Question

A manufacturer claims that its cleaneris 97% effective in killing bacteria. To test this claim,200 bacteria are placed in a dish and the cleaner is applied.

A.If the claim by the manufacturer is true, what is the probability thatat least 10 of the bacteria are still living after the application of the cleaner? Show all work. Give your answer as a sentence.

B.If you did see 10 or more bacteria still alive after the application of the cleaner, should this cause you to doubt the claim made by the manufacturer? Explain.

Answer 1

Let p denote the effectiveness of the cleaner in killing bacteria

Null hypothesis, H​​​​​​0 : p = 0.97

Alternative hypothesis, H​​​​​​a < 0.97

n = 200, p = 0.97

Thus, np = 194, n(1 - p) = 6

Let X denote the number of killed bacteria

Thus, X can be approximated to Normal distribution with Mean = 194 and standard deviation = = 2.41

A) Probability that atleast 10 of the bacteria are still living

= Probability that atmost 190 bacteria are killed

= P(X ≤ 190)

Using correction of continuity, the required probability = P(X < 190.5)

= P{Z < (190.5 - 194)/2.41}

= P(Z < -1.451)

= 0.0734

If the claim by the manufacturer is true, the probability that atleast 10 of the bacteria are still living after the application of the cleaner is 0.0734

B) Yes, it doubts the claim made by the manufacture since the required probability is greater than 0.05 which is the significance level considered.