per minute: approximately 0.26% of students

Key concepts

Standard Deviation

A standard deviation measures how data points vary from the mean in a dataset. It tells us how "spread out" the numbers are. If you have a low standard deviation, data points are close to the mean; a high standard deviation means they are spread out over a wide range.
For the breathing rates example, the standard deviation is 2.3 breaths per minute. This indicates the average amount by which students' breathing rates differ from the mean. To calculate standard deviation, you take each data point, find its distance from the mean, square it, average these squared differences, and then take the square root of that average.
Standard deviation helps us understand the variability of a dataset and is crucial for calculating z-scores, which tell us how far a particular point is from the mean in units of standard deviations.

Mean

The mean is the average of all numbers in a dataset. Add them up and divide by the total count of numbers. It's a simple and common measurement of the "center" of data. In our problem, the mean breathing rate for students is 12 breaths per minute.
Calculating the mean provides a quick overview of the data's central tendency. Knowing the mean is important because it serves as a reference point when we calculate z-scores.
When data is normally distributed, as in this example, the mean divides the distribution evenly. This makes it an anchor point for understanding how individual data points are related to the whole set.

Z-Score

A z-score tells you how many standard deviations a data point is from the mean. It's calculated using the formula \((X - μ)/σ\), where \(X\) is the value, \(μ\) is the mean, and \(σ\) is the standard deviation. Z-scores standardize data points for comparison, no matter the units.
In the breathing rates exercise, the z-scores for intervals are calculated to see how common or rare certain breathing rates are. For instance, if the z-score is -1, it means the value is one standard deviation below the mean.
Z-scores are used with the standard normal distribution table to find probabilities, helping us understand the proportion of data points within certain ranges. They help answer questions like "What fraction of students have breathing rates between 9.7 and 14.3 breaths per minute?"

Normal Distribution

A normal distribution, or Gaussian distribution, is a bell-shaped curve that is symmetric about the mean. It describes how data is spread, showing that most data points are close to the mean, with fewer and fewer as you move further away.
In this exercise, students' breathing rates follow a normal distribution with a mean of 12 and a standard deviation of 2.3. This means most students have breathing rates around 12, with fewer students having much higher or lower rates.
Understanding normal distribution helps us use z-scores to find the probability of certain events, like how likely a student's breathing rate falls within a specific range or beyond extreme values. This approach is widely used in statistics for making predictions based on data.