at a 95% confidence level, each problem increases the sample size and the population proportion decreases. Why?
To understand why, at a 95% confidence level, increasing the sample size leads to a decrease in the population proportion, we need to consider the concept of confidence intervals and sampling variability.
In statistical analysis, a confidence interval provides a range of values within which we can estimate the true population parameter with a certain level of confidence. The confidence level, in this case 95%, indicates the percentage of confidence we have in the interval containing the true population parameter.
When we increase the sample size, we are obtaining more data points from the population. This increased sample size leads to a more precise estimation of the population proportion. As the sample size increases, the sampling variability, or the spread of the sample proportions, decreases.
With a smaller sampling variability, the confidence interval becomes narrower. The narrower interval implies that we have more confidence in the estimated range of values for the population proportion. Consequently, the margin of error decreases, resulting in a more accurate estimate.
Now, to address the second part of the question, why the population proportion decreases, it's important to note that the population proportion itself does not change. The decrease refers to the width or range of the confidence interval, not the actual population proportion.
In summary, when we increase the sample size, the precision of our estimation improves, leading to a narrower confidence interval. This narrower interval provides a more accurate estimate of the population proportion, giving us greater confidence in our results. However, it is crucial to understand that the actual population proportion remains unchanged; it is the estimation that becomes more precise.
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