Question

Construct a relative frequency histogram for these 50 measurements using classes starting at 1.6 with a class width of .5. Then answer the questions. $$\begin{array}{llllllllll}3.1 & 4.9 & 2.8 & 3.6 & 2.5 & 4.5 & 3.5 & 3.7 & 4.1 & 4.9 \\ 2.9 & 2.1 & 3.5 & 4.0 & 3.7 & 2.7 & 4.0 & 4.4 & 3.7 & 4.2 \\ 3.8 & 6.2 & 2.5 & 2.9 & 2.8 & 5.1 & 1.8 & 5.6 & 2.2 & 3.4 \\ 2.5 & 3.6 & 5.1 & 4.8 & 1.6 & 3.6 & 6.1 & 4.7 & 3.9 & 3.9 \\ 4.3 & 5.7 & 3.7 & 4.6 & 4.0 & 5.6 & 4.9 & 4.2 & 3.1 & 3.9\end{array}$$ What is the probability that a measurement drawn at random from this set will be greater than or equal to \(3.6 ?\)

Short answer

Answer: The probability that a measurement drawn at random from this set will be greater than or equal to 3.6 is 0.58.

Step-by-step solution

Sort the measurements in ascending order

Arrange the given measurements in ascending order: $$ 1.6, 1.8, 2.1, 2.2, 2.5, 2.5, 2.5, 2.7, 2.8, 2.8, 2.9, 2.9, 3.1, 3.1, 3.4, 3.5, 3.5, 3.6, 3.6, 3.6, 3.7, 3.7, 3.7, 3.7, 3.8, 3.9, 3.9, 3.9, 3.9, 4.0, 4.0, 4.0, 4.1, 4.2, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.9, 4.9, 5.1, 5.1, 5.6, 5.6, 5.7, 6.1, 6.2 $$

Create classes for the histogram

The starting point is \(1.6\) with a class width of \(0.5\). We create classes until all data is covered: 1. \([1.6, 2.1)\) 2. \([2.1, 2.6)\) 3. \([2.6, 3.1)\) 4. \([3.1, 3.6)\) 5. \([3.6, 4.1)\) 6. \([4.1, 4.6)\) 7. \([4.6, 5.1)\) 8. \([5.1, 5.6)\) 9. \([5.6, 6.1)\) 10. \([6.1, 6.6)\)

Count the number of measurements in each class

Count the number of measurements in each class: 1. \([1.6, 2.1)\): 2 2. \([2.1, 2.6)\): 5 3. \([2.6, 3.1)\): 6 4. \([3.1, 3.6)\): 8 5. \([3.6, 4.1)\): 11 6. \([4.1, 4.6)\): 8 7. \([4.6, 5.1)\): 6 8. \([5.1, 5.6)\): 2 9. \([5.6, 6.1)\): 1 10. \([6.1, 6.6)\): 1

Calculate relative frequencies for each class

Divide the count in each class by \(50\) to get the relative frequency: 1. \([1.6, 2.1)\): \(\frac{2}{50} = 0.04\) 2. \([2.1, 2.6)\): \(\frac{5}{50} = 0.1\) 3. \([2.6, 3.1)\): \(\frac{6}{50} = 0.12\) 4. \([3.1, 3.6)\): \(\frac{8}{50} = 0.16\) 5. \([3.6, 4.1)\): \(\frac{11}{50} = 0.22\) 6. \([4.1, 4.6)\): \(\frac{8}{50} = 0.16\) 7. \([4.6, 5.1)\): \(\frac{6}{50} = 0.12\) 8. \([5.1, 5.6)\): \(\frac{2}{50} = 0.04\) 9. \([5.6, 6.1)\): \(\frac{1}{50} = 0.02\) 10. \([6.1, 6.6)\): \(\frac{1}{50} = 0.02\) Now, we have the relative frequencies for each class, and you can create the relative frequency histogram using this information.

Calculate probability

To find the probability that a measurement drawn at random from this set will be greater than or equal to \(3.6?\), sum the relative frequencies for all classes that have measurements greater than or equal to \(3.6\). $$ \begin{aligned} P(x \geq 3.6) &= 0.22 + 0.16 + 0.12 + 0.04 + 0.02 + 0.02 \\ &= 0.58 \end{aligned} $$ So, the probability that a measurement drawn at random from this set will be greater than or equal to \(3.6\) is \(0.58\).

Key concepts

Relative Frequency

Relative frequency is an essential concept in data analysis. It shows how often a particular class occurs relative to the total number of data points. To calculate it, you divide the frequency of a class by the total number of data points. For example, if there are 11 measurements in a class and 50 total measurements, the relative frequency is \( \frac{11}{50} = 0.22 \).
This value tells us the proportion of data within that specific class, which is crucial for understanding data behavior.

Class Interval

Class intervals define the range of values within which data is grouped in a histogram. They help organize data into manageable parts. Each class interval has a lower and upper bound, and the choice of class interval width, such as 0.5 in this example, affects data visualization.
Appropriate class intervals lead to a clearer understanding of data distribution, allowing us to see patterns or trends more easily. In our example, the class intervals start at 1.6 and increase by 0.5, forming ranges like [1.6, 2.1) and [2.1, 2.6).

Probability Calculation

Probability calculation involves finding the likelihood of a specific event. In the given exercise, we are interested in the probability of drawing a measurement greater than or equal to 3.6.
To find this probability, sum the relative frequencies of all relevant classes: \( 0.22 + 0.16 + 0.12 + 0.04 + 0.02 + 0.02 = 0.58 \).
This indicates a 58% chance that a randomly chosen measurement will be at least 3.6, showing how probability and relative frequency are connected.

Data Analysis

Data analysis transforms raw numbers into meaningful information. It involves sorting, organizing, and interpreting data to identify patterns and answers to specific questions.
By creating a relative frequency histogram, we visualize how data points are spread across different classes. This visualization aids in recognizing data trends and making informed decisions.
Detailed data analysis reveals deeper insights, such as finding probabilities, understanding data distributions, and making comparisons across groups.