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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. Soc. Tribologists Lubricat. Engrs., Oct. 1997: 19-24) reported the accompanying observations on \(x=\) TOST time (hr) and \(y=\) RBOT time (min) for 12 oil specimens. $$ \begin{array}{lrrrrrr} \text { TOST } & 4200 & 3600 & 3750 & 3675 & 4050 & 2770 \\ \text { RBOT } & 370 & 340 & 375 & 310 & 350 & 200 \\ \text { TOST } & 4870 & 4500 & 3450 & 2700 & 3750 & 3300 \\ \text { RBOT } & 400 & 375 & 285 & 225 & 345 & 285 \end{array} $$ a. Calculate and interpret the value of the sample correlation coefficient (as did 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 a scatter plot and 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 Correlation Coefficient

First, we calculate the sample means \( \bar{x} \) and \( \bar{y} \). Then we find the covariance between TOST and RBOT times using the formula \( \text{Cov}(x, y) = \frac{1}{n-1} \sum (x_i - \bar{x})(y_i - \bar{y}) \). Finally, the correlation coefficient is \( r = \frac{\text{Cov}(x, y)}{s_x s_y} \), where \( s_x \) and \( s_y \) are the standard deviations of TOST and RBOT times respectively. Upon calculation, we find \( r \approx 0.894 \).

02

Effect of Switching Variables

The correlation coefficient \( r \) is symmetric with respect to the two variables, meaning that if we switch \( x \) and \( y \), the value of \( r \) remains unchanged. Hence, \( r \approx 0.894 \) even if \( x = \text{RBOT} \) time and \( y = \text{TOST} \) time.

03

Effect of Scaling RBOT Time

If RBOT time is expressed in hours rather than minutes, it results in a scaling by a factor of 1/60. However, since correlation is scale-invariant, the value of the correlation coefficient \( r \) does not change with changes in the units of measurement. Therefore, \( r \) remains \( 0.894 \).

04

Construct Scatter and Normal Probability Plots

To visualize the relationship, create a scatter plot of TOST time versus RBOT time. A strong positive linear trend would indicate a strong positive correlation. Normal probability plots of the residuals from the linear regression can help verify normality; points should fall approximately along a straight line. Analysis confirms a strong positive linear relationship.

05

Hypothesis Testing for Linear Relationship

To test the hypothesis of a linear relationship, we use \( H_0: \rho = 0 \) vs \( H_1: \rho eq 0 \). Calculate the test statistic \( t = r \sqrt{\frac{n-2}{1-r^2}} \) with \( n-2 \) degrees of freedom. With \( r \approx 0.894 \) and \( n = 12 \), the calculated \( t \) is significant at common significance levels (e.g., \( \alpha = 0.05 \)). Thus, we reject \( H_0 \), confirming a significant linear relationship.

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Key Concepts

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

Linear Relationship Testing

In statistics, testing for a linear relationship between two variables is crucial to understanding their interaction. It involves determining whether an increase or decrease in one variable results in a proportional increase or decrease in another. One of the most common methods for this is calculating the correlation coefficient, denoted by \( r \). This coefficient ranges from -1 to 1, where values closer to 1 suggest a strong positive relationship, values near -1 indicate a strong negative relationship, and values around 0 imply no linear relationship.
To test the significance of this relationship, we perform hypothesis testing. For instance, we set up a null hypothesis \( H_0: \rho = 0 \), which assumes no linear relationship between the variables. We then use a test statistic, calculated from the sample data, to determine whether to reject this hypothesis. If the test statistic falls into the critical region at a given significance level (such as \( \alpha = 0.05 \)), we reject the null hypothesis, indicating a significant linear relationship.

Scatter Plot Analysis

A scatter plot is a graphical representation of two variables plotted on a Cartesian plane. Each point on the plot reflects an observation's positioning between those two variables. This visual tool helps identify potential relationships, patterns, or trends in data.
Scatter plots are especially useful in initial data exploration. They can show whether a potential linear relationship exists:

• A scatter plot with points clinging closely to a straight line indicates a strong linear relationship.
• If the points form no discernible pattern, the variables may not be linearly related.

Moreover, outliers or anomalies can be easily spotted, which might suggest the need for further investigation. When dealing with exercises like examining TOST and RBOT times, a scatter plot can quickly confirm if there's apparent linear association to explore further.

Normal Probability Plot

The normal probability plot is a specialized form of a scatter plot used in statistics to assess whether a data set is approximately normally distributed. In such a plot, the quantiles of the data are plotted against the quantiles of a standard normal distribution. If the data is normally distributed, the points will roughly form a straight diagonal line.
Creating a normal probability plot involves the following steps:

• Rank the data points from smallest to largest.
• Plot these data points against their corresponding expected quantiles from a normal distribution.

This analysis is beneficial in identifying deviations from normality, such as skewness or kurtosis in the data, which may affect statistical tests that assume normality. For TOST and RBOT times, examining the residuals in this way helps verify the assumption of normality in linear regression analyses and ensures that any conclusions drawn are valid.

Hypothesis Testing in Statistics

Hypothesis testing is a foundational concept in statistics used to infer statistical significance. It involves making an assumption (or hypothesis) and then determining the likelihood that this hypothesis is true based on sample data. The process comprises several steps:

• Formulate the null hypothesis \( H_0 \), which typically suggests no effect or relationship, and the alternative hypothesis \( H_1 \), which reflects the anticipated effect or relationship.
• Collect data and compute a relevant test statistic.
• Determine the threshold value for deciding whether to accept or reject the null hypothesis, utilizing significance levels like \( \alpha = 0.05 \).
• Make a decision based on the comparison of the test statistic to this threshold.

For example, testing the linear relationship between TOST and RBOT times involves hypothesizing that there's no correlation (\( H_0: \rho = 0 \)). If the resultant test statistic is beyond this critical value, we reject \( H_0 \), concluding a significant linear relationship between the variables. This robust method ensures that findings are statistically sound and not due to chance.