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.
Solve it by using log.
To solve this problem using logarithms, we can use the concept of complement probability. Let's start by finding the probability that no one rates the food as 'poor' in a sample of 12 customers.
The probability that a customer rates the food as 'poor' is 13%. Therefore, the probability that a customer does not rate the food as 'poor' is 100% - 13% = 87%.
In a sample of 12 customers, the probability that no one rates the food as 'poor' is:
(0.87)^12 ≈ 0.2825
Now, we want to find the smallest value of n for which the probability that at least 1 person rates the food as 'poor' is greater than 0.95. This means we need to find the smallest value of n for which:
1 - probability of no one rating the food as 'poor' > 0.95
1 - 0.2825 > 0.95
0.7175 > 0.95
To solve this inequality, we can take the logarithm of both sides. Let's assume "x" represents the smallest value of n we are looking for.
log(0.7175) > log(0.95)
To find the value of x, we need to rearrange the inequality:
x > log(0.95)/log(0.7175)
Using a calculator, we can find the value of x to be approximately 10.96.
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 11.