Answer:
Not enough evidence to reject Null hypothesis
Step-by-step explanation:
Solution:-
- A comparative study for bacterial growth in manual and electronic faucets is made.
- It is observed that there is a higher growth in electronic faucets due to slower flow rates, i.e electronic faucets are not thoroughly flushed; hence, giving more resident time for the scaled bacteria to grow.
- It is known that 15% of cultures from older faucets were tested positive for the Legionella bacteria.
- A study at John Hopkins was conducted on a sample n = 20 electronic faucets with the probability of bacteria growing in a faucet is 0.15.
- We will conduct a hypothesis for at-least half proportion of electronic faucets have cultured bacteria.
- State the hypothesis for the proportion of electronic faucets culturing Legionella bacteria:
Null Hypothesis: P = 0.15
Alternate hypothesis: P > 0.15
- To determine the test statistics for the study conducted at John hopkins. We had a sample size of n = 20, and the probability for a bacteria to grow in a faucet is 0.15.
- Denote random variable, X: The number of electronic faucets culturing Legionella bacteria.
- Since, the probability for a bacteria to grow in a faucet is independent for each new faucet. We will assume the RV " X " to follow binomial distribution with probability of success 0.15:
X ~ Bin ( 20 , 0.15 )
- We are to determine that at-least half of the sample is subjected to the said bacteria. This is the probability of P ( X ≥ 10 ).
- The pmf for a binomially distributed random variable X is given below:
Where,
p ( success ) = 0.15
p ( fail ) = 1 - p ( success ) = 1 - 0.15 = 0.85
- Use the pmf to determine the required test statistics:
- The probability that 10 or more electronic faucets is found to have Legionella bacteria growing is 0.003
- The test proportion of 10 and more electronic faucets have culturing bacteria is p = 0.003.
- Assuming normality of the population, the Z-statistics would be:
- If we were to test the claim to 90% level of confidence:
significance level (α) = 1 - CI = 1 - 0.9 = 0.1
- The rejection region Z-critical is defined by a right-tail:
- Compare the test statistics with the rejection criteria defined by the Z-critical:
Z-test < Z-critical
-1.84 < 1.28
Conclusion:
There is not enough evidence that the probability of Legionella bacteria growing in electronic faucets is greater than 15%.