Questions: According to an almanac, 70% of adult smokers started smoking before turning 18 years old. (a) Compute the mean and standard deviation of the random variable X, the number of smokers who started before 18 in 400 trials of the probability experiment. (b) Interpret the mean. (c) Would it be unusual to observe 360 smokers who started smoking before turning 18 years old in a random sample of 400 adult smokers? Why? (a) mux=280 sigmax=9.2 (Round to the nearest tenth as needed.) (b) What is the correct interpretation of the mean? A. It is expected that in 50% of random samples of 400 adult smokers, 280 will have started smoking before turning 18. B. It is expected that in a random sample of 400 adult smokers, 280 will have started smoking before turning 18. C. It is expected that in a random sample of 400 adult smokers, 280 will have started smoking after turning 18.

According to an almanac, 70% of adult smokers started smoking before turning 18 years old.
(a) Compute the mean and standard deviation of the random variable X, the number of smokers who started before 18 in 400 trials of the probability experiment.
(b) Interpret the mean.
(c) Would it be unusual to observe 360 smokers who started smoking before turning 18 years old in a random sample of 400 adult smokers? Why?
(a) mux=280
sigmax=9.2 (Round to the nearest tenth as needed.)
(b) What is the correct interpretation of the mean?
A. It is expected that in 50% of random samples of 400 adult smokers, 280 will have started smoking before turning 18.
B. It is expected that in a random sample of 400 adult smokers, 280 will have started smoking before turning 18.
C. It is expected that in a random sample of 400 adult smokers, 280 will have started smoking after turning 18.

Transcript text: According to an almanac, $70 \%$ of adult smokers started smoking before turning 18 years old. (a) Compute the mean and standard deviation of the random variable X , the number of smokers who started before 18 in 400 trials of the probability experiment. (b) Interpret the mean. (c) Would it be unusual to observe 360 smokers who started smoking before turning 18 years old in a random sample of 400 adult smokers? Why? (a) $\mu_{x}=280$ $\sigma_{\mathrm{x}}=9.2$ (Round to the nearest tenth as needed.) (b) What is the correct interpretation of the mean? A. It is expected that in $50 \%$ of random samples of 400 adult smokers, 280 will have started smoking before turning 18. B. It is expected that in a random sample of 400 adult smokers, 280 will have started smoking before turning 18. C. It is expected that in a random sample of 400 adult smokers, 280 will have started smoking after turning 18.

Solution

Solution Steps

Step 1: Calculating the mean (μ_x)

The mean (μ_x) is calculated as the product of the sample size (n) and the probability (p). \[\mu_x = n \times p = 400 \times 0.7 = 280\]

Step 2: Calculating the standard deviation (σ_x)

The standard deviation (σ_x) is calculated using the formula $\sqrt{n \times p \times (1-p)}$. \[\sigma_x = \sqrt{400 \times 0.7 \times (1 - 0.7)} = 9.2\]

Step 3: Interpreting the mean

The mean (μ_x) represents the expected number of individuals in the sample exhibiting the behavior, which is 280.

Step 4: Assessing unusualness

An observation is considered unusual if it falls outside the range defined by $\mu \pm 2\sigma$. For this problem, the range is [261.670, 298.330]. The specific observation of 360 is unusual as it falls outside the range.