Final answer:
The probability that at least 100 out of 350 people at a town meeting favor the construction of a municipal parking lot, with a presumed favorability rate of 34%, is approximately 0.9857, after calculating the normal approximation to the binomial distribution.
Step-by-step explanation:
To find the probability that at least 100 people at a town meeting favor the construction of a municipal parking lot, given that 34% of the residents in the town are estimated to favor it, we can use the normal approximation to the binomial distribution. Since the sample size is large (350 people), this method is applicable.
First, we need to determine the mean (μ) and standard deviation (σ) for the number of people who favor the parking lot. We use the following formulas where p is the proportion of success (0.34) and n is the sample size (350).
Mean (μ) = np = 350 * 0.34 = 119
- Standard Deviation (σ) = √(np(1-p))
- = √(350 * 0.34 * (1 - 0.34)) = √(78.54)
- ≈ 8.86
Next, we calculate the z-score for the value of 99.5 (using continuity correction for a value of 100) as follows:
Z = (X - μ) / σ = (99.5 - 119) / 8.86 ≈ -2.19 (rounded to two decimal places)
Using standard normal distribution tables or a calculator, we find the probability corresponding to the z-score of -2.19. This gives us the probability that fewer than 100 people favor the construction. To find the probability of at least 100 people favoring it, we subtract this probability from 1.
If P(Z < -2.19) = 0.0143, then the probability of at least 100 people favoring it is:
P(X ≥ 100) = 1 - P(Z < -2.19) = 1 - 0.0143 = 0.9857 (rounded to four decimal places)