According to a recent survey, 38% of americans get 6 hours or less of sleep each night. If 25 people are selected, find the probability that 14 or more people will get 6 hours or less sleep each night..
There are a few steps missing from the above solution. It ended too quickly:
If P(X>=a) then P(X-.5) so P(X=14-.5 or 13.5)
z=13.5-9.5/2.427 = 1.65 = .9505. Since the number is more than take 1-.9505 = .0495
The answer now matches the back of the book.
There are a number of ways to solve this problem. Here is one.
The standard deviation of a binominal is sqrt(n*p*q) where p is the probability of an event and q is 1-p. So, SD = sqrt(25*.38*.62) = 2.427
The expected number that 6 hours or less is .38*25 = 9.5. Now then (14-9.5)/2.427 = 1.85 standard deviations away from the mean.
Take it from here.
I have the answer in the back of my book and its coming out to be -1.65. I don't understand how is that they are getting that answer?
To find the probability that 14 or more people out of 25 will get 6 hours or less of sleep each night, we can use the binomial probability formula. This formula calculates the probability of a certain number of successes (in this case, people getting 6 hours or less of sleep) in a fixed number of trials (in this case, selecting 25 people).
The binomial probability formula is as follows:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k): probability of getting exactly k successes
- C(n, k): the number of combinations of n items taken k at a time (also known as "n choose k")
- p: probability of success on a single trial
- k: number of successes
- n: number of trials
In this case, the probability of each person getting 6 hours or less sleep is given as 38%, which can be written as 0.38. Therefore, p = 0.38.
The number of trials (n) is 25.
To find the probability of 14 or more people getting 6 hours or less sleep, we need to calculate the probability of 14, 15, 16, ..., up to 25 people having 6 hours or less sleep and sum them all up.
P(14 or more) = P(14) + P(15) + P(16) + ... + P(25)
We can calculate each probability using the binomial probability formula and sum them up to get the final probability.