Question

Handwashing vs. hand rubbing. The British Medical Journal (August 17, 2002) published the results of a study to compare the effectiveness of handwashing with soap and hand scrubbing with alcohol. Health care workers who used hand scrubbing had a mean bacterial count of 35 per hand with a standard deviation of 59. Health care workers who used handwashing had a mean bacterial count of 69 per hand with a standard deviation of 106. In a random sample of 50 health care workers, all using the same method of cleaning their hands, the mean bacterial count per hand, x̅, is less than 30. Give your opinion on whether the sample of workers used hand rubbing with alcohol ot handwashing with soap.

Answer 1

By Cental limit theorem, the sampling distribution of mean (x̅) will follow normal distribution for large sample size like 50.

Assuming the sample of workers used hand rubbing with alcohol,

Standard error of mean = 59 / = 8.34386

Probability that the mean bacterial count per hand, x̅, is less than 30

= P(x̅ < 30) = P[z < (30 - 35)/8.34386]

= P[z < -0.6]

= 0.2743

Assuming the sample of workers used handwashing

Standard error of mean = 106 / = 14.99066

Probability that the mean bacterial count per hand, x̅, is less than 30

= P(x̅ < 30) = P[z < (30 - 69)/14.99066]

= P[z < -2.60]

= 0.0047

Since the probability that the mean bacterial count per hand, x̅, is less than 30 is greater when workers used hand rubbing with alcohol,  my opinion is that the sample of workers used hand rubbing with alcohol.