# Mathematics

Question 1

Question 2

Question 3

Surfactants are chemical agents, such as detergents, that lower the surface tension of a liquid. Surfactants play an important role in the cleaning of contaminated soils. In an experiment to determine the effectiveness of a certain method for removing toluene from sand, the sand was washed with a surfactant, and then rinsed with de-ionized water. Of interest was the amount of toluene that came out in the rinse. In five such experiments, the amounts of toluene removed in the rinse cycle, expressed as a percentage of the total amount originally present, were 5, 4.8, 9, 10, and 7.3.

(a) Find a 95% confidence interval for the percentage of toluene removed in the rinse. (b) Conduct a hypothesis at the (Alpha)a = 5% level. Can you conclude that the mean amount of toluene removed in the rinse is less than 8%?

Question 4

Two formulations of a certain coating, designed to inhibit corrosion, are being tested. For each of eight pipes, half the pipe is coated with formulation A and the other half is coated with formulation B. Each pipe is exposed to a salt environment for 500 hours. Afterward, the corrosion loss (in μm) is measured for each formulation on each pipe.

 Pipe A B 1 197 204 2 161 182 3 144 140 4 162 178 5 185 183 6 154 163 7 136 156 8 130 143

Can you conclude that the mean amount of corrosion differs between the two formulations? Conduct a hypothesis test at the (Alpha)a = 10% significance level. (a) State the appropriate null and alternative hypotheses. (b) Compute the test statistic. (c) Compute the P-value (d) State the conclusion of the test in the context of this setting.

Question 5

A 95% confidence interval for μX – μY is (-0.3, 0.15). Based upon the data from which the confidence interval was constructed, someone wants to test H0: μX = μY versus Ha: μX 6= μY, at the (alpha) a = 5% significance level.

(a) Based upon the confidence interval, what is your conclusion of the hypothesis test? (Explain)

(b) Can we use the above confidence interval to conduct the hypothesis test at the (alpha) a = 10% level?

Why or why not?

Extra:

Suppose we have conducted a t-test for the differences of mean μX – μY , with (alpha) a = 0.10, and the Pvalue is 0.07. For each the following statements, say where the statement is true or false and explain why.

(a) There is a 7% probability that H0 is true. (b) If H0 is true, the probability of getting a test statistic at least as extreme as the value of ts that was actually obtained is 7%.

Question 6

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