According to a survey, $29 \%$ of residents of a country 25 years old or older had earned at least a bachelor's degree. You are performing a study and would like at loast 10 people in the study to have eamed at least a bachelor's degree. (a) How many residents of the country 25 years old or older do you expect to randomly select? (b) How many residents of the country 25 years old or older do you have to randomly select to have a probability 0.996 that the sample contains at least 10 who have earned at least a bachelor's degree? (a) The number of randomly selected residents is 35 . (Round up to the nearest integer.) (b) The number of randomly selected residents, with a probability 0.996 containing at least 10 who have eamed at least a bachelor's degree, is $\square$ (Round up to the nearest integer.)
Solution
Step 1 ：According to a survey, 29% of residents of a country 25 years old or older had earned at least a bachelor's degree. You are performing a study and would like at least 10 people in the study to have earned at least a bachelor's degree.
Step 2 ：(a) To find out how many residents of the country 25 years old or older you expect to randomly select, you can simply divide 10 by 0.29. This gives you approximately 34.48. Since you can't select a fraction of a person, you round up to the nearest integer, which is 35.
Step 3 ：(b) To find out how many residents of the country 25 years old or older you have to randomly select to have a probability 0.996 that the sample contains at least 10 who have earned at least a bachelor's degree, you need to use the cumulative distribution function (CDF) of the binomial distribution. You need to find the smallest number n such that the CDF of the binomial distribution with parameters n and 0.29 at 9 is less than or equal to 0.004 (1 - 0.996). This gives you approximately 64.48. Again, since you can't select a fraction of a person, you round up to the nearest integer, which is 65.
Step 4 ：Final Answer: (a) The number of randomly selected residents is \( \boxed{35} \). (b) The number of randomly selected residents, with a probability 0.996 containing at least 10 who have earned at least a bachelor's degree, is \( \boxed{65} \).
