Question

Preschool The ages (in months) at which 50 children were first enrolled in a preschool are listed below. $$\begin{array}{llllllllll}38 & 40 & 30 & 35 & 39 & 40 & 48 & 36 & 31 & 36 \\\47 & 35 & 34 & 43 & 41 & 36 & 41 & 43 & 48 & 40 \\\32 & 34 & 41 & 30 &46 & 35 & 40 & 30 & 46 & 37 \\\55 & 39 & 33 & 32 & 32 & 45 & 42 & 41 & 36 & 50 \\\42 & 50 & 37 & 39 & 33 & 45 & 38 & 46 & 36 & 31\end{array}$$ a. Construct a stem and leaf display for the data. b. Construct a relative frequency histogram for these data. Start the lower boundary of the first class at 30 and use a class width of 5 months. c. Compare the graphs in parts a and b. Are there any significant differences that would cause you to choose one as the better method for displaying the data? d. What proportion of the children were 35 months (2 years, 11 months) or older, but less than 45 months ( 3 years, 9 months) of age when first enrolled in preschool? e. If one child were selected at random from this group of children, what is the probability that the child was less than 50 months old ( 4 years, 2 months) when first enrolled in preschool?

Short answer

Answer: The probability is 0.58 or 58%.

Step-by-step solution

a. Construct a stem and leaf display

To create a stem and leaf display, first, we need to sort the given dataset in ascending order. Then, we can construct the display using the tens place as a 'stem' and the ones place as a 'leaf'. Sorted dataset: 30, 30, 30, 31, 31, 32, 32, 32, 33, 33, 34, 34, 35, 35, 35, 36, 36, 36, 36, 37, 37, 38, 38, 39, 39, 39, 40, 40, 40, 40, 41, 41, 41, 42, 42, 43, 43, 45, 45, 46, 46, 46, 47, 48, 48, 50, 50, 55 Stem and leaf display: 3 | 0 0 0 1 1 2 2 2 3 3 4 4 5 5 5 6 6 6 6 7 7 8 8 9 9 9 4 | 0 0 0 0 1 1 1 2 2 3 3 5 5 6 6 6 7 8 8 5 | 0 0 5

b. Construct a relative frequency histogram

To construct a relative frequency histogram, we need to create frequency classes and count the number of data points within each class. We are given a class width of 5 months, starting at 30. Frequency classes: - 30-34 - 35-39 - 40-44 - 45-49 - 50-54 - 55-59 Next, we count the data points within each class and calculate the relative frequency (actual frequency/total data points). Relative frequency table: 30-34: 12/50 = 0.24 35-39: 16/50 = 0.32 40-44: 13/50 = 0.26 45-49: 7/50 = 0.14 50-54: 2/50 = 0.04 55-59: 1/50 = 0.02 Now we can plot the histogram with the frequency classes on the x-axis and the relative frequencies on the y-axis.

c. Compare the graphs

Both the stem and leaf display and the relative frequency histogram are useful for understanding data. The stem and leaf display gives exact age values, which may be useful when comparing individual ages. However, it doesn't offer as clear a visualization of the overall distribution as the histogram. The relative frequency histogram helps identify the distribution of ages more intuitively, with the frequency classes grouped together. This allows for easier identification of trends, patterns, and center in the data. However, it lacks the granularity provided by the stem and leaf display. There is no significant difference that would cause choosing one as the better method for displaying the data, as both offer useful insights in different ways.

d. Proportion of children within a specific age range

To find the proportion of children aged between 35 months (2 years, 11 months) and 45 months (3 years, 9 months), we need to count the data points within this age range and divide by the total number of data points (50). There are 16 children aged 35-39 months and 13 children aged 40-44 months. So, there are a total of 29 children within the given range. The proportion is 29 children / 50 children = 0.58, or 58%.

e. Probability of a child under 50 months old

To find the probability that a randomly selected child is less than 50 months old when first enrolled in preschool, we count the data points with ages less than 50 months and divide by the total number of data points (50). In our data set, there are 48 children aged under 50 months old. The probability is 48 children / 50 children = 0.96, or 96%.

Key concepts

Stem and Leaf Plot

A stem and leaf plot is a simple way to organize and visualize data. It takes each data point, separates it into a "stem" and "leaf." The 'stem' is generally the leading digit(s), and the 'leaf' is usually the last digit.
For our problem, we have ages in months, ranging from 30 to 55. Here, the tens digit will serve as the stem, and the ones digit will be the leaf.
Creating a stem and leaf plot helps in quickly understanding the distribution of data, such as identifying clusters, gaps, and outliers. You can see individual data points as well as how often groups of numbers occur.
Remember to sort the data before plotting, as this gives a clear picture of the dataset's shape.

Relative Frequency Histogram

A relative frequency histogram represents how often data points fall within specified intervals. It's a bar graph where each bar's height is proportional to the frequency of data in that interval, showing the relative sizes of each group.
In our exercise, class intervals of 5 months were used, starting from 30, resulting in groups like 30-34, 35-39, etc. The relative frequency is calculated by dividing the number of observations in a class by the total number of observations.
This visualization makes it easy to see the overall distribution, patterns, and central tendencies in the data. Even though it doesn't convey detailed data precision, it provides a clearer picture of the data's overall spread.

Probability

Probability is the measure of the likelihood of an event occurring. It's expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
In the context of our exercise, we calculated the probability of a child being less than 50 months old when first enrolled in preschool. This was done by determining how many children fall under this age and dividing that by the total number of children.
Probability is fundamental in predicting outcomes and making informed decisions, be it through experiments, surveys, or data sets like the one in our exercise.

Data Visualization

Data visualization is the graphical representation of information and data. It helps in understanding data quickly by presenting it visually via graphs, charts, and plots.
In our scenario, both stem and leaf plots and histograms were used for data visualization purposes. They allowed us to represent the ages of children graphically, each with a unique way of showing the data’s characteristics.
By using visuals, complex data becomes easier to digest, making trends, patterns, and outliers more recognizable. Effective data visualization is crucial for communicating insights clearly and efficiently.