Question

Research psychologists are interested in finding out whether a person's breathing patterns are affected by a particular experimental treatment. To determine the general respiratory patterns of the \(n=30\) people in the study, the researchers collected some baseline measurements-the total ventilation in liters of air per minute adjusted for body size - for each person before the treatment. The data are shown here, along with some descriptive tools generated by MINITAB and MS Excel. \(\begin{array}{llllllllll}5.23 & 4.79 & 5.83 & 5.37 & 4.35 & 5.54 & 6.04 & 5.48 & 6.58 & 4.82 \\ 5.92 & 5.38 & 6.34 & 5.12 & 5.14 & 4.72 & 5.17 & 4.99 & 4.51 & 5.70 \\ 4.67 & 5.77 & 5.84 & 6.19 & 5.58 & 5.72 & 5.16 & 5.32 & 4.96 & 5.63\end{array}\) Descriptive Statistics: Liters \(\begin{array}{llll}\text { Variable } & \text { N } & \text { N* } & \text { Mean } & \text { SE Mean } & \text { StDey }\end{array}\) Liters \(\begin{array}{lllll}30 & 0 & 5,3953 & 0,0997 & 0,5462 & 2\end{array}\) \(\begin{array}{llll}\text { Minimum } & \text { Q1 Median } & \text { Q3 Variable Maximum }\end{array}\) \(\begin{array}{lllll}4.3500 & 4.9825 & 5.3750 & 5.7850 & \text { Liters } & 6.5800\end{array}\) Stem and Leaf Display: Liters Stem-and-leaf of Liters \(N=30\) Leaf unit \(=0.10\) \(\begin{array}{lll}1 & 4 & 3\end{array}\) \(\begin{array}{lll}2 & 4 & 5\end{array}\) \(5 \quad 4 \quad 677\) \(8 \quad 4 \quad 899\) \(\begin{array}{lll}12 & 5 & 1111\end{array}\) (4) 52333 \(\begin{array}{lll}14 & 5 & 455\end{array}\) 1156777 75889 \(4 \quad 6 \quad 01\) \(\begin{array}{lll}2 & 6 & 3\end{array}\) 165 MS Excel Descriptive Statistics \begin{tabular}{|lr|} \hline \multicolumn{2}{|c|} { Liters } \\ \hline Mean & 5.3953 \\ Standard Error & 0.0997 \\ Median & 5.3750 \\ Mode & #N/A \\\ Standard Deviation & 0.5462 \\ Sample Variance & 0.2983 \\ Kurtosis & 20.4069 \\ Skewness & 0.1301 \\ Range & 2.23 \\ Minimum & 4.35 \\ Maximum & 6.58 \\ Sum & 161.86 \\ Count & 30 \\ \hline \end{tabular} a. Summarize the characteristics of the data distribution using the computer output.b. Does the Empirical Rule provide a good description of the proportion of measurements that fall within two or three standard deviations of the mean? Explain. c. How large or small does a ventilation measurement have to be before it is considered unusual?

Short answer

Answer: The lower and upper bounds for identifying unusual ventilation measurements in the given dataset are 4.3029 liters and 6.4877 liters, respectively.

Step-by-step solution

Summarizing the data distribution

We will use the available data from both MINITAB and MS Excel outputs to summarize the data distribution. 1. Mean = 5.3953 2. Median = 5.3750 3. Standard Deviation = 0.5462 4. Range = 2.23 5. Minimum = 4.35 6. Maximum = 6.58 7. Skewness = 0.1301 8. Kurtosis = 20.4069 9. Sample Variance = 0.2983 10. Q1 (First quartile) = 4.9825 11. Q3 (Third quartile) = 5.7850

