Question

Categorical Data. In Exercises 23 and 24, use the given categorical data to construct the relative frequency distribution.

Births Natural births randomly selected from four hospitals in New York State occurred on the days of the week (in the order of Monday through Sunday) with these frequencies: 52, 66, 72, 57, 57, 43, 53. Does it appear that such births occur on the days of the week with equal frequency?

Short answer

The relative frequency distribution is constructed as follows:

Days	Relative Frequency
Monday	13%
Tuesday	17%
Wednesday	18%
Thursday	14%
Friday	14%
Saturday	11%
Sunday	13%

It can be said that births occur equally frequently on different days of the week.

Step-by-step solution

Given information

The frequency distribution of the number of births that occur on different days of the week is provided.

Define relative frequency

The relative frequency of a class interval in a grouped frequency distribution is obtained using the given formula:

$\text{Relative}\text{frequency}\text{of}\text{a}\text{class}\text{interval}=\frac{\text{Frequency}\text{of}\text{a}\text{class}\text{interval}}{\text{Sum}\text{of}\text{all}\text{frequencies}}\times 100\%$

Calculate relative frequency distribution

The table below shows the frequencies of births on the days of the week

Days	Frequency
Monday	52
Tuesday	66
Wednesday	72
Thursday	57
Friday	57
Saturday	43
Sunday	53

The sum of all frequencies is shown below:

$$\begin{gathered} \text{Sum}\text{of}\text{all}\text{frequencies}=52+66+72+57+57+43+53 \\ =400 \end{gathered}$$

Using the formula mentioned above, the relative frequencies for the different categories are calculated as shown below:

$$\begin{gathered} \text{Relative}{\text{freq}}_1=\frac{52}{400}\times 100\% \\ =13\% \\ \text{Relative}{\text{freq}}_2=\frac{66}{400}\times 100\% \\ =17\% \end{gathered}$$

$$\begin{gathered} \text{Relative}{\text{freq}}_3=\frac{72}{400}\times 100\% \\ =18\% \\ \text{Relative}{\text{freq}}_4=\frac{57}{400}\times 100\% \\ =14\% \end{gathered}$$

$$\begin{gathered} \text{Relative}{\text{freq}}_5=\frac{57}{400}\times 100\% \\ =14\% \\ \text{Relative}{\text{freq}}_6=\frac{43}{400}\times 100\% \\ =11\% \end{gathered}$$

$$\begin{gathered} \text{Relative}{\text{freq}}_7=\frac{53}{400}\times 100\% \\ =13\% \end{gathered}$$

Thus, the relative frequency distribution is constructed as follows:

Days	Relative Frequency
Monday	13%
Tuesday	17%
Wednesday	18%
Thursday	14%
Friday	14%
Saturday	11%
Sunday	13%
Total	100%

Analysis

It can be observed that almost all days have an approximately equal relative frequency.

Therefore, it can be said that the births occur equally frequently on different days of the week.