91Ó°ÊÓ

Temperature transducers of a certain type are shipped in batches of 50 . A sample of 60 batches was selected, and the number of transducers in each batch not conforming to design specifications was determined, resulting in the following data: \(\begin{array}{llllllllllllllllllll}2 & 1 & 2 & 4 & 0 & 1 & 3 & 2 & 0 & 5 & 3 & 3 & 1 & 3 & 2 & 4 & 7 & 0 & 2 & 3 \\ 0 & 4 & 2 & 1 & 3 & 1 & 1 & 3 & 4 & 1 & 2 & 3 & 2 & 2 & 8 & 4 & 5 & 1 & 3 & 1 \\ 5 & 0 & 2 & 3 & 2 & 1 & 0 & 6 & 4 & 2 & 1 & 6 & 0 & 3 & 3 & 3 & 6 & 1 & 2 & 3\end{array}\) a. Determine frequencies and relative frequencies for the observed values of \(x=\) number of nonconforming transducers in a batch. b. What proportion of batches in the sample have at most five nonconforming transducers? What proportion have fewer than five? What proportion have at least five nonconforming units? c. Draw a histogram of the data using relative frequency on the vertical scale, and comment on its features.

Step by step solution

01

Tally Frequencies for Each x Value

First, list all the possible values of \(x\), which is the number of nonconforming transducers in a batch. Then, count how many times each \(x\) value appears in the data set. The possible \(x\) values from the data are 0, 1, 2, 3, 4, 5, 6, 7, and 8. For example, \(x = 2\) appears 12 times in the data.

02

Calculate Relative Frequencies

For each \(x\) value, divide its frequency by the total number of batches (60) to find the relative frequency. For example, if \(x = 2\) has a frequency of 12, then its relative frequency is \( \frac{12}{60} = 0.20 \). Repeat this for each \(x\) value.

03

Calculate Proportions for Comparison

Identify the number of batches with at most five nonconforming transducers, those with fewer than five, and those with at least five. Sum the frequencies of \(x\) values 0 through 5 for at most 5, 0 through 4 for fewer than 5, and 5 through 8 for at least 5. Divide each by the total (60) to find each proportion.

04

Create a Histogram

On the x-axis, plot the number of nonconforming transducers, and on the y-axis, plot the relative frequencies. For each \(x\) value, draw a bar representing its relative frequency. This histogram provides a visual representation of the distribution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Frequency Distribution

In data analysis, understanding how data points are distributed is crucial. The frequency distribution is one way to achieve this. It involves counting how often each distinct data point, or category, appears in a dataset.
In our example of temperature transducers, each batch is counted for how many nonconforming transducers it contains. From the data provided, we create a list of all possible outcomes, which are the numbers 0 through 8, representing the count of nonconforming transducers in each batch.
To create a frequency distribution, we tally how many batches fall into each category. This gives us a quick snapshot of where most data points lie, showing us trends or patterns. Frequency distribution is foundational to recognize data patterns.

Histogram

A histogram is a type of bar chart that represents the frequency distribution of a dataset visually. It’s an excellent tool to quickly convey how the data is spread across different categories.
In our exercise, once we have the relative frequencies calculated, we use them to plot a histogram.
The x-axis will represent the number of nonconforming transducers in each batch, ranging from 0 to 8, while the y-axis will show the relative frequency of each of these numbers, indicating the proportion of each within the sample of batches.
Each bar’s height in the histogram corresponds to the relative frequency of that category. By examining the heights, we can see which outcomes occur more frequently. In the data about temperature transducers, the histogram lets us see at a glance which categories of nonconforming items occur most often, providing insight into potential quality control issues.

Proportions

Understanding proportions in datasets allows us to comprehend segments of the data concerning the whole.
In the temperature transducer context, once we have determined frequency, we can calculate proportions to answer specific questions. For example, we might want to know what fraction of batches had five or fewer nonconforming transducers.
Here we sum the number of batches in this category and divide by the total number of batches, obtaining a fraction or proportion. This helps in determining compliance or quality issues across batches.
Proportions offer a relative measure of frequency, assisting us in making comparisons and interpreting the data in line with specifications or standards.

Data Analysis

Data analysis involves examining, cleaning, and modeling data with the aim of discovering useful information, informing conclusions, and supporting decision-making.
It starts with collecting data and understanding its structure—a process exemplified by our frequency distribution and histogram.
The next step involves summarizing this information through calculations like means, medians, and our focus here, the proportions.
For our temperature transducers, we identified issues through the number of nonconformities—steps like tallying, calculating frequencies, and visualizing them in a histogram facilitate deeper understanding.
Through detailed data analysis, we find significant patterns or anomalies that inform the quality and standards of the batches, driving better production processes and decisions.