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Chapter 12: Problem 58

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. $$ \begin{array}{lrrrrrr} \text { TOST } & 4200 & 3600 & 3750 & 3675 & 4050 & 2770 \\ \text { RBOT } & 370 & 340 & 375 & 310 & 350 & 200 \\ \text { TOST } & 4870 & 4500 & 3450 & 2700 & 3750 & 3300 \\ \text { RBOT } & 400 & 375 & 285 & 225 & 345 & 285 \end{array} $$ 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 normal probability plots and comment. e. Carry out a test of hypotheses to decide whether RBOT time and TOST time are linearly related.

Short Answer

Expert verified
The correlation coefficient between TOST and RBOT times is unaffected by switching variables or changing RBOT units to hours. It suggests a potential linear relationship subject to hypothesis testing and significance.

Step by step solution

01

Organize Data

First, we need to list the data in terms of two variable lists for easier calculation. For the TOST time \(x\) (in hours), our data is [4200, 3600, 3750, 3675, 4050, 2770, 4870, 4500, 3450, 2700, 3750, 3300]. For the RBOT time \(y\) (in minutes), our data is [370, 340, 375, 310, 350, 200, 400, 375, 285, 225, 345, 285].
02

Calculate Means

Calculate the mean of the TOST time \(\bar{x}\) and the RBOT time \(\bar{y}\). \(\bar{x} = \frac{\sum x_i}{n} = \frac{\sum [4200, 3600, 3750, 3675, 4050, 2770, 4870, 4500, 3450, 2700, 3750, 3300]}{12}\) and \(\bar{y} = \frac{\sum [370, 340, 375, 310, 350, 200, 400, 375, 285, 225, 345, 285]}{12}\).
03

Compute Variances and Covariances

Compute the variances \(s_x^2\), \(s_y^2\) and the covariance \(s_{xy}\). Variance for x is \(s_x^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}\) and variance for y is \(s_y^2 = \frac{\sum (y_i - \bar{y})^2}{n-1}\). Covariance is \(s_{xy} = \frac{\sum ((x_i - \bar{x})(y_i - \bar{y}))}{n-1}\).
04

Calculate Correlation Coefficient \(r\)

The sample correlation coefficient \(r\) is calculated using \(r = \frac{s_{xy}}{s_x s_y}\), where\(s_x = \sqrt{s_x^2}\) and \(s_y = \sqrt{s_y^2}\).
05

Interpret Correlation

A value of \(r\) close to 1 or -1 indicates a strong linear relationship. A value near 0 represents a weak linear relationship. Interpret \(r\) based on its calculated value.
06

Effect of Switching x and y

The sample correlation coefficient \(r\) is symmetric with respect to the variables, meaning the correlation between \(x\) and \(y\) is the same as that between \(y\) and \(x\). Thus, switching them does not affect \(r\).
07

Effect of Changing Units

Changing the units of RBOT time from minutes to hours would scale \(y\) by the factor of 1/60. Since the correlation coefficient \(r\) is unitless and scale-invariant, \(r\) remains the same regardless of changing units.
08

Construct Normal Probability Plots

Construct normal probability plots for both TOST and RBOT times to check for normality. This involves plotting the values against a theoretical normal distribution in order to determine if they form an approximately straight line.
09

Comment on Normal Probability Plots

If the data points closely follow a straight line, it indicates that the data is approximately normally distributed. Any deviance may indicate non-normality.
10

Formulate Hypotheses

Set up the null hypothesis \(H_0\): RBOT time and TOST time are not linearly related, and the alternative hypothesis \(H_a\): RBOT time and TOST time are linearly related.
11

Hypothesis Test

Perform a hypothesis test for correlation. Use the t-statistic formula \(t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\), where \(n\) is the number of pairs. Compare with the critical value from t-distribution with \(n-2\) degrees of freedom.
12

Conclusion of Hypothesis Test

If the calculated t-statistic is greater than the critical value, or if the p-value is small, reject \(H_0\). Otherwise, fail to reject \(H_0\). This determines if there is a significant linear relationship.

