Time/CNN voter polls monitored public opinion for the presidential candidates during the 2000 presidential election campaign. One Time/CNN poll conducted by Yankelovich Partners, Inc., used a sample of n = 589 likely voters (Time, June 26, 2000). Assume the population proportion for a presidential candidate is p = 0.50. The questions below concern the sample proportion of likely voters favoring the presidential candidate.

What is the probability the Time/CNN poll will provide a sample proportion within plus-or-minus 0.04 of the population proportion? ____________

What is the probability the Time/CNN poll will provide a sample proportion within plus-or-minus 0.03 of the population proportion? ____________

What is the probability the Time/CNN poll will provide a sample proportion within plus-or-minus 0.02 of the population proportion? ____________

Suppose the sample size is increased from n = 589 to n = 1,178. With this new, larger sample size, what is the probability the Time/CNN poll will provide a sample proportion within plus-or-minus 0.02 of the population proportion? ____________

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To find the probabilities in this scenario, we can use the formula for the standard error of the sample proportion, which is given by:

SE = sqrt(p*(1-p)/n)

where p is the population proportion and n is the sample size.

Let's calculate the probabilities step by step:

1) Probability with a sample size of 589 and a margin of error of 0.04:
Using the formula, we have:
SE = sqrt(0.50*(1-0.50)/589) = 0.0249
The margin of error is 0.04, which means the proportion can vary by 0.04 on both sides of the population proportion. So, the range for the sample proportion is 0.50 +/- 0.04.
To find the probability, we need to find the proportion of the standard normal distribution that falls within this range. This can be done by finding the area under the normal curve between the boundaries.
Using a standard normal distribution table or a calculator, the probability is approximately 0.879.

2) Probability with a sample size of 589 and a margin of error of 0.03:
Using the same standard error, we have:
SE = 0.0249
The margin of error is 0.03, so the range for the sample proportion is 0.50 +/- 0.03.
Calculating the probability using the normal distribution, we find it to be approximately 0.765.

3) Probability with a sample size of 589 and a margin of error of 0.02:
Again, using the same standard error, we have:
SE = 0.0249
The margin of error is 0.02, so the range for the sample proportion is 0.50 +/- 0.02.
Calculating the probability using the normal distribution, we find it to be approximately 0.606.

4) Probability with a sample size of 1,178 and a margin of error of 0.02:
With a larger sample size, the standard error will decrease. To calculate the new standard error, we use the formula with the new sample size:
SE = sqrt(0.50*(1-0.50)/1178) = 0.0176
Using the same margin of error of 0.02, the range for the sample proportion is 0.50 +/- 0.02.
Calculating the probability using the normal distribution, we find it to be approximately 0.778.

Note: The probabilities calculated here assume that the sampling is done randomly and without bias.