According to a survey, 21% of residents in a country over age 25 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 a bachelor's degree.
a)How many residents of the country do you plan t randomly select? This is 10/0.21 = 48
But I don't get the next part....
b)How many residents of the country do you have to randomly select to have a probability 0.89 that the sample contains at least 10 who have earned a bachelor's degree?
To find out the number of residents you need to randomly select to have a probability of 0.89 that the sample contains at least 10 individuals with a bachelor's degree, you can use the concept of binomial probability.
The probability of selecting at least 10 individuals with a bachelor's degree can be calculated using the binomial distribution formula. The formula is:
P(X ≥ k) = 1 - [P(X = 0) + P(X = 1) + ... + P(X = k-1)]
where P(X = k) is the probability of getting exactly k successes (in this case, individuals with a bachelor's degree) in a sample.
Given that the probability of an individual having a bachelor's degree is 0.21, and you want to find the number of residents you need to select to have a probability of 0.89 of getting at least 10 individuals with a bachelor's degree, you need to solve for the value of k in the above formula.
To simplify the calculation, you can use a statistical calculator or software that can calculate cumulative binomial probabilities. By using these tools and inputting the values: probability of success (0.21), number of trials (k), and the desired cumulative probability (0.89), you can find the required number of residents to be randomly selected.
Alternatively, if you prefer to do it manually, you can start with a low number of residents, calculate the cumulative probability using the above formula, and gradually increase the number of residents until the cumulative probability reaches or exceeds the desired 0.89.
You may have to perform multiple calculations to find the exact number of residents required, but it will be an iterative process to approximate the number that satisfies the condition. Using techniques such as trial and error or using an iterative method in a spreadsheet tool will help you find the required number of residents efficiently.
To find the number of residents of the country you have to randomly select to have a probability of 0.89 that the sample contains at least 10 people who have earned a bachelor's degree, we can use the concept of the binomial distribution.
Let's denote the number of residents you have to select as 'x'. The probability of selecting at least 10 people with a bachelor's degree can be found using the binomial probability formula:
P(X ≥ 10) = 1 - P(X < 10)
In this case, the probability of selecting a person with a bachelor's degree is 0.21, and the probability of not selecting a person with a bachelor's degree is (1-0.21) = 0.79.
So, P(X < 10) = C(0, x) * (0.21^0) * (0.79^(x-0)) + C(1, x) * (0.21^1) * (0.79^(x-1)) + ... + C(9, x) * (0.21^9) * (0.79^(x-9))
To simplify this calculation, you can use a statistical software or a calculator that has built-in functions to compute binomial probabilities. Alternatively, you can use trial and error by incrementing 'x' until you find the desired probability of 0.89.
For example, starting from 'x = 50', you can calculate P(X < 10) and check if it is less than 0.11 (1-0.89). If it is less, increase 'x' by 1 and recalculate until you find the desired probability.
Note: This calculation assumes that the sampling is done with replacement, and the population size is large enough relative to the sample size that sampling without replacement has negligible effect.