An engineer is investigating the diameter of crystals that are produced in a man

Question

An engineer is investigating the diameter of crystals that are produced in a manufacturing process. In the first experiment, the temperature of 90 °F is used and the diameters of 35 crystals are measured. The sample mean of ! = 22.73 µm and the sample deviation of Sx = 5.20 are observed. In the second experiment the temperature of 130 °F is used and the diameters of 35 crystals are measured, with the sample mean of ! = 12.66 µm and the sample deviation of Sy = 3.06 µm. The engineer decides that it is not appropriate to assume that the variances of the crystal diameters are the same under both sets of experimental conditions. a. Test the hypothesis that the average crystal diameter does not depend on the temperature. Obtain the corresponding p-value and comment on the results. b. Construct a 99% two-sided confidence interval for the difference between the average crystal diameters at the two temperatures. c. If a 99% two-sided confidence interval for the difference between the average crystal diameters at the two temperatures is required to have a length no greater than 4 µm, how much additional sampling would you recommend? (minitab preferable)

Explanation / Answer

Two-Sample T-Test and CI

Sample   N   Mean StDev SE Mean
1       35 22.73   5.20     0.88
2       35 12.66   3.06     0.52

Difference = (1) - (2)
Estimate for difference: 10.07
99% CI for difference: (7.35, 12.79)
T-Test of difference = 0 (vs ): T-Value = 9.87 P-Value = 0.000 DF = 55

a)

p-value = 0.000 < 0.01

hence we reject the null and conclude that the average crystal diameter depends on the temperature.

b)

99% CI for difference: (7.35, 12.79)

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