The article “Wear in Boundary Lubrication” (S. Hsu, R.Munro, and M. Shen, Journal of Engineering Tribology, 2002: 427-441) discuses several experiments involving various lubricants. In one experiment, 45 steel balls lubricated with purified paraffin were subjected to a 40 kg load at 600 rpm for 60 minutes. The average wear, measured by the reduction in diameter, was 673.2 μm, and the standard deviation was 14.9 μm. Assume that the specification for a lubricant is that the mean wear be less than 675 μm. Find the P– value for testingH0 : μ≥675 versus H1 :μ<675. What can we conclude based on the P– value?

To find the P-value, we first need to calculate the test statistic. This is a one-sample t-test because we are comparing one sample mean (673.2 μm) to a hypothesized population mean (675 μm).

The formula for the test statistic in a one-sample t-test is:

t = (X̄ - μ0) / (s/√n)

where: X̄ is the sample mean (673.2 μm), μ0 is the hypothesized population mean (675 μm), s is the sample standard deviation (14.9 μm), and n is the sample size (45).

Substituting the given values into the formula, we get:

t = (673.2 - 675) / (14.9/√45) = -0.68

Next, we find the P-value associated with this t-value. The P-value is the probability of observing a t-value as extreme as -0.68, assuming the null hypothesis is true.

We look up the P-value in a t-distribution table or use a statistical software. Since this is a one-tailed test (because of the alternative hypothesis H1 :μ<675), we look at the one-tailed P-value.

Assuming a significance level of 0.05, if the P-value is less than 0.05, we reject the null hypothesis. If the P-value is greater than 0.05, we fail to reject the null hypothesis.

Without the exact P-value, we can't make a definitive conclusion. However, if the P-value is less than 0.05, we can conclude that the mean wear is significantly less than 675 μm. If the P-value is greater than 0.05, we cannot conclude that the mean wear is significantly less than 675 μm.