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

List the Unique Values and Count Frequencies

First, identify the unique values in the data set representing the number of nonconforming transducers per batch. These unique values are \(0, 1, 2, 3, 4, 5, 6, 7, 8\). Next, count how many times each value appears in the data. This will give us the frequency for each unique value. For example, 0 appears 6 times, 1 appears 10 times, etc.

02

Calculate Relative Frequencies

Relative frequency can be found by dividing each frequency by the total number of observations (60 batches). For example, if the frequency of a value is 6, the relative frequency is \(\frac{6}{60} = 0.1\). Apply this calculation to each unique value.

03

Calculate Proportion of Batches with Nonconforming Units

To find the proportion of batches with at most five nonconforming transducers, sum the relative frequencies for values \(0\) through \(5\). To find fewer than five, sum the relative frequencies for values \(0\) through \(4\). For at least five, sum the relative frequencies from \(5\) through \(8\).

04

Construct a Histogram of Relative Frequencies

Create a histogram with the number of nonconforming units on the x-axis and relative frequency on the y-axis. Each bar's height should correspond to the relative frequency calculated in Step 2. Discuss the histogram's shape, such as if it is left-skewed or right-skewed based on the distribution of frequencies.

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Key Concepts

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

Relative Frequency

The concept of relative frequency is a crucial part of understanding data distributions. Instead of just noting how many times a particular outcome happened, relative frequency gives that count in the context of the entire dataset. This helps us understand how significant or common each outcome is relative to others.

To calculate the relative frequency, follow these steps:

• Identify the frequency of each unique outcome. For instance, if you observe a number appearing 6 times in a dataset of 60, its frequency is 6.
• Compute the total number of observations in your data set, which in this exercise is 60 batches.
• Divide the frequency of each outcome by the total number of observations to get the relative frequency. If the frequency is 6, then the relative frequency is \( \frac{6}{60} = 0.1 \).

Relative frequency thus provides a proportion that allows for easy comparison across different outcomes. It always sums up to 1 (or 100%) when you account for every possible outcome, demonstrating that you're covering the entire data set.

Histogram

A histogram is a type of bar graph used to visualize the distribution of data. It's particularly helpful in showing the underlying frequency distribution of a set of continuous data.

When constructing a histogram:

• Place the data values on the x-axis, which usually represents the range of outcomes. In this case, the x-axis might be labeled with the number of nonconforming transducers in each batch, ranging from 0 to 8.
• The y-axis represents the relative frequency of these outcomes.
• Each bar in the histogram should reflect the relative frequency calculated previously, with their height or length depicting this frequency proportionality.

The histogram helps in visually assessing the distribution of nonconformity. For instance, if more batches have fewer nonconforming units, the histogram will show longer bars at the lower end of the x-axis. Analyzing the shape gives insights into the data, like whether it's skewed to one side or concentrated around a particular value. Skewness will tell us if our data has a bias towards higher or lower frequencies of nonconformance.

Proportion Calculation

Proportion calculation in data analysis allows us to understand how much of the data falls under a certain category or range in relation to the whole dataset. Calculating proportions helps answer specific questions about data segments.

Here's how to calculate proportions in this context:

• To determine the proportion of batches with at most five nonconforming units, sum up all relative frequencies for values from 0 to 5.
• For batches with fewer than five nonconforming units, sum relative frequencies from 0 to 4. This gives insight into how many batches have relatively good conformity to specifications.
• Proportion for at least five nonconforming units considers relative frequencies from 5 to the highest value, here 8. This shows the spread of nonconformity towards more significant numbers.

Understanding these proportions can guide quality analysis and improvement initiatives, by showing areas where most issues are concentrated and helping identify whether the majority of data meets desired specifications. Such analysis is central to maintaining and improving product quality across batches.