Question

A mechanical engineer wishes to compare strength properties of steel beams with similar beams made with a particular alloy. Each beam will be set in horizontal position with a support on each end, a force of 2500 lbs will be applied at the centre, and the deflection will be measured. From past experience with such beams, the engineer is willing to assume that the true standard deviation of deflection for both types of beam is .05 inches. Because the alloy is more expensive, the engineer wishes to test at level α = .01 whether or not it has a smaller average deflection than the steel beam.

(a) What would be the power if there were 25 beams of each type used and the difference in true average deflection favours the alloy by .04 inches?

(b) What size of samples are appropriate if the desired power is .95 when the difference in true average deflection favours the alloy by .04 inches?

Answer 1

a)

Power and Sample Size

2-Sample t Test

Testing mean 1 = mean 2 (versus <)
Calculating power for mean 1 = mean 2 + difference
α = 0.01 Assumed standard deviation = 0.05

            Sample
Difference    Size     Power
     -0.04      25 0.663511

The sample size is for each group.

power = 0.6635

b)

Level of significance,

Type II error probability,

From the table:

the critical value at 1% level of significance,

the critical value at 5% probability,

Explanation | Hint for next step

The critical value of z can be obtained from the standard normal table using the desired level of significance and the tail of test.

From the z-table, &

The critical value tells the number of standards deviations away is the result from the mean.

Probability of type I error denoted by

Probability of type II error denoted by

Type I error is to falsely infer the existence of something that is not there, while a type II error is to falsely infer the absence of something that is.

From the given information,

The sample size is calculated as,

The required sample size with the provided information is 50