A polling organization is attempting to determine how voters feel about a new tax in their county. Unbeknownst to the pollers, 45% of voters in the county support the new tax, which means the results of a large unbiased surveys will be normally distributed with a mean of 0.45. If they ask 100 people how they feel about the tax, the proportion of support will be approximately normal with a mean of 45% supporting and a standard deviation of 4.97%. If they ask 500 people how they feel about the tax, the proportion of support will be approximately normal with a mean of 45% and a standard deviation of 2.22% (a) If the polling organization asks 100 people if they support the new tax, what is the probability that they get a proportion supporting between 43% and 50%?
(b) If the polling organization asks 500 people if they support the new tax, what is the probability that they get a proportion supporting between 43% and 50%?
(c) If the polling organization asks 100 people if they support the new tax, what is the probability that a majority of people asked do support the new tax?
(d) If the polling organization asks 500 people if they support the new tax, what is the probability that a majority of people asked do support the new tax?
(e) What do the previous two questions demonstrate about our ability to discern the true feelings of voters in the county in the context of variability and random sampling?