Confidence Interval for a population proportion | Solved Problems

This paragraph introduces the concept of estimating a population proportion using confidence intervals. It explains that a point estimate is a single value estimate for the unknown population proportion, represented by the sample proportion p-hat, which is calculated as the number of successes (x) divided by the sample size (n). The sample proportion of failures is denoted as q-hat. Using a sample of 500 adults where 420 responded 'yes' to a question, the sample proportion is calculated as 0.84. This value serves as an estimate for the population proportion but does not guarantee that exactly 84% of the population will respond 'yes'. To make inferences about the population, a confidence interval is constructed, which is a range of values expected to capture the population parameter (p) at a given confidence level. The confidence level indicates the proportion of times the interval is expected to capture the population parameter with repeated sampling. The confidence interval for p is calculated using the formula: sample proportion ± (z-critical value * standard error of the sample proportion). The standard error is an estimate of the standard deviation of the distribution of sample proportions, representing how much we expect a sample proportion to differ from the true population proportion on average. The confidence interval is valid when the sample size is large enough to meet the minimum required number of successes and failures, which is typically 5 or more. The z-critical value is determined based on the desired confidence level, with common confidence intervals constructed at the 95% level unless stated otherwise. The video demonstrates constructing confidence intervals at unconventional levels, such as 85% and 94%, and explains the process of calculating the standard error, finding the z-critical value, and determining the margin of error to establish the confidence interval's limits. It concludes by interpreting the confidence interval and explaining that if multiple such intervals were constructed from random samples of the population, about 85% of them would be expected to contain the true population proportion in the long run.

This paragraph discusses the impact of the confidence level and sample size on the width of a confidence interval. It explains that as the confidence level increases, the z-critical value also increases, leading to a larger margin of error and consequently a wider confidence interval. Conversely, a lower confidence level results in a narrower interval. The sample size is another factor that affects the width of the confidence interval. Since the sample size is in the denominator of the standard error formula, increasing the sample size decreases the standard error, which in turn reduces the margin of error and narrows the confidence interval. This aligns with the logic that larger samples are expected to yield more precise estimates. The video provides an example to illustrate this point, stating that a sample size of 200 would produce a wider confidence interval than a sample size of 500, but a narrower one than a sample size of 100. The paragraph concludes by summarizing the key takeaways from the video and thanking the viewer for watching.