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The minimum injection pressure (psi) for injection molding specimens of high amylose corn was determined for eight different specimens (higher pressure corresponds to greater processing difficulty), resulting in the following observations (from "Thermoplastic Starch Blends with a Polyethylene-Co-Vinyl Alcohol: Processability and Physical Properties,"Polymer Engrg. \& Sci., 1994: 17-23): \(\begin{array}{llllllll}15.0 & 13.0 & 18.0 & 14.5 & 12.0 & 11.0 & 8.9 & 8.0\end{array}\) a. Determine the values of the sample mean, sample median, and \(12.5 \%\) trimmed mean, and compare these values. b. By how much could the smallest sample observation, currently \(8.0\), be increased without affecting the value of the sample median? c. Suppose we want the values of the sample mean and median when the observations are expressed in kilograms per square inch (ksi) rather than psi. Is it necessary to reexpress each observation in ksi, or can the values calculated in part (a) be used directly? [Hint: \(\mathrm{l} \mathrm{kg}=2.2 \mathrm{lb} .]\)

Step by step solution

01

Calculate Sample Mean

To calculate the sample mean, sum up all the observations and divide by the number of observations. \[ \text{Sample Mean} = \frac{15.0 + 13.0 + 18.0 + 14.5 + 12.0 + 11.0 + 8.9 + 8.0}{8} = \frac{100.4}{8} = 12.55 \text{ psi} \]

02

Determine Sample Median

First, order the observations from smallest to largest: \( [8.0, 8.9, 11.0, 12.0, 13.0, 14.5, 15.0, 18.0] \). Since there are 8 observations, the sample median is the average of the 4th and 5th values: \[ \text{Sample Median} = \frac{12.0 + 13.0}{2} = 12.5 \text{ psi} \]

03

Calculate 12.5% Trimmed Mean

First, determine how many observations to trim: \(0.125 \times 8 = 1\). Thus, trim the smallest and largest observations. The trimmed dataset is \([8.9, 11.0, 12.0, 13.0, 14.5, 15.0] \).Calculate the mean using only the trimmed data: \[ \text{Trimmed Mean} = \frac{8.9 + 11.0 + 12.0 + 13.0 + 14.5 + 15.0}{6} = \frac{74.4}{6} = 12.4 \text{ psi} \]

04

Analyze Comparison of Measures

The sample mean is \(12.55\) psi, the median is \(12.5\) psi, and the 12.5% trimmed mean is \(12.4\) psi. These measures give us similar results, but the trimmed mean is slightly lower due to the exclusion of extreme values.

05

Evaluate Impact of Changing Smallest Observation

Since the median is the average of the 4th and 5th ordered observations \(12.0\) and \(13.0\), changing the smallest value\(8.0\) will not affect the median as long as the smallest observation remains less than \(12.0\), which ensures the position of 4th and 5th observations remain the same.

06

Convert to Kilograms per Square Inch (ksi)

To convert psi to ksi, recall that \(1\text{ psi} = \frac{1}{1000}\) \( ext{ksi} \). Thus, we can convert the previously calculated mean and median values directly. Mean: \( 12.55 \text{ psi} = 0.01255 \text{ ksi} \). Median: \(12.5 \text{ psi} = 0.0125 \text{ ksi}\). No need to recalculate individually; conversion scales directly.

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Key Concepts

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

Sample Mean

The sample mean is a fundamental concept in statistics. It represents the average of a set of observations, providing a measure of central tendency. To calculate the sample mean, you simply sum up all the individual observations and then divide this sum by the number of observations.

For instance, if you have the following pressure readings: 15.0 psi, 13.0 psi, 18.0 psi, 14.5 psi, 12.0 psi, 11.0 psi, 8.9 psi, and 8.0 psi, you would calculate the mean as follows:

• First, sum up all the observations: 15.0 + 13.0 + 18.0 + 14.5 + 12.0 + 11.0 + 8.9 + 8.0 = 100.4 psi.
• Then, divide by the number of observations: 100.4 / 8 = 12.55 psi.

The sample mean provides a quick snapshot of the dataset's central value and is extremely useful for making comparisons.

Sample Median

The sample median is another important measure of central tendency. Unlike the mean, which considers all observations, the median is the middle value in an ordered dataset. This makes it more robust in the presence of extreme values.

To find the median, arrange the observations in ascending order: 8.0, 8.9, 11.0, 12.0, 13.0, 14.5, 15.0, 18.0. With 8 observations, the median is the average of the 4th and 5th values, since there is no single middle number. Thus, the median is calculated as:

• 
  (12.0 + 13.0)/2 = 12.5 psi.

The median captures the center of the dataset in terms of order, rather than value, making it especially useful when dealing with skewed data.

Trimmed Mean

A trimmed mean is a variation of the mean that seeks to mitigate the impact of outliers. By removing a specific percentage of the smallest and largest values, it offers a more stable measure of central tendency for datasets with extreme values.

In our dataset, a 12.5% trimmed mean instructs us to remove the smallest and largest values, resulting in the trimmed set: 8.9, 11.0, 12.0, 13.0, 14.5, 15.0. Then, you calculate the mean of these remaining values:

• Sum is 8.9 + 11.0 + 12.0 + 13.0 + 14.5 + 15.0 = 74.4.
• Divide by the number of observations left, which is 6, resulting in a trimmed mean of 12.4 psi.

The trimmed mean is often more representative of the dataset's usual center, assuming you have a few extreme measurements.

Data Conversion

Data conversion is a frequently needed process in statistics, often involving the change of measurement units. This can be important when combining datasets from various sources or when standardizing the data for analysis.

The exercise involves converting from psi to ksi. Since 1 psi equals 1/1000 ksi, we can directly convert the calculated mean and median from psi to ksi without recalculating everything individually. So, for the mean:

• 12.55 psi becomes 0.01255 ksi.

And for the median:

• 12.5 psi becomes 0.0125 ksi.

This simplicity in unit conversion ensures that all statistical properties remain proportionally the same, allowing for an easy translation across different scales.