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The Turbine Oil Oxidation Test (TOST) and the Rotating Bomb Oxidation Test (RBOT) are two different procedures for evaluating the oxidation stability of steam turbine oils. The article "Dependence of Oxidation Stability of Steam Turbine Oil on Base Oil Composition" ( \(J\). of the Society of Tribologists and Lubrication Engrs., Oct. 1997: 19-24) reported the accompanying observations on \(x=\) TOST time (hr) and \(y=\) RBOT time (min) for 12 oil specimens. $$ \begin{array}{l|rrrrrr} \text { TOST } & 4200 & 3600 & 3750 & 3675 & 4050 & 2770 \\ \hline \text { RBOT } & 370 & 340 & 375 & 310 & 350 & 200 \\ \text { TOST } & 4870 & 4500 & 3450 & 2700 & 3750 & 3300 \\ \hline \text { RBOT } & 400 & 375 & 285 & 225 & 345 & 285 \end{array} $$ a. Calculate and interpret the value of the sample correlation coefficient (as do the article's authors). b. How would the value of \(r\) be affected if we had let \(x=\) RBOT time and \(y=\) TOST time? c. How would the value of \(r\) be affected if RBOT time were expressed in hours? d. Construct normal probability plots and comment. e. Carry out a test of hypotheses to decide whether RBOT time and TOST time are linearly related.

Step by step solution

01

Calculate the Means

First, calculate the mean of the TOST times \(\bar{x}\) and the mean of the RBOT times \(\bar{y}\). For TOST: \(\bar{x} = \frac{4200 + 3600 + 3750 + 3675 + 4050 + 2770 + 4870 + 4500 + 3450 + 2700 + 3750 + 3300}{12}\). For RBOT: \(\bar{y} = \frac{370 + 340 + 375 + 310 + 350 + 200 + 400 + 375 + 285 + 225 + 345 + 285}{12}\).

02

Compute Sample Covariance

Calculate the covariance \(S_{xy}\) using the formula \(S_{xy} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})\). Replace \(x_i\) and \(y_i\) with each observation from the data set.

03

Compute Sample Standard Deviations

Compute the standard deviations for both TOST \(S_x\) and RBOT \(S_y\) using \(S_x = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n}(x_i - \bar{x})^2}\) and \(S_y = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n}(y_i - \bar{y})^2}\).

04

Calculate the Sample Correlation Coefficient

Use the formula \(r = \frac{S_{xy}}{S_x S_y}\) to get the sample correlation coefficient. This coefficient measures the strength and direction of the linear relationship between TOST and RBOT times.

05

Interpretation of Correlation

If \(r\) is close to 1, there is a strong positive linear relationship. If \(r\) is close to -1, there’s a strong negative linear relationship. If \(r\) is close to 0, there’s little to no linear relationship.

06

Value of r when Variables Switch

The correlation coefficient \(r\) is invariant to the switch of variables \(x\) and \(y\). Hence the value of \(r\) remains the same if \(x\) is RBOT time and \(y\) is TOST time.

07

Effect of Time Conversion

Converting RBOT time from minutes to hours involves dividing each RBOT value by 60. This transformation does not affect the value of \(r\) since correlation is a standardized measure that is not impacted by changes in scale.

08

Constructing Normal Probability Plots

Create normal probability plots for both TOST and RBOT data to assess normality. Plot the sorted data against the z-scores of a standard normal distribution. If the points lie roughly along a straight line, the normality assumption holds.

09

Hypothesis Test for Linearity

Conduct a hypothesis test for linearity using the correlation coefficient \(r\). The null hypothesis \(H_0: \rho = 0\) indicates no linear relationship. Compute the test statistic \(t = r \sqrt{\frac{n-2}{1-r^2}}\) and compare with the critical t-value from the t-distribution table with \(n-2\) degrees of freedom. A significant t-value (p-value less than 0.05) rejects \(H_0\), confirming a linear relationship.

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Key Concepts

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

Sample Covariance

Sample covariance is a statistical measure that reveals how two variables change together. When you calculate the sample covariance, you're essentially checking for a relationship between two datasets. In our exercise, we are looking at TOST time in hours and RBOT time in minutes for different oil samples.

• Positive covariance means that as one variable increases, the other tends to increase as well.
• Negative covariance suggests that as one variable increases, the other tends to decrease.

The formula for sample covariance is \[S_{xy} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})\]where \(n\) is the number of paired observations.
Calculating sample covariance requires computing the mean of both variables first. Once you have these means, you determine how each observation deviates from its mean and multiply these deviations for each pair of observations. The sum of these products, divided by the number of observations, gives you the sample covariance. This tells us how changes in TOST times may relate to changes in RBOT times.

Standard Deviation

Standard deviation is a key measure of variability or spread in a set of data. It tells us how much individual data points differ from the mean of the dataset. In simpler terms, it shows how spread out the numbers are. The exercise involves calculating the standard deviation for both TOST and RBOT times.
To calculate the standard deviation, use the formula:\[S_x = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n}(x_i - \bar{x})^2}\]This formula measures the average distance of each data point from the mean.

• A small standard deviation means that the values tend to be close to the mean.
• A large standard deviation indicates a wide range of values.

By calculating standard deviation, you get a clearer idea of how much variability there is in the TOST and RBOT measurements. Understanding this spread helps to interpret the correlation coefficient more accurately.

Hypothesis Testing

Hypothesis testing in statistics allows us to draw conclusions about population parameters based on a sample set. For our exercise, hypothesis testing is used to determine whether there's a linear relationship between TOST and RBOT times.

• The null hypothesis \(H_0\) posits that there is no linear relationship: \( \rho = 0 \).
• The alternative hypothesis \(H_a\) suggests that a linear relationship does exist: \( \rho eq 0 \).

We use the test statistic \(t\) calculated from the correlation coefficient:\[t = r \sqrt{\frac{n-2}{1-r^2}}\]Where \(r\) is the correlation coefficient, and \(n-2\) is the degrees of freedom. This statistic helps us decide whether to reject the null hypothesis. If the calculated \(t\) value leads to a p-value lower than 0.05, we reject the null hypothesis, suggesting a significant linear relationship. This process aids in making informed decisions based on data evidence rather than guesswork.

Normal Probability Plot

A normal probability plot is a graphical tool to assess if a dataset follows a normal distribution. In the exercise, creating normal probability plots for TOST and RBOT times helps check the normality assumption.

• To create this plot, you first sort your data in ascending order.
• Then, plot the data against a standard normal distribution's z-scores.

If the plot displays the points forming roughly a straight line, your data is approximately normally distributed.
Normal distribution is a key assumption in many statistical analyses, including correlation and regression. By ensuring normality through plots, we confirm that our statistical tests will perform as expected. Using normal probability plots is a simple yet effective technique to verify data distribution characteristics before diving into more complex analysis.