91Ó°ÊÓ

Understanding the calculation of the sample mean and standard deviation is crucial in probability and statistics, particularly when working with sampling distributions. The sample mean, denoted as \( \mu \), is a measure of the central tendency of the data. It represents the average outcome we might expect. In our exercise, where 65% of voters favor a policy, and we sample 225 voters, the mean is calculated by multiplying the sample size \( n \) by the probability of a voter favoring the policy \( p \):

\[ \mu = n \times p \]

For our case, this yields:

\[ \mu = 225 \times 0.65 = 146.25 \]

The standard deviation, symbolized as \( \sigma \), quantifies the amount of variation or dispersion in the sample. The formula to find the standard deviation for a binomial distribution is:

\[ \sigma = \sqrt{n \times p \times (1-p)} \]

Applying it to our scenario:

\[ \sigma = \sqrt{225 \times 0.65 \times (1 - 0.65)} \approx 6.864 \]

This calculation sets up the next steps in our problem, allowing us to assess the likelihood of different outcomes within our sample.

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. It's a dimensionless quantity that helps us determine the likelihood of a specific outcome.

To calculate the z-score, we use the formula:

\[ z = \frac{(x - \mu)}{\sigma} \]

In the context of our voter sample, we computed the z-scores for different thresholds of voters favoring the waiting period to purchase a handgun. For 150 voters, the calculation was:

\[ z = \frac{(150 - 146.25)}{6.864} \approx 0.546 \]

This z-score tells us that 150 is approximately 0.546 standard deviations above the mean. We also computed the z-score for more than 150 and fewer than 125, giving us z-scores that can be used with a z-table to find associated probabilities.

The z-table is a reference table that provides the probability of a z-score occurring by chance under a standard normal distribution. The table is a fundamental tool for finding the proportion of values to the left of a specified z-score.

To find the probability related to each z-score:

• For at least 150 favoring the policy, we looked up the z-score of 0.546 in the z-table, which gave us a value of 0.7079. The probability we need is the complement since we want 'at least', so we subtract it from 1 to get the upper tail.
• For more than 150, we do the same with a z-score of 0.619, resulting in a table value of 0.7324, and again subtract it from 1 for the upper tail probability.
• For fewer than 125, the z-score of -3.16 corresponds to 0.0008 in the z-table, which directly gives us the probability since it's less than the mean.

These probabilities help us understand the likelihood of observing at least or more than a certain number of voters in favor, or fewer than another, within our sample.