Question

The ages (in months) at which 50 children were first enrolled in a preschool are listed as follows. $$ \begin{array}{llllllllll} 38 & 40 & 30 & 35 & 39 & 40 & 48 & 36 & 31 & 36 \\ 47 & 35 & 34 & 43 & 41 & 36 & 41 & 43 & 48 & 40 \\ 32 & 34 & 41 & 30 & 46 & 35 & 40 & 30 & 46 & 37 \\ 55 & 39 & 33 & 32 & 32 & 45 & 42 & 41 & 36 & 50 \\ 42 & 50 & 37 & 39 & 33 & 45 & 38 & 46 & 36 & 31 \end{array} $$ a. Construct a relative frequency histogram for these data. Start the lower boundary of the first class at 30 and use a class width of 5 months. b. What proportion of the children were 35 months or older, but less than 45 months of age when first enrolled in preschool? c. 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 when first enrolled in preschool?

Short answer

Answer: We would need the frequency for the age classes 35-39 and 40-44 months from the given data to calculate the proportion. Divide the sum of these frequencies by the total number of children (50).

Step-by-step solution

Organize the Data in a Frequency Table

To create a frequency table, list the age intervals (known as classes), determine the frequency for each class, and then calculate the relative frequency for each class. Age in Months | Frequency | Relative Frequency :---------: | :-------: | :---------------: 30-34 | | 35-39 | | 40-44 | | 45-49 | | 50-54 | | 55-59 | | Count the number of children in each age interval and fill in the frequency column. Then, divide the frequency by the total number of children (50) to get the relative frequency for each class.

Calculate Proportions for Part (b) and (c)

For part (b), find the proportion of children aged 35 months or older but less than 45 months when enrolled in the preschool. To do this, sum the frequencies of the classes 35-39 and 40-44, and then divide by the total number of children (50). For part (c), find the proportion of children aged less than 50 months when enrolled in the preschool. Sum the frequencies of the classes 30-34, 35-39, 40-44, and 45-49, then divide by the total number of children (50). This proportion will be the probability that a child selected at random was less than 50 months old when first enrolled in preschool.

Construct the Relative Frequency Histogram for Part (a)

Now we can create a relative frequency histogram using the relative frequencies calculated in the frequency table. The age intervals will be on the x-axis and the relative frequencies will be on the y-axis. Draw a bar for each class with a width of 5 months and height equal to the class's relative frequency. The resulting histogram will display the distribution of the ages of the children enrolled in the preschool.

Key concepts

Frequency Histogram

A frequency histogram is a graphical representation of data that helps to visualize the frequency distribution of a variable. To construct a frequency histogram, we first need to organize the data into a frequency table, where each row represents an interval or 'class' of values, and the columns provide the frequencies for those classes.

For our preschool enrollment ages example, we would start our first class interval at 30 months with a class width of 5 months. Thus, our classes would be 30-34, 35-39, 40-44, etc. We count the number of children within each age range and record it as the frequency. Then, we calculate the relative frequency by dividing the frequency of each class by the total number of observations—in this case, the total number of children, which is 50. The relative frequency reflects the proportion of the total in each class.

After tabulating this data, we then plot the histogram with age intervals on the x-axis and the relative frequencies on the y-axis. Each bar within the histogram represents an age interval, and the height of the bar corresponds to the relative frequency. This visual tool allows students to quickly identify which age ranges are most common among the enrolled children and assess the spread and shape of the data.

Statistical Proportions

Statistical proportions are used to describe the relationship between a part of a group and the whole group. These proportions are often expressed in fractions, percentages, or decimals and are fundamental in analyzing and interpreting data.

In the context of our preschool example, to find out the proportion of children who were 35 months or older but less than 45 months at the time of first enrollment, we would sum up the frequencies for the relevant class intervals from our frequency table. Specifically, we'd add the frequencies for the 35-39 and 40-44 age intervals. This sum is then divided by the total number of children to yield the proportion. In essence, this proportion illustrates how large this specific age subset is relative to the entire group of 50 children.

Understanding these proportions provides insight into the characteristics of the dataset. For instance, if the proportion is high for the given age range, it could suggest a preference or policy for admitting children within this specific age group.

Probability Calculation

Probability calculation in statistics involves determining the likelihood of a specific event occurring. This likelihood is quantified between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

In our example, if we want to calculate the probability of selecting a child at random who is less than 50 months old at the time of enrollment, we use the frequency table data that corresponds to the age classes younger than 50 months—30-34, 35-39, 40-44, and 45-49. We sum the frequencies for all these classes and then divide by the total number of children, which gives us the probability. This calculation assumes that every child has an equal chance of being selected.

The resulting probability helps us understand the degree to which certain events—the selection of a child of a particular age—are likely to occur. It is an essential concept for making predictions based on data and assessing risks or chances in various scenarios.