Question

The number of births in randomly chosen hours of a day is as follows. $$ 4,0,6,5,2,1,2,0,4,3 . $$ Use this data to estimate the proportion of hours that had two or fewer births.

Short answer

The proportion of hours that had two or fewer births is \(60\% \). To get this, 6 hours (the number of hours with two or fewer births) were divided by the total number of hours (10), resulting in 0.6. This was then multiplied by 100 to be expressed as a percentage.

Step-by-step solution

Understanding the Data

The data given represents the number of births that happened in randomly chosen hours of a day. Each number shown corresponds to the number of births that occurred within that particular hour.

Counting the Hours

The next step is to count the total number of hours, represented by the total number of items in the list. In this case, there are 10 hours in total, as there are 10 numbers given.

Identifying the Relevant Hours

After obtaining the total number of hours, the next step is to identify the number of hours that had two or fewer births. This involves counting the number of instances in the data where the amount is 2 or less.

Calculating the Proportion

Once you have both numbers (total number of hours and number of hours with two or fewer births), you can calculate the proportion by dividing the latter by the former. The result represents the proportion of hours that had two or fewer births in a day.

Expressing the Proportion as a Percentage

Finally, to express this proportion as a percentage, simply multiply the result by 100. This final step gives a more intuitive and understandable representation of the proportion of hours that experienced two or fewer births.

Key concepts

Proportion Estimation

Proportion estimation is a statistical approach used to determine the ratio of a particular subset of data relative to the whole set. In the context of our example, we estimate the proportion of hours during a day that had two or fewer births.

To initiate this process, we analyze our data set which consists of the number of births during different hours in a day: \begin{align*}4, & 0, 6, 5, 2, 1, 2, 0, 4, 3\text{.}ewlineewline\rightarrow &\text{Total hours} = 10 ewline\rightarrow &\text{Hours with ≤ 2 births} = 5end{align*}By counting the instances where the number of births is two or fewer, we found out that such a condition was met in 5 out of the 10 recorded hours. To estimate the proportion, we must divide the number of hours with two or fewer births (5) by the total number of hours (10):\[\text{Proportion} = \frac{\text{Hours with ≤ 2 births}}{\text{Total hours}} = \frac{5}{10} = 0.5\]Next, to provide a more intuitive understanding, this proportion is converted into a percentage by multiplying by 100, yielding a result of 50%. This implies that, on the day observed, approximately 50% of the time, there were two or fewer births each hour.

Descriptive Statistics

Descriptive statistics are a collection of procedures that summarize and describe the characteristics of a data set. These include measures such as the mean, median, mode, range, and standard deviation, which help in understanding the distribution and central tendency of the data.

• The mean provides an average value of the data points.


• The median represents the middle value when the data is ordered.


• The mode indicates the most frequently occurring value.


• The range exposes the spread between the lowest and highest values.


• The standard deviation measures the variability or dispersion of a data set.

In our example, we can observe that the data is quite varied, illustrating different numbers of births per hour. While we focused on the count of hours with two or fewer births for proportion estimation, descriptive statistics would invite us to also consider the average births per hour, the most common number of births, and how varied the number of births are throughout the day. Through such analysis, we build a comprehensive picture of the birth patterns, enhancing our understanding beyond a simple proportion.

Quantitative Data

Quantitative data refers to information that can be quantified and expressed numerically, allowing for more precise and objective analysis. It contrasts with qualitative data, which is descriptive and subjective in nature.The data set in our example, consisting of the numbers of births per hour, is a perfect example of quantitative data. Each number represents a countable feature—specifically, the count of births—which is a discrete form of quantitative data. Here's how it can be useful in statistical analysis:

• Facilitates computation of various statistics, such as means and proportions.


• Enables the generation of graphs and charts for visual data representation.


• Allows for the application of advanced analytical methods, like regression analysis, to identify trends and make predictions.

By dealing with quantitative data, researchers and analysts can perform a wide range of statistical examinations to draw meaningful conclusions and insights from the data.