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
Repost.
To solve this problem using logarithms, we'll need to find the smallest value of n for which the probability that at least one person will rate the food as 'poor' is greater than 0.95.
Let's first calculate the probability that a single customer rates the food as 'poor'. We know that 13% of customers rated the food as 'poor', so the probability of a single customer rating it as 'poor' is 0.13.
The probability that a single customer does not rate the food as 'poor' is equal to 1 minus the probability that they do rate it as 'poor'. Therefore, the probability that a single customer does not rate the food as 'poor' is 1 - 0.13 = 0.87.
Now, let's consider the probability that none of the n customers rate the food as 'poor'. Since the samples are taken randomly, the probability that a single customer does not rate the food as 'poor' applies to all n customers. Therefore, the probability that none of the n customers rate the food as 'poor' is (0.87)^n.
We want to find the smallest value of n for which the probability that at least one person will rate the food as 'poor' is greater than 0.95. In other words, we want to find the smallest value of n for which 1 - (0.87)^n > 0.95.
To solve this equation, we can take the logarithm of both sides. Using the logarithmic property log(1 - x) = -log(x), we can rewrite the equation as -log((0.87)^n) > log(0.05).
Simplifying further, we get n * log(0.87) < log(0.05). Dividing both sides by log(0.87), we obtain n > log(0.05) / log(0.87).
Now, we can plug the values into a calculator to find the smallest value of n.
Using logarithms, the smallest value of n for which the probability that at least one person will rate the food as 'poor' is greater than 0.95 can be obtained by evaluating n > log(0.05) / log(0.87).