Final answer:
To find the probability that between 35 and 40 adults consumed alcohol before turning 21 years old in a random sample of 50 adult Americans, we use the binomial distribution and the standard normal distribution. We calculate the probability of an individual adult consuming alcohol before turning 21, convert the range into standard scores, find the probabilities associated with those scores, and subtract to find the desired probability.
Step-by-step explanation:
To find the probability that between 35 and 40 adults consumed alcohol before turning 21 years old, we first need to calculate the probability of an individual adult consuming alcohol before turning 21 years old. According to the survey, 68% of adult Americans consumed alcohol before turning 21 years old. This means that the probability of an individual adult consuming alcohol before turning 21 is 0.68.
Now, we can use this probability to calculate the probability that between 35 and 40 adults consumed alcohol before turning 21 years old in a random sample of 50 adult Americans. Since the sample size is large (50), we can approximate the distribution of the number of adults consuming alcohol before turning 21 with a normal distribution.
Using the formula for the standard deviation of a binomial distribution, we can calculate the standard deviation of the number of adults consuming alcohol before turning 21 in the sample:
Standard deviation = sqrt(n * p * (1 - p)) = sqrt(50 * 0.68 * (1 - 0.68)) = sqrt(10.32) ≈ 3.21
Next, we can convert the range of 35 to 40 adults consuming alcohol before turning 21 into standard scores (z-scores) using the formula:
z = (x - mean) / standard deviation
For x = 35, z = (35 - 50) / 3.21 ≈ -4.67
For x = 40, z = (40 - 50) / 3.21 ≈ -3.11
Using a standard normal distribution table or a calculator, we can find the probabilities associated with these z-scores:
Probability for z = -4.67 is approximately 0.000009528
Probability for z = -3.11 is approximately 0.001454852
Finally, we can calculate the probability that between 35 and 40 adults consumed alcohol before turning 21 by subtracting the probability for z = -4.67 from the probability for z = -3.11:
Probability = 0.001454852 - 0.000009528 ≈ 0.001445325
Therefore, the probability that between 35 and 40 adult Americans consumed alcohol before turning 21 years old in a random sample of 50 is approximately 0.001445325.