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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 Engr. and Science, 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: \(\mathrm{lkg}=2.2 \mathrm{lb} .]\)

Step by step solution

01

Calculate the Sample Mean

The sample mean is the average of all the observations. To find it, sum all the data and divide by the number of observations:\[\bar{x} = \frac{15.0 + 13.0 + 18.0 + 14.5 + 12.0 + 11.0 + 8.9 + 8.0}{8}\]Carry out the sum:\[\bar{x} = \frac{100.4}{8} = 12.55\]

02

Calculate the Sample Median

To determine the sample median, first order the data from smallest to largest. The ordered data is:\[ 8.0, 8.9, 11.0, 12.0, 13.0, 14.5, 15.0, 18.0 \]Since there are 8 observations (an even number), the median is the average of the 4th and 5th values:\[\text{Median} = \frac{12.0 + 13.0}{2} = 12.5\]

03

Calculate the 12.5% Trimmed Mean

To calculate the 12.5% trimmed mean, you need to remove the smallest and largest data points. For 8 data points, cutting 12.5% would mean removing exactly one value from each end (rounded value from 8 * 0.125 = 1).Remove 8.0 and 18.0, then sum the remaining data and divide by the new number of observations:\[\text{Trimmed Mean} = \frac{8.9 + 11.0 + 12.0 + 13.0 + 14.5 + 15.0}{6}\]Carry out the sum:\[\text{Trimmed Mean} = \frac{74.4}{6} = 12.4\]

04

Analyze Effect on Sample Median with Increase in Smallest Observation

The median is affected by the central values of an ordered list. As long as the smallest observation, 8.0, is increased but less than or equal to the current fourth value (12.0), it doesn't affect the ordered order up to the 4th and 5th values, so the median remains unchanged at 12.5.

05

Determine if Conversion to ksi Affects Sample Mean or Median

To convert psi values to ksi, divide by 1000. Both the sample mean and median will be the same factor of 1000 smaller. Thus, the relative ordering of data and the average remain unchanged in terms of proportion. Therefore, you can use the psi mean or median value and simply divide by 1000. Final mean in ksi: 12.55 / 1000 = 0.01255 ksi. Final median in ksi: 12.5 / 1000 = 0.0125 ksi.

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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 statistical measure that gives us the average of our data set. It is particularly useful for understanding the central tendency of the data. Calculating the sample mean is straightforward: we sum up all individual observations and divide by the total number of observations. For example, with data points such as 15.0, 13.0, 18.0, and others, you add these values up and divide by their count (in this case, 8), yielding a mean value.

• Formula: \( \bar{x} = \frac{\Sigma x_i}{n} \)
• Where \( \Sigma x_i \) is the sum of all data points, and \( n \) is the number of data points.

The sample mean provides a single representative number, summarizing an entire dataset, thus enabling you to quickly grasp the overall level of the data points.

Sample Median

The sample median is another measure of central tendency, which highlights the middle value in a dataset. Unlike the mean, the median is not influenced by extreme outliers, making it particularly useful in datasets with skewed distributions. To find the median, you first need to arrange the data in ascending order. If the number of observations is odd, the median is the middle number. For a dataset with an even number of points, like our example, the median is the average of the two middle numbers.

• Ordered Data Example: 8.0, 8.9, 11.0, 12.0, 13.0, 14.5, 15.0, 18.0
• Median: \( \frac{12.0 + 13.0}{2} = 12.5 \)

Observing how the median is calculated helps us understand how it can remain stable regardless of fluctuations at the extremes of the data set.

Trimmed Mean

The trimmed mean is a modified version of the sample mean that excludes a specified smallest and largest portion of the dataset before calculating the average. This trimming process reduces the impact of outliers or extreme values on the mean. An example would be the 12.5% trimmed mean. Here, we remove the lowest and highest 12.5% of data points, and calculate the mean using the remaining data.

Benefits of Using Trimmed Mean

• Reduces influence of extreme values.
• Provides a more robust average in datasets with outliers.

By carefully calculating a trimmed mean, analysts can obtain a more representative central value that still reflects the majority of the data points, but is less skewed by extremes.

Data Conversion

Data conversion is an essential step when dealing with units of measurement, such as converting psi to ksi in our example. This conversion involves multiplying or dividing by a factor to switch between units. Here, to convert psi (pounds per square inch) to ksi (kilo pounds per square inch), we divide the psi value by 1000.

• Conversion Factor: 1 ksi = 1000 psi
• Example: 12.55 psi to ksi is \( \frac{12.55}{1000} = 0.01255 \) ksi.

When converting data, particularly for statistical calculations like the mean or median, it is crucial to ensure that the relative relationships between data points remain intact. The conversion factor ensures that even after scaling measurements, key characteristics like the ordering and proportions are not lost. Thus, allowing calculations to be effectively applied across units.