A certain manufacturer claims that they produce 50% brown candies. Sam plans to buy a large family size bag of these candies and Kerry plans to buy a small fun size bag. Which bag is more likely to have more than 70% brown candies?Group of answer choicesSam, because there are more candies, so his bag can have more brown candies.Sam, Because there is more variability in the proportion of browns among larger samples.Kerry, because there is more variability in the proportion of browns among smaller samples.Kerry, because most small bags will have more than 50% brown candies.Both have the same chance because they are both random samples.
Question
A certain manufacturer claims that they produce 50% brown candies. Sam plans to buy a large family size bag of these candies and Kerry plans to buy a small fun size bag. Which bag is more likely to have more than 70% brown candies?Group of answer choicesSam, because there are more candies, so his bag can have more brown candies.Sam, Because there is more variability in the proportion of browns among larger samples.Kerry, because there is more variability in the proportion of browns among smaller samples.Kerry, because most small bags will have more than 50% brown candies.Both have the same chance because they are both random samples.
Solution 1
The correct answer is: Kerry, because there is more variability in the proportion of browns among smaller samples.
Here's why:
The proportion of brown candies in a bag is a binomial proportion. The variability of a binomial proportion is given by the formula sqrt[p(1-p)/n], where p is the proportion of successes (in this case, brown candies) and n is the sample size (in this case, the number of candies in the bag).
As the sample size increases, the variability of the proportion decreases. This means that larger samples are more likely to be close to the true proportion (50% in this case). Smaller samples, on the other hand, are more likely to deviate from the true proportion.
Therefore, it is more likely for a small bag (Kerry's bag) to have more than 70% brown candies than a large bag (Sam's bag).
Solution 2
The correct answer is: Kerry, because there is more variability in the proportion of browns among smaller samples.
Here's why:
The proportion of brown candies in a bag is a binomial proportion. The variability of a binomial proportion is given by the formula sqrt[p(1-p)/n], where p is the proportion of success (in this case, the proportion of brown candies, which is 0.5) and n is the sample size (the number of candies in the bag).
As the sample size increases, the denominator of this fraction gets larger, which makes the overall fraction (and thus the variability) smaller. Therefore, smaller samples have more variability than larger samples.
So, if we're looking for a bag that is more likely to have an unusually high proportion of brown candies (more than 70%), we're more likely to find that in a bag with more variability, which would be the smaller bag that Kerry plans to buy.
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