Suppose that 7.6% of all people are allergic to the H1N1 vaccine. A sample of 517 people are selected
a. Calculate the probability that less than 8% of the sample will be allergic to vaccine.
Mean = np = 517 * .076 = 39.292
Standard deviation = √npq = √(517 * .076 * .924) = √(36.3058) = 6.025
Note: q = 1 - p
x = .08 * 517 = 41.36
Use z-scores:
z = (x - mean)/sd
Therefore:
z = (41.36 - 39.292)/(6.025) = 0.34
Check a z-table for the probability using the z-score calculated above. Remember that the problem is asking "less than" when looking at the table.
I hope this helps.
To calculate the probability that less than 8% of the sample will be allergic to the vaccine, we can use the binomial distribution formula.
The binomial distribution formula is given by:
P(X = k) = (n C k) * p^k * (1-p)^(n-k)
Where:
P(X = k) represents the probability of getting exactly k successes (in this case, k people being allergic to the vaccine),
n is the sample size,
k is the number of successes, and
p is the probability of success.
In this case, n = 517, p = 0.076 (7.6% expressed as a decimal), and we want to calculate the probability of getting less than 8% allergic to the vaccine, so k < 0.08 * n.
To calculate the probability that less than 8% of the sample will be allergic to the vaccine, we need to sum the probabilities for k = 0, 1, 2, ..., (0.08 * n - 1).
P(less than 8% allergic) = P(X < 0.08 * n) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 0.08 * n - 1)
Let's calculate this using a statistical software or calculator.