Answer:
The mean and standard deviation of the number of adults out of 200 who get news on an average day are mu = 68 and sigma = 5.36, respectively, assuming we can use the normal distribution to approximate the binomial distribution.
So the probability that at least 85 people say they get no news on an average day is approximately 0.0013 or 0.13%.
Explanation:
Since the survey found that 34% of US adults ages 18-24 get news on an average day, we can assume that the probability of a randomly selected adult in this age group getting news on an average day is p = 0.34. Therefore, the number of adults out of 200 who get news on an average day follows a binomial distribution with parameters n = 200 and p = 0.34.
To use the normal distribution to approximate this binomial distribution, we need to check if the conditions for doing so are met. These conditions are:
np >= 10
n(1-p) >= 10
Here, np = 200 x 0.34 = 68 and n(1-p) = 200 x 0.66 = 132. Both of these values are greater than 10, so the conditions are met.
Now, we can approximate the binomial distribution with a normal distribution with mean mu = np = 68 and standard deviation sigma = sqrt(np(1-p)) = sqrt(200 x 0.34 x 0.66) = 5.36.
Therefore, the mean and standard deviation of the number of adults out of 200 who get news on an average day are mu = 68 and sigma = 5.36, respectively, assuming we can use the normal distribution to approximate the binomial distribution.
Using the previous information, to find the probability that at least 85 people say they get no news on an average day.
Let X be the number of people out of 200 who say they get no news on an average day. We want to find the probability that X is greater than or equal to 85.
Since the probability of any one person saying they get no news on an average day is q = 1 - p = 0.66, we can use the binomial distribution with parameters n = 200 and p = 0.34 to model the number of people who say they get news on an average day.
The probability of at least 85 people saying they get no news on an average day can be calculated using the complement rule:
P(X >= 85) = 1 - P(X < 85)
To use the normal distribution to approximate the binomial distribution, we need to standardize the variable X.
Z = (X - mu) / sigma
where mu = np = 68 and sigma = sqrt(npq) = 5.36, as calculated in the previous question.
Using the continuity correction, we can adjust the upper bound to P(X < 84.5) since we want the probability of at least 85 people saying they get no news.
Z = (84.5 - 68) / 5.36 = 3.00
Using a standard normal distribution table or calculator, we can find that P(Z < 3.00) = 0.9987.
Therefore, the probability of at least 85 people saying they get no news on an average day is:
P(X >= 85) = 1 - P(X < 85)
≈ 1 - P(Z < 3.00)
= 1 - 0.9987
≈ 0.0013
So the probability that at least 85 people say they get no news on an average day is approximately 0.0013 or 0.13%.