Question

Suppose a mechanical engineering company is interested in patenting a new device that is intended to serve as the next generation of breast cancer screening beyond mammography. A key measurement statistic for the new device is its false positive rate. Suppose the current false positive rate for mammography is 12% with a σ of 3%, and the company tests the device on a SRS of 245 patients that results in an average false positive rate of 6.5%.

A) Can we assume that the new device improves upon the false positive rate for mammography? (Use α=0.05)

B) State and interpret your results.

		A) Ho: µ=.12, Ha: µ>0.12, Z statistic=250.13 pvalue of <0.001 1) There is sufficient evidence to reject the null hypothesis that the average false positive result for the new screening method is equal to 0.12. The new screening tool has a much higher false positive rate based on the results from the sample.
		A) Ho: µ=.12, Ha: µ<0.12, Z statistic=175.87 pvalue of <0.0001 2) There is not sufficient evidence to reject the null hypothesis that the average false positive result for the new screening method is equal to 0.12. The new screening tool has a much lower false positive rate based on the results from the sample.
		A) Ho: µ=.12, Ha: µ<0.12, Z statistic=276.53 pvalue of <0.0001 3) There is sufficient evidence to reject the null hypothesis that the average false positive result for the new screening method is equal to 0.12. The new screening tool has a much lower false positive rate based on the results from the sample.

Answer 1

Answer)

As the population standard deviation is known, we can use standard normal z table to conduct the test

Ho : u = 0.12

Ha : u < 0.12 (as false positive will improve if detecting false positive will go down)

Test statistics z = (sample mean - claimed mean)/(s.d/√n)

N = 245

Sample mean = 0.065

Claimed = 0.12

S.d = 0.03

Z = −28.696205711

From z table, p(z<-28.696) = 0

That is P-value is < 0.0001

B)

Ho: µ=.12, Ha: µ<0.12, Z statistic=276.53 pvalue of <0.0001

3) There is sufficient evidence to reject the null hypothesis that the average false positive result for the new screening method is equal to 0.12. The new screening tool has a much lower false positive rate based on the results from the sample.

Is correct

As we reject the null hypothesis when p-value is less than alpha

As 0 is less than 0.05

We reject Ho