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Key Concepts

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

Understanding Linear Relationships
A linear relationship between two variables means that one variable changes at a constant rate when the other changes. In mathematical terms, if you were to plot this relationship on a graph, it would form a straight line. This is crucial when analyzing data, especially in studies like the one with TOST and RBOT times.
  • The strength of this relationship is measured using the correlation coefficient, denoted as \(r\).
  • Values of \(r\) close to 1 or -1 indicate a strong linear relationship, while a value near 0 suggests a weak relationship.
  • A positive \(r\) means as one variable increases, so does the other. Conversely, a negative \(r\) means as one goes up, the other goes down.
Understanding linear relationships helps in predicting the behavior of one variable based on another, which is vital in fields like engineering, economics, and natural sciences.
Exploring Normal Distribution
Normal distribution is a critical concept in statistics, often referred to as the Gaussian distribution. It's a continuous probability distribution characterized by its bell-shaped curve.
  • For a dataset, being normally distributed means that most of the data points are concentrated around the mean, symmetrically distributed.
  • In the context of TOST and RBOT times, checking the normality can help validate if the assumptions for certain statistical tests are met.

Using Normal Probability Plots

A normal probability plot is used to graphically assess if a dataset satisfies the normal distribution assumption. When the data points lie approximately along a straight line, it suggests normality. Deviations from this line may indicate that the data is not normally distributed. This can guide decisions on the appropriate statistical tests to use.
Diving into Hypothesis Testing
Hypothesis testing is a method used to decide if there is enough evidence to accept or reject a specific claim, usually about a population parameter.
  • The null hypothesis \(H_0\) often represents no effect or no relationship, while the alternative hypothesis \(H_a\) suggests the effect or relationship exists.
  • In the case of the RBOT and TOST times, testing helps determine if a linear relationship exists between them.

Conducting the Test

To test the hypothesis regarding the correlation, we use the t-statistic derived from the correlation coefficient. This involves:
  • Calculating the t-statistic using the formula \( t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \), where \(n\) is the sample size.
  • Comparing this with a critical value from the t-distribution table, typically at a 95% confidence level for a given degree of freedom \(n-2\).
The result guides whether to reject \(H_0\) in favor of \(H_a\), suggesting a significant linear relationship.
Understanding the t-statistic
The t-statistic is a crucial component in statistical hypothesis testing. It measures how far the sample statistic lies from the null hypothesis, relative to its standard deviation.
  • This helps determine how likely it is that the observed data occurred under the null hypothesis which, in correlation tests, assumes no relationship between the variables.
  • For correlation, the t-statistic is used to test whether the correlation coefficient \(r\) is significantly different from zero.

Interpreting the t-statistic

Once the t-statistic is calculated, it is compared to a critical value at a chosen significance level (like 0.05) from the t-distribution. If the t-statistic is greater than the critical value or the p-value is less than the significance level, it suggests strong evidence against the null hypothesis, indicating a significant linear relationship between the variables.

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Most popular questions from this chapter

The article "Chronological Trend in Blood Lead Levels" (N. Engl. J. Med., 1983: 1373-1377) gives the following data on \(y=\) average blood lead level of white children age 6 months to 5 years and \(x=\) amount of lead used in gasoline production (in 1000 tons) for ten 6-month periods: $$ \begin{array}{l|ccccc} x & 48 & 59 & 79 & 80 & 95 \\ \hline y & 9.3 & 11.0 & 12.8 & 14.1 & 13.6 \\ x & 95 & 97 & 102 & 102 & 107 \\ \hline y & 13.8 & 14.6 & 14.6 & 16.0 & 18.2 \end{array} $$ a. Construct separate normal probability plots for \(x\) and \(y\). Do you think it is reasonable to assume that the \((x, y)\) pairs are from a bivariate normal population? b. Does the data provide sufficient evidence to conclude that there is a linear relationship between blood lead level and the amount of lead used in gasoline production? Use \(\alpha=.01\).

