Final answer:
To estimate the probability that less than 25% of those polled prefer Nisar, John, and Rohini in a municipal election, we can use the central limit theorem. Based on the given information, we can calculate the z-score and use a standard normal distribution table to find the probability. The probability turns out to be 1.
Step-by-step explanation:
To estimate the probability that less than 25% of those polled prefer Nisar, John, and Rohini, we can use the central limit theorem. Based on the given information, 50% of the population supports Arjun, 20% supports Sarika, and the rest are split between Nisaar, John, and Rohini. Since the poll asks 400 random people, we can assume that it represents a large enough sample. To calculate the probability, we need to find the z-score and use a standard normal distribution table to find the probability.
First, let's calculate the proportion of people who support Nisar, John, and Rohini. Since the rest of the population is split between them, we can assume an equal distribution. So the proportion would be (1 - (50% + 20%)) / 3 = 10%.
Next, we need to find the standard deviation by multiplying the proportion by the square root of (1 - proportion) and dividing it by the square root of the sample size. In this case, the standard deviation would be sqrt((10% * 90%) / 400) = 0.033.
Now, let's calculate the z-score by subtracting the desired percentage (25%) from the mean proportion (10%) and dividing it by the standard deviation. The formula is (observed proportion - mean proportion) / standard deviation. So the z-score would be (0.25 - 0.10) / 0.033 = 4.55.
Using a standard normal distribution table, we can find that the probability of a z-score less than 4.55 is approximately 1. Therefore, the probability of less than 25% of those polled prefer Nisar, John, and Rohini is 1.