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

Short Answer: Using Tchebysheff's Theorem, we can conclude that at least 75% of the data falls within 2 standard deviations and 88.89% within 3 standard deviations from the mean. The suitability of the Empirical Rule for these data is uncertain, as we don't know if the data has a normal distribution. However, the percentages according to the Empirical Rule are 68% within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. In the case of 4 measurements, the Empirical Rule is not suitable as the sample size is too small and we cannot determine if the data follows a normal distribution.

Step-by-step solution

State Tchebysheff's Theorem

Tchebysheff's theorem states that at least \((1 - \frac{1}{k^2})\) of the data will fall within \(k\) standard deviations (\(k\sigma\)) from the mean \(\mu\).

Calculate the Proportions for Tchebysheff's Theorem

Using the given values of mean and standard deviation, let's calculate the proportions for different values of \(k\): For \(k=1\), at least \((1 - \frac{1}{1^2}) = 0\) (0% data within 1 standard deviation) For \(k=2\), at least \((1 - \frac{1}{2^2}) = 0.75\) (75% data within 2 standard deviations) For \(k=3\), at least \((1 - \frac{1}{3^2}) = 0.8889\) (88.89% data within 3 standard deviations) #a. Tchebysheff's Theorem Conclusion# Using Tchebysheff's Theorem, we can say that at least 75% of the data should be within 2 standard deviations and at least 88.89% of the data should be within 3 standard deviations from the mean. #b. Empirical Rule#

State the Empirical Rule

The Empirical Rule states that in a normal distribution, the following hold: Approximately 68% of the data fall within 1 standard deviation from the mean, Approximately 95% of the data fall within 2 standard deviations from the mean, Approximately 99.7% of the data fall within 3 standard deviations from the mean.

Assess the Suitability of the Empirical Rule

Since we don't have information about the shape of the distribution, we cannot conclude that the Empirical Rule is suitable for describing these data. However, we will mention the percentages according to the Empirical Rule for comparison: 68% of the data within 1 standard deviation, 95% of the data within 2 standard deviations, 99.7% of the data within 3 standard deviations. #b. Empirical Rule Conclusion# Uncertainty exists about the suitability of the Empirical Rule for these data since we don't know if the data has a normal distribution. However, the Empirical Rule suggests 68% within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. #c. Empirical Rule for n=4 Measurements#

Observe the Data

Given readings: 0.15, 0.19, 0.17, and 0.15.

Assess the Suitability of the Empirical Rule

The Empirical Rule is generally applicable to large datasets that are normally distributed. In this case, we only have 4 measurements, which is not enough to determine if the data follows a normal distribution. Additionally, the small sample size makes the Empirical Rule less suitable since it can't effectively describe such a small dataset. #c. Empirical Rule for n=4 Measurements Conclusion# The Empirical Rule is not suitable for describing the n=4 measurements because the sample size is too small and we cannot determine if the data follows a normal distribution.