work Chapter 6
Question 50, 6.6.11-T ,
HW Score: $13.73 \%, 7$ of 51 points
Points: 0 of 1
Use a normal approximation to find the probability of the indicated number of voters. In this case, assume that 121 eligible voters aged $18-24$ are randomly selected. Suppose a previous study showed that among eligible voters aged $18-24,22 \%$ of them voted.
Probability that exactly 32 voted
The probability that exactly 32 of 121 eligible voters voted is (Round to four decimal places as needed.)
Answer
Final Answer: The probability that exactly 32 out of 121 eligible voters voted is approximately \(\boxed{0.0436}\).
Step 1 ：The problem is asking for the probability that exactly 32 out of 121 eligible voters voted, given that the probability of a voter voting is 22%. This is a binomial distribution problem, but since the number of trials (n=121) is large, we can use a normal approximation to the binomial distribution.
Step 2 ：The mean of the binomial distribution is given by np, and the standard deviation is given by sqrt(np(1-p)), where n is the number of trials and p is the probability of success. In this case, n = 121 and p = 0.22, so the mean is \(121 \times 0.22 = 26.62\) and the standard deviation is \(\sqrt{121 \times 0.22 \times (1 - 0.22)} = 4.556709338985756\).
Step 3 ：We can then use the normal distribution to find the probability that the number of successes is exactly 32. However, since we are looking for the probability of exactly 32 successes, we need to use a continuity correction and look for the probability that the number of successes is between 31.5 and 32.5.
Step 4 ：The z-scores for 31.5 and 32.5 are calculated as \(z1 = \frac{31.5 - 26.62}{4.556709338985756} = 1.0709482736255023\) and \(z2 = \frac{32.5 - 26.62}{4.556709338985756} = 1.2904048870733513\) respectively.
Step 5 ：Using the standard normal distribution table, we find the probabilities corresponding to these z-scores and subtract them to find the probability that the number of successes is between 31.5 and 32.5. This gives us a probability of 0.043641286124332934.
Step 6 ：Final Answer: The probability that exactly 32 out of 121 eligible voters voted is approximately \(\boxed{0.0436}\).
