Final answer:
Moussa's hypothesis testing of dentist gum recommendations must meet three conditions: randomness of the sample, large enough sample size for the normal approximation to the binomial distribution, and a population at least 10 times larger than the sample size.
Step-by-step explanation:
Moussa saw a commercial on television that claimed 9 out of 10 dentists recommend using a specific brand of chewing gum, but he suspected the true proportion was lower. To test his hypothesis, he collected data from a sample of 50 dentists and wanted to perform a hypothesis test with the null hypothesis H0: p=0.9 and the alternative hypothesis Ha: p<0.9, where p is the proportion of all dentists that recommend this brand of chewing gum.
For such a hypothesis test, the conditions Moussa's sample must meet typically include the following:
- The sample must be random.
- The sample size must be large enough to justify the use of the normal approximation to the binomial distribution. Specifically, np and n(1-p) should both be greater than 5, where n is the sample size and p is the hypothesized proportion.
- The population from which the sample is drawn should be at least 10 times larger than the sample size to minimize the impact of the sample on the population proportion.
In Moussa's case, if his sample of 50 dentists was selected randomly, the sample would meet the randomness condition. Assuming the true proportion is 0.9, his expected number of 'successes' (dentists who recommend the gum) would be 45 (np = 50*0.9), and the expected number of 'failures' would be 5 (n(1-p) = 50*0.1), both of which satisfy the condition for large enough sample size for the normal approximation. Lastly, if there are more than 500 dentists in the population (10 times the sample size), the third condition would also be met.