In a restaurant 13% ;customers rated the food as 'poor', 22% of the customer rated the food as 'satisfactory' and 65% rated it as 'good'.A random sample of 12 customers who went for a meal at restaurant was taken.On a separate occasion, a random sample of n customers who went for a meal at the restaurant was taken.Find the smallest value of n for which the probability that at least 1 person will rate the food as 'poor' is greater than 0.95.
To find the smallest value of n for which the probability that at least 1 person will rate the food as 'poor' is greater than 0.95, we can use the concept of complementary probability.
The probability that at least 1 person will rate the food as 'poor' is equal to 1 minus the probability that none of the customers rate the food as 'poor'.
Let's break down the problem:
We know that 13% of customers rated the food as 'poor'. Therefore, the probability that a single customer rates the food as 'poor' is 0.13.
The probability that a single customer does not rate the food as 'poor' is 1 - 0.13 = 0.87.
For a sample of 12 customers, the probability that none of them rates the food as 'poor' can be calculated using the probability of a single customer not rating the food as 'poor'. We raise this probability to the power of 12 (number of customers in the sample):
P(no one rates the food as 'poor') = (0.87)^12
To find the smallest value of n, we need to solve the following equation:
1 - P(no one rates the food as 'poor') > 0.95
1 - (0.87)^12 > 0.95
Now, we can solve for n:
(0.87)^12 < 0.05
n * log(0.87) < log(0.05)
n > log(0.05) / log(0.87)
Using a calculator, we can find that n ≈ 31.
Therefore, the smallest value of n for which the probability that at least 1 person will rate the food as 'poor' is greater than 0.95 is approximately 31.