Final answer:
The expected number of smokers who started before turning 18 out of a sample of 200 is 140. Observing 160 such smokers would be unusual, as it is more than 3 standard deviations from the expected number.
Step-by-step explanation:
A student posed two questions related to statistical expectations. Firstly, we are asked to calculate the expected number of smokers who started before the age of 18 within a sample of 200 adults, given that 70% of adult smokers started at a younger age. Secondly, we analyze whether observing 160 adult smokers who started before 18 in the sample would be unusual.
Answer to Part (a)
To calculate the expected number of adult smokers who started smoking before turning 18 years old from a sample of 200, we multiply the size of the sample by the probability provided:
Expected number = 200 adults × 70% = 140 adults.
Answer to Part (b)
To ascertain if observing 160 smokers in the sample is unusual, we can look at the standard deviation for the given probability distribution and then assess how many standard deviations the number 160 is from the expected value. For a binomial distribution, which is applicable here, the standard deviation (σ) is calculated as:
σ = √(n × p × (1-p)), where n is the sample size and p is the probability. For this scenario,
σ = √(200 × 0.70 × 0.30) ≈ 6.48.
The number of standard deviations from the mean is calculated as:
Z-score = (Observation - Expected number) / σ = (160 - 140) / 6.48 ≈ 3.09.
A Z-score of 3.09 indicates that the observation is more than 3 standard deviations away from the expected value, which in normal circumstances is considered unusual, as it happens less than 1% of the time under the normal distribution assumption.