Question

Preschool The ages (in months) at which 50 children were first enrolled in a $$\begin{aligned}&\text { preschool are listed below. }\\\&\begin{array}{llllllllll}38 & 40 & 30 & 35 & 39 & 40 & 48 & 36 & 31 & 36 \\\47 & 35 & 34 & 43 &41 & 36 & 41 & 43 & 48 & 40 \\\32 & 34 & 41 & 30 & 46 & 35 & 40 & 30 & 46 & 37 \\\55 & 39 & 33 & 32 & 32 & 45 & 42 & 41 & 36 & 50 \\\42 & 50 &37 & 39 & 33 & 45 & 38 & 46 & 36 & 31\end{array}\end{aligned}$$ 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

Question: Compare the stem and leaf display and the relative frequency histogram created with a lower boundary of 30 and class width of 5 months to determine if there are any significant differences that would lead us to choose one method over the other when analyzing the age data of 50 children when they first enrolled in preschool. Also, calculate the proportion of children who were 35 months or older but younger than 45 months when they first enrolled in preschool, and find the probability that a randomly selected child would be less than 50 months old.

Step-by-step solution

Sort the Data

Arrange the ages in ascending order.

Create Stems and Leaves

Organize the data into stems (representing groups). Each group should be 10 months apart. Then, list the leaves for each stem (the last digit of each data point in the group). #b. Constructing a Relative Frequency Histogram#

Determine the Classes

Create classes with a lower boundary of 30 and a class width of 5 months.

Count the occurrences in each class

Count how many ages fall into each class.

Calculate Relative Frequencies

Divide the frequency of each class by the total number of data points (50) to find the relative frequency.

Draw the Histogram

Create a histogram using the classes and their relative frequencies, with the classes on the x-axis and relative frequencies on the y-axis. #c. Comparing the Graphs and Determining any Differences# Examine both graphs and note any differences that might make one method better than the other for displaying the data. #d. Proportion of children 35 months or older but younger than 45 months#

Count the number of children in this age range

Determine how many children were 35-44 months old when they first enrolled in preschool.

Calculate the proportion

Divide the number of children aged 35-44 months by the total number of children (50). #e. Probability of a randomly selected child being less than 50 months old#

Count the number of children less than 50 months

Determine how many children were less than 50 months old when they first enrolled.

Calculate the probability

Divide the number of children less than 50 months by the total number of children (50).

Key concepts

Stem-and-Leaf Display

A stem-and-leaf display is a simple way to organize and visualize a dataset. Each data point is split into a "stem" and a "leaf." For example, if you have the number 46, the stem is 4, and the leaf is 6. The idea is to arrange data to show its distribution.
When creating a stem-and-leaf display, the stems represent groups like tens, hundreds, etc. The leaves are the individual units forming the data. Here, it would help categorize data in 10-month intervals.
This visualization helps in quickly identifying the distribution, the mode, and any outliers. It's particularly useful for finding clusters or gaps in the data. Moreover, unlike histograms, it retains the original data values.

Relative Frequency Histogram

A relative frequency histogram is a visual representation of the distribution of a dataset. It's similar to a regular histogram, but it uses relative frequency rather than absolute frequency. Relative frequency is the proportion, or fraction, of the data that falls within each class.
To construct a relative frequency histogram, follow these steps:

• Create class intervals with specified boundaries and widths.
• Count the number of data points falling within each interval.
• Calculate the relative frequency by dividing the frequency by the total number of data points.
• Plot the class intervals on the x-axis and the relative frequencies on the y-axis.

The main benefit of a relative frequency histogram is that it makes it easier to compare different datasets, as the y-axis is scaled in terms of proportion rather than raw counts.

Probability

Probability measures the likelihood of an event occurring. In context, it relates to selecting one data point and finding the chance it fits a certain category.
To find the probability, first count the favorable outcomes. For example, if we want to know how likely a child is less than 50 months old, count those children. Next, divide this by the total population size. Formulaically, probability is calculated as:\[ P(Event) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \] This provides a value between 0 and 1, where 0 means the event cannot happen, and 1 means it certainly will. Probability helps in making predictions about random events based on past data.

Proportions

Proportion is a part, share, or number considered in comparative relation to a whole. It's used for understanding and comparing fractions of datasets.
In this context, it's about dividing the children into groups based on their age enrollment. If you're analyzing what fraction of children are between 35 and 44 months old, you calculate the proportion by dividing the count of such children by the total count of children.

• Proportions help in understanding the relative size of different categories within a dataset.
• A typical formula for proportion is: \( \, \frac{\text{Number in category}}{\text{Total number}} \, \).

By analyzing proportions, one can infer patterns, trends, or distributions within the larger dataset, which aids in statistical analysis and decision-making.