According to a study conducted by a statistical organization, the proportion of americans who are satisfied with the way things are going in their lives is 0.82. Suppose that a random sample of 105 Americans is obtained.
(Please answer the below and round to four decimals places).
(a) What is the probability that at leaset 80 Americans in the sample are satisfied with their lives? _______
(b) What is the probability that 77 or fewer Americans in the sample are satisfied is?___________
dank memes
To calculate the probability in these scenarios, we can use the binomial probability formula. The formula for the probability of getting exactly k successes in n trials, with a probability of success p in each trial, is:
P(X = k) = (nCk) * p^k * (1 - p)^(n-k)
Where nCk represents the number of combinations of n items taken k at a time.
Let's calculate the probabilities in each case:
(a) What is the probability that at least 80 Americans in the sample are satisfied with their lives?
To find the probability of at least 80 Americans being satisfied, we need to calculate the probabilities for 80, 81, 82, ..., up to the maximum possible number in the sample, which is 105.
P(X >= 80) = P(X = 80) + P(X = 81) + P(X = 82) + ... + P(X = 105)
Using the binomial probability formula, we can calculate each probability individually and then sum them up.
(b) What is the probability that 77 or fewer Americans in the sample are satisfied?
To find the probability of 77 or fewer Americans being satisfied, we need to calculate the probabilities for 0, 1, 2, ..., up to 77.
P(X <= 77) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 77)
Again, using the binomial probability formula, we can calculate each probability individually and then sum them up.
Now, you can use these formulas and the given information (proportion of Americans satisfied with their lives) to calculate the probabilities by substituting the appropriate values into the formulas.