Question

Suppose that a particular candidate for public office is in fact favored by \(48 \%\) of all registered voters in the district. A polling organization will take a random sample of 500 voters and will use \(p\), the sample proportion, to estimate \(\pi\). What is the approximate probability that \(p\) will be greater than .5, causing the polling organization to incorrectly predict the result of the upcoming election?

Short answer

The approximate probability that \(p\) will be greater than .5 comes down to finding the Z-Score and then using that Z-Score to look up the related probability in a standard normal distribution table. Depending on the Z-Score calculated, this probability may change, so the exact numerical output may not be provided here.

Step-by-step solution

Calculate Standard Error

The first step is to calculate the standard error of the sample proportion \(p\). Standard error (SE) is given by the formula \[SE = \sqrt{\frac{\pi(1-\pi)}{n}}\] where \(\pi = 0.48\) (population proportion or the true proportion of voters favoring the candidate) and \(n = 500\) (sample size). Substituting the values, calculate the SE.

Find the Z Score

Next, calculate the Z score for .5. The Z score helps us understand how far away a data point is from the mean in terms of standard deviations. We calculate the Z score by using the formula \[Z = \frac{p-\pi}{SE}\] where \(p = 0.5\) (the value for which we are finding the probability), \(\pi = 0.48\) and SE is the value calculated in the first step. Calculate the Z score by substituting the appropriate values.

Use Z-Score to Find Probability

Once we have our Z score from step 2, we can find the corresponding probability from a standard normal distribution table, which tells us the probability that a statistic is observed below, above, or between values. Since we need the probability that \(p\) is greater than .5 (i.e., our Z score is greater than the calculated value), we need to look up the complementary probability (1 minus the value found on the table) as Z-Tables typically give the area to the left of the given Z-Score.

Key concepts

Standard Error Calculation

Understanding the standard error (SE) is crucial for interpreting the variability of a sample statistic. In probability, the standard error of the sample proportion is a measure of how much we would expect the sample proportion to vary from sample to sample. The formula to calculate the standard error is:

\[SE = \sqrt{\frac{\pi(1-\pi)}{n}}\]

where \(\pi\) is the population proportion and \(n\) is the sample size. For a population proportion of 48% or 0.48, and a sample size of 500, the formula becomes:
\[SE = \sqrt{\frac{0.48(1-0.48)}{500}}\]
To calculate the standard error, simply plug these numbers into the formula and solve for SE. The resulting SE gives us an idea of how much random sampling error we might expect if we were to take numerous samples of the same size from the same population. In other words, it tells us how 'spread out' the sample proportions are likely to be around the true population proportion.

Z Score Computation

A Z score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. In the context of our exercise, the Z score computation involves how the sample proportion compares to what we expect from the population proportion.

To calculate the Z score, we use the formula:
\[Z = \frac{p-\pi}{SE}\]
This equation requires the sample proportion (\(p\)), the population proportion (\(\pi\)), and the standard error (SE), which we computed from the previous section. For a sample proportion of 50% or 0.5, and a population proportion of 48%, with the previously calculated SE, the Z score can be computed as:
\[Z = \frac{0.5-0.48}{SE}\]
This Z score tells us how many standard errors the sample proportion is above or below the population proportion. A positive Z score indicates that the sample proportion is above the population mean, while a negative Z score indicates it is below.

Normal Distribution Table

The normal distribution table, also known as the Z-table, is a reference for statisticians to determine the probability of a given Z score occurring within a normal distribution. This table shows the cumulative probability associated with a Z score up to a certain point to the left of the Z score in the distribution.

To use it, you first calculate your Z score using the formula from the previous step. Then, you locate the Z score on the Z-table to find the corresponding probability. However, keep in mind that standard Z-tables give the area under the curve to the left of the Z score. If you are interested in the probability of the sample proportion being greater than a certain value, you will need to look at the complementary probability, which is simply 1 minus the probability found in the Z-table.
For example, if our Z score is 2.00 and we need the probability that a value is more than 2 standard deviations away from the mean, we would look at the complementary probability:
1 - (cumulative probability for Z = 2.00 from Z-table). This would give us the area under the normal distribution curve to the right of Z = 2.00, which is what we are interested in for our exercise.