Final answer:
To determine the probability that fewer than 320 out of 750 adults over 65 suffer from certain mental health conditions, the normal approximation to the binomial distribution is used. After calculating the mean and standard deviation, the Z-score is found and the corresponding cumulative probability from the standard normal distribution table is used to answer the question.
Step-by-step explanation:
To calculate the probability that fewer than 320 out of 750 adults over 65 in the study suffer from one or more of the mental health conditions, we can use the normal approximation to the binomial distribution, as the sample size is large. We are given that the probability of an adult over 65 suffering from one of the conditions is 45%, or 0.45.
First, we calculate the mean (μ) and standard deviation (σ) of the binomial distribution:
Mean, μ = n * p = 750 * 0.45
Standard deviation, σ = sqrt(n * p * (1 - p))
These calculations give us:
μ = 337.5
σ ≈ 13.94
Next, we find the Z-score for 319.5 (using 319.5 instead of 320 for continuity correction):
Z = (X - μ) / σ = (319.5 - 337.5) / 13.94
Calculating the above gives us a Z-score, which we then look up in the standard normal distribution table to find the cumulative probability for that Z-score.
Finally, we interpret this cumulative probability as the probability that fewer than 320 adults suffer from one or more of the conditions.