Question

1.26 \text { A Recurring IIIness The length of 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: Given the data on the length of time (in months) between the onset of a particular illness and its recurrence for 50 patients, create a relative frequency histogram, describe the shape of the histogram as roughly symmetric, skewed right, or skewed left, and calculate the fraction of recurrence times less than or equal to 10 months.

Step-by-step solution

Organize the data and calculate the relative frequencies

To construct the histogram, we need to divide the data into intervals (also called bins) and calculate the relative frequency for each interval. To do that, first, we should decide on the number of bins. A common choice is to use the \(\sqrt{n}\) rule, in this case, \(\sqrt{50} \approx 7\). So, let's use 7 bins. Now, we find the range of the data (max-min) and divide by the number of bins to find the width of each bin. After that, sort the data into these bins and calculate the relative frequency of each bin, which is the count of data points in that bin divided by the total number of data points.

Create the histogram

Create the histogram using the intervals/bins and their corresponding relative frequencies calculated in the previous step. Make sure to label the x-axis with the corresponding bin values and y-axis with the relative frequencies.

Analyze the histogram and describe its shape

Look at the shape of the histogram created in Step 2. If the data in the histogram is roughly symmetric, with a peak in the middle and tails on both directions, describe it as symmetric. If the histogram has a tail elongated to the right (with a larger fraction of recurrence times above the average), describe it as skewed right. If the histogram has a tail elongated to the left (with a larger fraction of recurrence times below the average), describe it as skewed left.

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

Count the number of recurrence times in the data set that are less than or equal to 10 months. Divide this count by the total number of data points (50), giving the fraction of recurrence times less than or equal to 10 months.

Key concepts

Relative Frequency Histogram

A relative frequency histogram is an essential tool in descriptive statistics that allows us to understand how often various data values occur within a dataset. It visualizes the distribution of the data by displaying the relative frequencies of different intervals, also known as bins. The relative frequency of a bin is calculated by dividing the number of data points in that bin by the total number of data points. For example, if we divide our dataset into 7 bins, each bin's relative frequency will show what proportion of the dataset falls into that specific range. The vertical axis of the histogram represents these relative frequencies, while the horizontal axis shows the intervals or bins. This method provides a clear, visual representation of data distribution, making it easier to see where most of the data points lie. A larger section on the histogram indicates that more data points fall within that specific range of values.

Data Distribution

Understanding the data distribution is crucial in statistical analysis. It refers to how data points are spread across different values in a dataset. With the help of a histogram, such as the one we created in the previous section, we can visualize the distribution pattern. There are common shapes for data distributions:

• Symmetric: Where data points are evenly distributed, often resembling a bell curve.
• Skewed Right: Where the majority of the data points are concentrated towards the left, with a tail extending to the right.
• Skewed Left: Where most of the data points are concentrated towards the right, with a tail extending to the left.

Examining the histogram allows us to describe and understand the spread. For instance, if most recurrence times are on one end, leaving a longer tail on the opposite side, we can determine whether the dataset is skewed or symmetrical.

Statistical Analysis

Statistical analysis involves techniques and methods to interpret and extract meaningful information from a dataset. For example, in analyzing recurrence times of the illness, we aren't just interested in creating a histogram, but also in interpreting the meaning of various statistical measures. Furthermore, we can compute the fraction of recurrence times that are less than or equal to a particular value, such as 10 months. This involves counting how many data points fall below this threshold and dividing that by the total number of observations. Statistical analysis provides insights beyond basic visual representations, allowing for a deeper understanding of patterns, trends, and relationships within the dataset. This information is valuable for making informed decisions and predictions.