91Ó°ÊÓ

In statistics, the sample proportion is a key measure that represents the fraction of a particular outcome within a sample. For the example of electronic calculators, we're figuring out the fraction of defective calculators from a batch of 1200. Initially, we count the number of defectives, which is eight out of 1200. This means the sample proportion, denoted as \( \hat{p} \), is calculated as:\[ \hat{p} = \frac{8}{1200} = 0.0067 \]This value tells us that approximately 0.67% of the calculators in this sample are defective. Understanding how to compute the sample proportion is crucial as it serves as a foundation for estimating the population proportion through confidence intervals.

The standard error is an essential component when calculating confidence intervals. It measures the variability of a sample statistic. For the sample proportion, the standard error helps us understand how much the sample proportion \( \hat{p} \) is likely to vary from the true population proportion.To calculate the standard error for a sample proportion, use the formula:\[ \text{SE} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]In the calculator example, \( \hat{p} = 0.0067 \), and \( n = 1200 \). Plugging these values into the formula yields:\[ \text{SE} = \sqrt{\frac{0.0067 \times 0.9933}{1200}} = 0.0023 \]The standard error of 0.0023 indicates a small degree of variability, which is expected given the large sample size. This relatively low value of the standard error implies high precision in our sample estimate.

The population proportion represents the fraction of individuals or items within an entire population that possess a particular characteristic. In this case, it would be the proportion of defective calculators among all calculators produced, not just the ones sampled.A sample proportion is used as an estimate of the population proportion, but since we can't examine every calculator, there's always some uncertainty. This uncertainty is why we calculate confidence intervals—to estimate where the true population proportion might lie. While the sample gave us \( \hat{p} = 0.0067 \), the true population proportion is what we aim to estimate, acknowledging that it might be different.

The critical value is an important concept in the computation of confidence intervals. It helps determine how wide the interval will be by accounting for the level of confidence we desire in our estimate.In a normal distribution, the critical value corresponds to the value that cuts off a set percentage of the distribution's tails. For a 95% confidence interval, you often use a Z-value of \( 1.96 \). This reflects the idea that about 95% of sample proportions will fall within \( 1.96 \) standard errors of the population proportion on either side.Thus, we calculate the confidence interval as follows:\[ \hat{p} \pm Z \cdot \text{SE} = 0.0067 \pm 1.96 \times 0.0023 \]This results in an interval of approximately \([0.0022, 0.0112]\), suggesting that the true population proportion is likely to fall within this range with 95% confidence. Using critical values helps us make inferences about population parameters with a specified level of certainty.