Suppose that 30% of the nation's TV viewers are watching a particular presidential press conference. Typically, a Nielsen survey will sample about 1000 viewers. Suppose random sampling. What is the probability that at most 28% of the viewers in the sample are watching the press conference?
To find the probability that at most 28% of the viewers in the sample are watching the press conference, we can use the binomial distribution formula.
The binomial distribution is often used to model the probability of a certain number of successes (in this case, viewers watching the press conference) in a fixed number of trials (the sample size).
In this scenario, we have a success rate (probability of viewers watching the press conference) of 30%. Let's denote this as p = 0.3. The sample size is 1000 viewers, denoted as n = 1000.
The probability of at most 28% of viewers watching the press conference can be calculated using the cumulative distribution function (CDF) of the binomial distribution up to x = 280 (28% of 1000):
P(X ≤ 280) = Σ(k=0 to 280)((nCk) * p^k * (1-p)^(n-k))
To compute this probability, we use statistical software or online calculators with the binomial distribution. Alternatively, we can use an approximation formula like the normal approximation to the binomial:
Since n is large (n = 1000), we can approximate the binomial distribution by the normal distribution with mean (µ) and standard deviation (σ):
µ = n * p
σ = √(n * p * (1 - p))
Now, we can convert our problem into a probability statement using the normal distribution:
P(X ≤ 280) ≈ P(Z ≤ (280 - µ) / σ)
We need to calculate the z-score, which is the number of standard deviations above or below the mean:
Z = (280 - µ) / σ
By substituting the values of µ, σ, and x, we can calculate the Z-score. After finding the Z-score, we can consult a standard normal distribution table (or use software/calculators) to find the probability associated with that Z-score.
In this case, you have all the information needed. The Z-score can be calculated using the equation mentioned above and then converted into the probability using a standard normal distribution table or statistical software.