The article "Some Field Experience in the Use of an Accelerated Method in Estimating 28-Day Strength of Concrete" (J. Amer. Concrete Institute, 1969: 895) considered regressing \(y=28\)-day standard-cured strength (psi) against \(x=\) accelerated strength (psi). Suppose the equation of the true regression line is \(y=1800+1.3 x\). a. What is the expected value of 28-day strength when accelerated strength \(=2500\) ? b. By how much can we expect 28-day strength to change when accelerated strength increases by 1 psi? c. Answer part (b) for an increase of \(100 \mathrm{psi}\). d. Answer part (b) for a decrease of \(100 \mathrm{psi}\).

The following summary statistics were obtained from study that used regression analysis to investigate the relationship between pavement deflection and surface temperature of the pavement at various locations on a state highway. Here \(x=\) temperature \(\left({ }^{\circ} \mathrm{F}\right)\) and \(y=\) deflection adjustment factor \((y \geq 0)\) : $$ \begin{aligned} &n=15 \quad \sum x_{i}=1425 \quad \sum y_{i}=10.68 \\ &\sum x_{i}^{2}=139,037.25 \quad \sum x_{i} y_{i}=987.645 \\ &\sum y_{i}^{2}=7.8518 \end{aligned} $$ (Many more than 15 observations were made in the study; the reference is "Flexible Pavement Evaluation and Rehabilitation," Transportation Eng. J., 1977: 75-85.) a. Compute \(\hat{\beta}_{1}, \hat{\beta}_{0}\), and the equation of the estimated regression line. Graph the estimated line. b. What is the estimate of expected change in the deflection adjustment factor when temperature is increased by \(1^{\circ} \mathrm{F}\) ? c. Suppose temperature were measured in \({ }^{\circ} \mathrm{C}\) rather than in \({ }^{\circ} \mathrm{F}\). What would be the estimated regression line? Answer part (b) for an increase of \(1^{\circ} \mathrm{C}\). [Hint: \({ }^{\circ} \mathrm{F}=(9 / 5)^{\circ} \mathrm{C}+32\); now substitute for the "old \(x\) " in terms of the "new \(x\)."] d. If a \(200^{\circ} \mathrm{F}\) surface temperature were within the realm of possibility, would you use the estimated line of part (a) to predict deflection factor for this temperature? Why or why not?

Suppose the expected cost of a production run is related to the size of the run by the equation \(y=4000+10 x\). Let \(Y\) denote an observation on the cost of a run. If the variables size and cost are related according to the simple linear regression model, could it be the case that \(P(Y>5500\) when \(x=\) \(100)=.05\) and \(P(Y>6500\) when \(x=200)=.10\) ? Explain.

The accompanying data on \(x=\) diesel oil consumption rate measured by the drain-weigh method and \(y=\) rate measured by the CI-trace method, both in \(\mathrm{g} / \mathrm{hr}\), was read from a graph in the article "A New Measurement Method of Diesel Engine Oil Consumption Rate" (J. Society Auto Engr., 1985: 28-33). $$ \begin{array}{l|ccccccccccccc} x & 4 & 5 & 8 & 11 & 12 & 16 & 17 & 20 & 22 & 28 & 30 & 31 & 39 \\ \hline y & 5 & 7 & 10 & 10 & 14 & 15 & 13 & 25 & 20 & 24 & 31 & 28 & 39 \end{array} $$ a. Assuming that \(x\) and \(y\) are related by the simple linear regression model, carry out a test to decide whether it is plausible that on average the change in the rate measured by the CI-trace method is identical to the change in the rate measured by the drain-weigh method. b. Calculate and interpret the value of the sample correlation coefficient.
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