According to a recent survey, 38% of americans get 8 hours or less of sleep. if 25 people are selected, find the probability that 14 or more people will get 8 hours or less of sleep.
To find the probability that 14 or more people out of 25 will get 8 hours or less sleep, we need to calculate the probability for each possible outcome and then sum them up.
This problem involves a binomial distribution because we are interested in the number of successes (people getting 8 hours or less sleep) out of a fixed number of trials (25 people). The probability of success in one trial is given as 38% (0.38).
To calculate the probability, we can use the binomial formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of getting exactly k successes,
C(n, k) is the number of combinations of n items taken k at a time,
p is the probability of success in one trial,
k is the number of successes, and
n is the number of trials.
In this case, we want to find the probability that 14 or more people get 8 hours or less sleep. So, we need to calculate the probabilities for 14, 15, 16, ..., up to 25 people.
Let's calculate the probability for each case and then sum them up:
P(X ≥ 14) = P(X = 14) + P(X = 15) + P(X = 16) + ... + P(X = 25)
P(X = k) = C(25, k) * 0.38^k * (1-0.38)^(25-k)
Now, we can substitute the values and calculate the probabilities for each case:
P(X = 14) = C(25, 14) * 0.38^14 * (1-0.38)^(25-14)
P(X = 15) = C(25, 15) * 0.38^15 * (1-0.38)^(25-15)
P(X = 16) = C(25, 16) * 0.38^16 * (1-0.38)^(25-16)
...
P(X = 25) = C(25, 25) * 0.38^25 * (1-0.38)^(25-25)
Finally, we sum up all the individual probabilities to obtain the probability that 14 or more people will get 8 hours or less sleep.