Betty’s Bite-Size Candies are packaged in bags. The number of candies per bag is normally distributed, with a mean of 50 candies and a standard deviation of 3. At a quality control checkpoint, a sample of bags is checked, and 12 bags contain fewer than 47 candies. How many bags were probably taken as samples?
We can use the normal distribution to find the probability of getting 12 bags with fewer than 47 candies out of a sample of bags. First, we need to calculate the z-score for 47 candies:
z = (47 - 50) / 3 = -1
Now we use a standard normal distribution table or calculator to find the probability of getting a z-score of -1 or less, which is 0.1587. This means that there is a 15.87% chance of getting a bag with fewer than 47 candies in a random sample.
To find the number of bags that were probably taken as samples, we can use the formula:
n = (z * σ / E)^2
where n is the sample size, z is the z-score for the desired level of confidence (we'll assume z = 1.96 for a 95% confidence level), σ is the standard deviation, and E is the maximum error we are willing to tolerate (we'll assume E = 0.05).
Substituting the values, we get:
n = (1.96 * 3 / 0.05)^2 = 2662.56
Rounding up to the nearest whole number, we get:
n = 2663
Therefore, we can estimate that around 2663 bags were probably taken as samples to get 12 bags with fewer than 47 candies.