Checking if Empirical Rule applies

The Empirical Rule states that for data with a normal (bell-shaped) distribution, approximately 68% of the data lies within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. First, let's find the ranges for the 1st, 2nd, and 3rd standard deviations: 1. Mean ± 1 SD: [5.3953 - 0.5462, 5.3953+0.5462] = [4.8491, 5.9415] 2. Mean ± 2 SD: [5.3953 - 2×0.5462, 5.3953+2×0.5462] = [4.3029, 6.4877] 3. Mean ± 3 SD: [5.3953 - 3×0.5462, 5.3953+3×0.5462] = [3.7567, 7.0339] Now, let's count how many of the data points fall within these ranges: 1. Mean ± 1 SD: 19 out of 30 data points (63.33%) 2. Mean ± 2 SD: 29 out of 30 data points (96.67%) 3. Mean ± 3 SD: 30 out of 30 data points (100%) Looking at the calculated percentages, the Empirical Rule provides a reasonably good description of the proportion of measurements within 2 or 3 standard deviations of the mean but differs slightly for one standard deviation (63.33% instead of 68%). Considering that the dataset is fairly small, (n=30), we can accept this slight deviation.

Identifying unusual ventilation measurements

We'll consider a ventilation measurement unusual if it lies outside the range of 2 standard deviations from the mean, as 95% of the data is expected to fall within this range. Using the mean and standard deviation calculated earlier, the lower and upper bounds for 2 standard deviations are: Lower bound: 5.3953 - 2×0.5462 = 4.3029 liters Upper bound: 5.3953 + 2×0.5462 = 6.4877 liters Therefore, a ventilation measurement is considered unusual if it is smaller than 4.3029 liters or larger than 6.4877 liters.

Key concepts

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, is a simple guideline for understanding how data is dispersed in a normal distribution. It states that:

• Approximately 68% of data falls within one standard deviation ( 1σ ) of the mean.
• About 95% of data lies within two standard deviations ( 2σ ) from the mean.
• Nearly 99.7% of data is within three standard deviations ( 3σ ) of the mean.

This rule is quite handy when you want to quickly assess where most of your data points lie and detect any unusual observations. For instance, if a large portion of your data significantly deviates from these boundaries, your data might not be normally distributed. However, always remember, the Empirical Rule truly applies to perfectly normal distributions, so minor discrepancies can occur with smaller datasets.

Data Distribution

Data distribution refers to how data points are spread across different values on a scale. A commonly used format to describe these patterns is a histogram, stem-and-leaf display, or box plot. These descriptive tools help convey if data is symmetric, skewed, or has outliers.

In the provided study's dataset, the mean and median of about 5.39 liters and 5.375 liters respectively suggest a rather symmetric distribution. This is further supported by a low skewness of 0.1301, indicating only a slight right skew. Moreover, descriptive statistics like quartiles (Q1 and Q3) and the range help one understand data spread. Quartiles divide the data set into four equal parts, showing spread specifically around the median. For this dataset, incorporating these descriptive tools provides a solid sense of data's overall spread and center balance.

Standard Deviation

Standard deviation is a measure that quantifies the amount of variation or dispersion in a dataset. A lower standard deviation means the data points tend to be close to the mean of the set, while a higher standard deviation indicates that the data points are spread out over a larger range.

In our dataset, the standard deviation is 0.5462. This indicates the average distance each data point is from the mean (5.3953). Observing this measure helps us determine how concentrated the data is around the mean. If most data points fall within one standard deviation of the mean, it confirms a less varied distribution. However, excessive values or gaps beyond this range are less common and might be points of interest such as potential outliers or data collection anomalies.

Normal Distribution

A normal distribution, often called a bell curve due to its iconic shape, is where most data points cluster around a central mean, with frequency decreasing symmetrically towards the extremes. It is characterized by its bell shape, which is defined by its mean (the highest point) and standard deviation (the width).

This statistical distribution is foundational in probability theory and statistics due to the Central Limit Theorem, which explains why it appears in various natural phenomena.

• Characteristics of a normal distribution include symmetry about the mean.
• The mean, median, and mode are all equal.
• The distribution is asymptotic, never touching the x-axis.

In the provided exercise, although the data shows a slight skew and higher kurtosis, it mostly models a normal distribution with a bell-like shape, suitable for applying the Empirical Rule.