91Ó°ÊÓ

Understanding the sample proportion is crucial in statistics when trying to make inferences about a population. In the context of our exercise, we begin by identifying how many out of the total sample of households own firearms. This is your number of successes. For our study, this number is 133 households.

To find the sample proportion, denoted as \( \hat{p} \), you use the formula:\[\hat{p} = \frac{x}{n}\]where \( x \) is the number of successes and \( n \) is the sample size.

In our example, there were 133 households owning firearms out of a sample of 539. Plugging in these numbers, we calculate:\[\hat{p} = \frac{133}{539} \approx 0.2468\]This means that approximately 24.68% of the sampled households own at least one firearm. The sample proportion helps provide a baseline for subsequent calculations like standard error and confidence intervals.

The standard error is an essential concept, as it measures how much the sample proportion \( \hat{p} \) is expected to vary from the true population proportion. Standard error gives us insight into the precision of our sample proportion.

To calculate the standard error of the sample proportion, use the following formula:\[SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]This formula incorporates the sample proportion and the sample size. For our study, \( \hat{p} \) is 0.2468 and \( n \) is 539.

Plug these values into the formula:\[SE = \sqrt{\frac{0.2468 \times (1 - 0.2468)}{539}} \approx 0.0192\]The result of 0.0192 indicates the amount of variation or "spread" in the sample proportion if you took many samples. The smaller the standard error, the more our sample proportion is a good estimate of the true population proportion.

The Z-value is a critical component in constructing confidence intervals, especially when the sample size is large. It relates to the desired confidence level, representing how many standard deviations a data point is from the mean.

For a one-sided confidence interval like in our problem, the Z-value reflects where we fall on the standard normal distribution curve to achieve a 95% confidence level.

Typically, a two-sided 95% confidence interval uses a Z-value of 1.96, but since we are only interested in the lower bound, we use a Z-value of 1.645. This lower Z-value gives a more one-sided approach, focusing solely on the lower limit of the confidence interval.

It's important to use the correct Z-value to ensure the interval calculated accurately reflects the specified level of confidence.

Once you have the sample proportion, standard error, and the Z-value, you're ready to calculate the lower confidence bound for the population proportion. The lower confidence bound provides us with a range, showing how confident we can be that the actual proportion is at least a specific value.

The formula for calculating the lower confidence bound is:\[\hat{p} - Z \times SE\]In our example, \( \hat{p} \) is 0.2468, the Z-value is 1.645, and the standard error is 0.0192.

Substitute the values:\[0.2468 - 1.645 \times 0.0192 \approx 0.215\]Thus, the lower confidence bound is approximately 0.215 or 21.5%. This tells us with 95% confidence that at least 21.5% of all households in the city own at least one firearm.

This interval provides assurance about the real proportion of households owning firearms in the larger population, based on our sample.