According to a Harris Interactive poll conducted in 2009, 20% of American adults smoke. In a random sample of 500 American adults, what is the probability that no more than 100 smoke?
Find mean and standard deviation.
mean = np = (500)(.20) = ?
standard deviation = √npq = √(500)(.20)(.80) = ?
Note: q = 1 - p
Once you have calculated the mean and standard deviation above, use z-scores to find the probability:
z = (x - mean)/sd
Use 100 for x.
After you find the z-score, check a z-table to determine your probability.
I hope this will help get you started.
To calculate the probability that no more than 100 out of 500 American adults smoke, we need to use the binomial probability formula.
The formula for the probability of exactly x successes in n trials, where each trial has a probability p of success, is:
P(x) = (n C x) * p^x * (1 - p)^(n - x)
In this case, x represents the number of smokers in our sample, n is the sample size (500), and p is the proportion of American adults who smoke (0.20).
To find the probability that no more than 100 out of 500 smoke, we can calculate the sum of probabilities for x = 0 to 100:
P(0) + P(1) + P(2) + ... + P(100)
Using a statistical software or a calculator that has a binomial probability function is the easiest way to obtain the result. However, if you don't have access to such tools, you can approximate the probability using the normal approximation to the binomial distribution.
For a large sample size (n) and a proportion (p) that is not close to 0 or 1, the binomial distribution can be approximated by a normal distribution with mean μ = np and standard deviation σ = sqrt(np(1-p)).
In this case, μ = 500 * 0.20 = 100 and σ = sqrt(500 * 0.20 * (1 - 0.20)) = 9.49.
We want to find the probability that no more than 100 smokers occur. To do this, we need to calculate the z-score:
z = (x - μ) / σ
For x = 100:
z = (100 - 100) / 9.49 = 0
We then use a standard normal distribution table or calculator to find the probability associated with this z-score. In this case, the probability P(Z <= 0) is 0.5000.
So, the approximate probability that no more than 100 out of 500 American adults smoke is approximately 0.5000.
To find the probability of no more than 100 adults smoking out of a random sample of 500 American adults, we can use the binomial probability formula.
The formula for calculating the probability of x successes (in this case, no more than 100 smokers) in n trials (in this case, the sample size of 500 adults) is:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
- P(x) is the probability of having x successes
- nCx is the combination formula, also known as "n choose x"
- p is the probability of success in a single trial (in this case, the proportion of American adults who smoke)
- x is the number of successes
- n is the number of trials
In this case:
- n = 500 (sample size)
- p = 0.20 (proportion of American adults who smoke)
- x can be 0, 1, 2, 3,..., 100 (no more than 100 smokers)
To calculate the probability of no more than 100 smokers (x ≤ 100), we need to sum the probabilities for x = 0, x = 1, x = 2, ..., x = 100.
P(no more than 100) = P(x=0) + P(x=1) + P(x=2) + ... + P(x=100)
Let's calculate the probability step by step:
P(x=0) = (500C0) * (0.20^0) * (1-0.20)^(500-0)
P(x=1) = (500C1) * (0.20^1) * (1-0.20)^(500-1)
P(x=2) = (500C2) * (0.20^2) * (1-0.20)^(500-2)
.
.
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P(x=100) = (500C100) * (0.20^100) * (1-0.20)^(500-100)
Finally, we can sum all of these individual probabilities to get the total probability of no more than 100 smokers.
P(no more than 100) = P(x=0) + P(x=1) + P(x=2) + ... + P(x=100)