Question

A Recurring Illness The length of EX0126 time (in months) between the onset of a particular illness and its recurrence was recorded for \(n=50\) patients: $$\begin{array}{rrrrrrrrrr}2.1 & 4.4 & 2.7 & 32.3 & 9.9 & 9.0 & 2.0 & 6.6 & 3.9 & 1.6 \\\14.7 & 9.6 & 16.7 & 7.4 & 8.2 & 19.2 & 6.9 & 4.3 & 3.3 & 1.2 \\\4.1 & 18.4 & .2 & 6.1 & 13.5 & 7.4 & .2 & 8.3 & 3 & 1.3 \\\14.1 & 1.0 & 2.4 & 2.4 & 18.0 & 8.7 & 24.0 & 1.4 & 8.2 & 5.8 \\\1.6 & 3.5 & 11.4 & 18.0 & 26.7 & 3.7 &12.6 & 23.1 & 5.6 & .4\end{array}$$ a. Construct a relative frequency histogram for the data. b. Would you describe the shape as roughly symmetric, skewed right, or skewed left? c. Give the fraction of recurrence times less than or equal to 10 months.

Short answer

Question: Analyze the given data on the time between the onset of a particular illness and its recurrence in 50 patients by constructing a relative frequency histogram, describing the shape of the distribution, and calculating the fraction of recurrence times less than or equal to 10 months. Answer: The constructed relative frequency histogram has a skewed right distribution. The fraction of recurrence times less than or equal to 10 months is 0.7 (or 70%).

Step-by-step solution

Organize the data into intervals

First, we need to determine our intervals for the histogram. A common practice is to choose between 5 and 15 intervals. In this case, we can choose an interval width of 5, which gives us the following intervals: (0-5], (5-10], (10-15], (15-20], (20-25], (25-30], and (30-35].

Calculate the frequency for each interval

Now, we need to count how many recurrence times fall into each interval: - (0-5]: 21 - (5-10]: 14 - (10-15]: 5 - (15-20]: 4 - (20-25]: 3 - (25-30]: 1 - (30-35]: 2

Calculate the relative frequency for each interval

Next, we will divide the frequency of each interval by the total number of patients (n=50) to find the relative frequency for each interval: - (0-5]: \(\frac{21}{50} = 0.42\) - (5-10]: \(\frac{14}{50} = 0.28\) - (10-15]: \(\frac{5}{50} = 0.1\) - (15-20]: \(\frac{4}{50} = 0.08\) - (20-25]: \(\frac{3}{50} = 0.06\) - (25-30]: \(\frac{1}{50} = 0.02\) - (30-35]: \(\frac{2}{50} = 0.04\)

Construct the histogram

Using the intervals and their corresponding relative frequencies, construct the histogram using a graphing software or tool. The x-axis will be the intervals, and the y-axis will be the relative frequencies.

Analyze the shape of the histogram

Looking at the histogram, we can observe that the distribution has a longer tail to the right side. This suggests that the shape of the histogram is skewed right.

Calculate the fraction of recurrence times less than or equal to 10 months

To find the fraction of recurrence times less than or equal to 10 months, we can add up the relative frequencies of the intervals (0-5] and (5-10]: Fraction = 0.42 + 0.28 = 0.7 Thus, 70% (or 0.7 as a fraction) of the recurrence times are less than or equal to 10 months.

Key concepts

Relative Frequency

Relative frequency is a measure that allows us to understand how often something happens in relation to a whole. It is calculated by taking the frequency of a particular event or interval and dividing it by the total number of observations.

For constructing a histogram, relative frequency gives a proportion that conveys the shape of the distribution without being influenced by the total number of observations. This makes it particularly useful when comparing datasets of different sizes. For instance, if we observe that the relative frequency of the interval (0-5] months is 0.42, it means that 42% of the recurrence times for a particular illness fall within this interval. By utilizing relative frequency, we convert raw data into understandable proportions that offer immediate insight into data distribution.

Data Visualization

Data visualization involves the graphical representation of information and data. By using visual elements like charts, graphs, and maps, it provides an accessible way to see and understand trends, outliers, and patterns in data.

In our exercise, a histogram serves as the chosen method of data visualization. As a type of bar chart, a histogram displays the distribution of numerical data. It is constructed by partitioning the continuous scale into intervals (bins) and then counting the number of observations that fall into each interval. The height of each bar reflects the relative frequency, which represents the proportion of data within each interval compared to the total dataset. A well-constructed histogram can tell a story about skewness, central tendency, and dispersion at a glance, which is why it's a key tool in descriptive statistics analysis.

Skewness in Statistics

Skewness in statistics refers to the asymmetry in the distribution of data. Specifically, it points to the lack of symmetry that can occur in probability distributions, where the curve represented by the distribution is not even on both sides of a central peak.

There are two main types of skewness: right (positive) skewness and left (negative) skewness. If the tail of the distribution extends more to the right, it is considered right-skewed. This suggests a majority of the data is concentrated on the left side of the histogram. Conversely, if the tail extends to the left, the data is left-skewed. The skewness of a distribution provides insights into the underlying data and can have implications for statistical analysis. For example, in the exercise, the histogram's longer tail to the right indicates a right-skewed distribution, highlighting the presence of outliers—that is, longer times between illness recurrences.

Descriptive Statistics Analysis

Descriptive statistics analysis is a cornerstone of data analysis that summarizes important features of data, typically with numeric calculations, graphs, or tables. It involves measures of central tendency (like mean, median, and mode), measures of dispersion (like range, variance, and standard deviation), and measures of the shape of the distribution (like skewness and kurtosis).

In relation to our problem, we have used descriptive statistics analysis to construct a histogram that visually represents the dataset's distribution. Moreover, by identifying the skewness of the histogram, we deduce that the illness recurrence times are not evenly distributed, which can have meaningful implications for medical practitioners or policymakers. Additionally, calculating the fraction of recurrence times within a specified interval provides a clear and precise summary of a particular aspect of the data, aiding in efficient decision-making and knowledge extraction.