A preliminary survey shows that 35% of college students smoke. In a class of 42 students, what is the probability that more than half the students smoke? Use Appendix B.1 for the z-values. (Round the z-value to 2 decimal places and the final answer to 4 decimal places.)
0.0207
0.207
To find the probability that more than half of the students smoke, we first need to analyze the given information and compute the required probability using the normal distribution and z-values from Appendix B.1.
Let's break down the problem step by step:
Step 1: Define the variables
Let p be the proportion of college students who smoke. We are given that p = 0.35.
Let n be the number of students in the class. We are given that n = 42.
Step 2: Calculate the mean and standard deviation
The mean (µ) of the binomial distribution is given by µ = np. Therefore, µ = 42 * 0.35 = 14.7.
The standard deviation (σ) of the binomial distribution is calculated as σ = √(np(1 - p)). Therefore, σ = √(42 * 0.35 * (1 - 0.35)) ≈ 3.276.
Step 3: Transform the problem to a normal distribution
To use the z-values from Appendix B.1, we need to transform the binomial distribution with parameters µ and σ to a standard normal distribution with mean 0 and standard deviation 1. This is done using the standardization formula:
z = (x - µ) / σ
In this case, we want to find the probability that more than half the students smoke, which is equivalent to finding the probability that X > 21, where X follows a binomial distribution with parameters µ and σ.
So, we standardize X as follows:
z = (21 - 14.7) / 3.276 ≈ 1.926
Step 4: Lookup the z-value in Appendix B.1
Looking up the z-value 1.926 in Appendix B.1, we find that the corresponding cumulative probability is approximately 0.9732.
Step 5: Calculate the final probability
Since we are interested in the probability that more than half the students smoke, we need to subtract the cumulative probability from 1:
P(X > 21) = 1 - P(X ≤ 21) = 1 - 0.9732 ≈ 0.0268
Therefore, the probability that more than half the students in the class smoke is approximately 0.0268.
Note: The final answer is rounded to 4 decimal places as required.