Question

An analytical chemist wanted to use electrolysis to determine the number of moles of cupric ions in a given volume of solution. The solution was partitioned into \(n=30\) portions of .2 milliliter each, and each of the portions was tested. The average number of moles of cupric ions for the \(n=30\) portions was found to be .17 mole; the standard deviation was .01 mole. a. Describe the distribution of the measurements for the \(n=30\) portions of the solution using Tchebysheff's Theorem. b. Describe the distribution of the measurements for the \(n=30\) portions of the solution using the Empirical Rule. (Do you expect the Empirical Rule to be suitable for describing these data?) c. Suppose the chemist had used only \(n=4\) portions of the solution for the experiment and obtained the readings \(.15, .19, .17,\) and \(.15 .\) Would the Empirical Rule be suitable for describing the \(n=4\) measurements? Why?

Short answer

Question: Based on the provided information, describe the applicability of Tchebysheff's Theorem and the Empirical Rule in discussing the distribution of measurements for \(n=30\) and \(n=4\) portions. Answer: Tchebysheff's Theorem can be applied to the \(n=30\) portions, as it does not make any assumptions about the distribution. According to Tchebysheff's Theorem, at least 75% of the data lies within \(.02\) moles of the mean, and at least 89% of the data lies within \(.03\) moles of the mean. The Empirical Rule can only be applied if the data follows a normal distribution, which is not confirmed. However, if assumed to follow a normal distribution, approximately 68% of the \(n=30\) portions data falls within \(.17\pm.01\) moles, about 95% falls within \(.17\pm.02\) moles, and about 99.7% within \(.17\pm.03\) moles. For the \(n=4\) measurements, the Empirical Rule would not be suitable, as the sample size is too small and may not represent a normal distribution.

Step-by-step solution

Identify the mean and standard deviation for \(n=30\) portions

The mean of cupric ions for \(n=30\) portions is .17 moles and the standard deviation is .01 moles.

Apply Tchebysheff's Theorem for \(k = 1, 2\), and \(3\)

For \(k = 1\), at least \(1 - \frac{1}{1^2} = 0\)% of the data lies within \(.01\) moles of the mean (.17 moles). For \(k = 2\), at least \(1 - \frac{1}{2^2} = 75\)% of the data lies within \(.02\) moles of the mean. For \(k = 3\), at least \(1 - \frac{1}{3^2} = 89\)% of the data lies within \(.03\) moles of the mean. b. Using the Empirical Rule for \(n=30\) portions

Apply the Empirical Rule

Since we don't have information to assume a normal distribution, we cannot say for certain whether the Empirical Rule will be suitable or not. However, assuming the data has a normal distribution, we can state the following: - Approximately 68% of the data falls within one standard deviation of the mean (\(.17 \pm .01\) moles). - Approximately 95% of the data falls within two standard deviations of the mean (\(.17 \pm .02\) moles). - Approximately 99.7% of the data falls within three standard deviations of the mean (\(.17 \pm .03\) moles). c. Empirical Rule suitability for \(n=4\) portions

Assess the possibility of using the Empirical Rule for \(n=4\) measurements

The Empirical Rule would not be suitable for the \(n=4\) measurements since the sample size is too small. The Empirical Rule is typically applied to large samples, preferably from a normal distribution. For small samples, the distribution is not very clear, and the conclusions drawn from the Empirical Rule may not be accurate.

Key concepts

Tchebysheff's Theorem

Tchebysheff's Theorem is an important tool in statistics that helps us understand how data is distributed, especially when the distribution type is unknown. This theorem gives us insights into the proportion of values that fall within a certain number of standard deviations from the mean.
It doesn't require the data to follow a particular distribution, like normal distribution, which makes it robust and versatile.
The formula for Tchebysheff's Theorem states that for any dataset, regardless of its distribution, at least \(1 - \frac{1}{k^2}\) of the data values lie within \(k\) standard deviations of the mean. To apply this concept, let's consider the problem at hand:

• For \(k = 1\), at least 0% of the data are within one standard deviation. This is trivial but is part of the theorem's consistency.
• For \(k = 2\), at least \(75\)% of the data points are within two standard deviations of the mean.
• For \(k = 3\), at least \(89\)% of the data points are within three standard deviations.

Using Tchebysheff's Theorem allows us to gauge data spread even when we don't assume any specific statistical distribution.

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, is specific to data that follows a normal distribution. This rule is especially useful for datasets that we suspect of having a bell-shaped curve, like a normal distribution, because it provides detailed information about how data points are spread around the mean.
The Empirical Rule indicates:

• About 68% of the data falls within one standard deviation of the mean.
• About 95% falls within two standard deviations.
• Nearly 99.7% falls within three standard deviations.

In the context of the exercise, assuming a normal distribution can give a clearer picture:- If the dataset of 30 portions were approximately normally distributed, these proportions provide a quick overview of data spread around the mean of \(0.17\) moles. However, applying the Empirical Rule requires caution when the distribution isn't verified as normal, particularly in smaller datasets like \(n=4\) portions. Such a small sample size makes the rule unreliable, as normal distribution assumptions do not hold true easily with less data.

Standard Deviation

Standard deviation is a fundamental concept in statistics that measures the amount of variation or dispersion present in a set of values. It tells you how much the individual data points deviate, on average, from the mean. This is crucial, as it helps in understanding how "spread out" a dataset is.
To calculate the standard deviation, one needs to

• Find the mean of the data points.
• Calculate the deviation of each data point from the mean and square it.
• Take the mean of these squared deviations.
• And finally, take the square root of this mean.

In the exercise provided, the standard deviation was \(0.01\) moles. This low value indicates that the measurements of cupric ions are closely clustered around the mean of \(0.17\) moles. The standard deviation provides insights into the reliability and precision of the data collected, which is especially significant in analytical experiments where consistent readings are essential.