It is known that 60% of people do not respond to mailed questionnaires. In a sample of 20 questionnaires mailed, what is the probability that more than 12 people will respond?
probability that 13 respond
p(13)= C(20,13) (.4^13) (.6^7)
= 77520 * 6.71 * 10^-6 * 2.8 * 10^-2
= 1.46 * 10^-2
and for 14
p(14) = C(20,14)(.4^14)(.6^6)
38760 * 2.68 * 10^-6 * 4.67 *10^-2
=.485 * 10^-2
etc for binomial distribution
p(15) = 15504 * .4^15 * .6^5
= .129 * 10^-2
p(16) = 4845 *.4^16 *.6^4
= .027 *10^-2
p(17) = 1140 * .4^17 * .6^3
=.0042 *10^-2
p(18) = 190 * .4^18 * .6*2
=.00047 *10^-2
forget p(19) and p(20), getting too small
add them up
2.1*10^-2 = .02
By the way, n = 20 is high enough so we could use the continuous version, the normal distribution, to approximate and see if we are way off base.
mean no response = 20 * .6 = 12
sigma = sqrt (20*.6*.4) = 2.2
To get more than 12 responding on the continuous distribution we want to be between 12 and 13 responding or between 7 and 8 not responding, call it 7.5
(7.5-12)/2.2 = -2 sigmas from mean
which is .023 in table of F(z) versus z on normal function table.
Well, statistically speaking, the probability of more than 12 people responding to the mailed questionnaires depends on a few factors. If we assume a 60% non-response rate, we're left with a 40% response rate. Using binomial probability, we can calculate the chances of getting more than 12 responses out of 20 questionnaires.
Now, let's do the math: P(X > 12) = 1 - P(X ≤ 12)
Calculating it step by step wouldn't be very funny, so let's just say it's quite unlikely but not impossible. Maybe the questionnaires were filled with some hilarious jokes, and people couldn't resist responding!
But hey, if you're willing to accept the challenge of number-crunching, you can use a binomial distribution formula or a statistical calculator to get the exact probability. Good luck with the numbers, and may the humor be with you!
To find the probability that more than 12 people will respond to the mailed questionnaires, we can use the binomial distribution formula.
The binomial distribution formula is: P(X = k) = C(n, k) * p^k * (1 - p)^(n - k), where:
- P(X = k) is the probability of getting exactly k successes,
- n is the number of trials (20 questionnaires mailed),
- k is the number of successes (more than 12 people responding),
- p is the probability of success (60% or 0.6), and
- C(n, k) is the number of combinations of n items taken k at a time.
In this case, we need to calculate the probability of getting more than 12 people (k) responding out of the 20 questionnaires (n).
We will use the formula to calculate the probabilities for k = 13, 14, 15, 16, 17, 18, 19, and 20 and sum them up.
P(X > 12) = P(X = 13) + P(X = 14) + P(X = 15) + P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20)
Now let's calculate it step-by-step.
Step 1: Calculate the probability for each value of k.
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
For k = 13:
P(X = 13) = C(20, 13) * (0.6)^13 * (1 - 0.6)^(20 - 13)
For k = 14:
P(X = 14) = C(20, 14) * (0.6)^14 * (1 - 0.6)^(20 - 14)
For k = 15:
P(X = 15) = C(20, 15) * (0.6)^15 * (1 - 0.6)^(20 - 15)
For k = 16:
P(X = 16) = C(20, 16) * (0.6)^16 * (1 - 0.6)^(20 - 16)
For k = 17:
P(X = 17) = C(20, 17) * (0.6)^17 * (1 - 0.6)^(20 - 17)
For k = 18:
P(X = 18) = C(20, 18) * (0.6)^18 * (1 - 0.6)^(20 - 18)
For k = 19:
P(X = 19) = C(20, 19) * (0.6)^19 * (1 - 0.6)^(20 - 19)
For k = 20:
P(X = 20) = C(20, 20) * (0.6)^20 * (1 - 0.6)^(20 - 20)
Step 2: Sum up the probabilities.
P(X > 12) = P(X = 13) + P(X = 14) + P(X = 15) + P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20)
Now let's calculate these values.
To determine the probability that more than 12 people will respond to the mailed questionnaires, we need to use the binomial probability formula.
The binomial probability formula is given by:
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 possible combinations of n items taken k at a time,
- p is the probability of success in a single trial,
- (1-p) is the probability of failure in a single trial,
- n is the number of trials.
In this case, n = 20 (the number of questionnaires mailed), k represents the number of people who respond, and p = 0.4 (since 60% do not respond, the probability of success is 1 - 0.6 = 0.4).
To find the probability that more than 12 people will respond, we need to calculate the sum of probabilities for k = 13, 14, ..., up to 20.
P(X > 12) = P(X = 13) + P(X = 14) + ... + P(X = 20)
Let's calculate this step by step.
First, let's find the probability for each individual value:
P(X = k) = C(20, k) * (0.4)^k * (0.6)^(20-k)
For k = 13:
P(X = 13) = C(20, 13) * (0.4)^13 * (0.6)^(20-13)
Using the combination formula, C(20, 13) = 20! / (13!(20-13)!), we can calculate the value.
Repeat this calculation for k = 14, 15, ..., 20, and sum up the probabilities:
P(X > 12) = P(X = 13) + P(X = 14) + ... + P(X = 20)
This will give us the probability that more than 12 people will respond to the mailed questionnaires.