91Ó°ÊÓ

Confidence intervals are a way to estimate a population parameter based on a sample statistic. They provide a range of values within which the true parameter is likely to fall, with a certain level of confidence. The level of confidence is typically expressed as a percentage, such as 90%, 95%, or 99%. For example, a 90% confidence interval suggests that if we were to take 100 different samples and construct a confidence interval from each of them, approximately 90 of the intervals would contain the true population parameter.

The width of the confidence interval is influenced by the sample size, standard deviation, and the confidence level itself. - A larger sample size tends to give a more precise (narrower) interval. - A higher confidence level results in a wider interval because we require more certainty. - A smaller variance or standard deviation in the sample also narrows the interval. In the original exercise, the assertion about the width of the confidence interval relates to the differences between 90% and 99% intervals, where a 90% confidence interval is narrower due to a smaller critical value compared to a 99% interval.

The standard error (SE) measures the variability or dispersion of a sample statistic from the actual population parameter. It is essentially the standard deviation of the sampling distribution, and it gets smaller as the sample size increases.

Mathematically, it is calculated as:\[ SE = \frac{\sigma}{\sqrt{n}} \]Where \( \sigma \) is the standard deviation of the population, and \( n \) is the sample size. - A larger sample size (\( n \)) reduces the standard error, resulting in a more accurate estimate of the population parameter. - Conversely, a smaller sample size increases the standard error, making the estimation less reliable.In the context of the original exercise, the standard error of 2.4% allows us to understand the precision of the 52% estimate of Twitter users. The statement related to 97.6% is misleading because the SE does not directly inform us about the proportion of the population surveyed.

Sample size is a critical component in statistics as it directly influences the reliability and precision of statistical estimates. A larger sample size tends to increase the accuracy of estimates by providing more data that can reflect the population more accurately. - When the sample size increases, the standard error generally decreases, which leads to a narrower confidence interval, leading to more precise estimates of the population parameter. - A small sample size might not represent the population well and can lead to larger errors in estimation. From the exercise, it was noted that tampering with the sample size would directly affect the standard error. Collecting less data (reducing sample size) results in a larger standard error, contrary to the claim made in the original question.

A Z-test for proportions is a statistical method used to determine if there is a significant difference between the observed sample proportion and a known proportion in the population. - It is used when the data follows a normal distribution, and the sample size is large enough.- The Z-test helps us decide whether to reject the null hypothesis by comparing a calculated Z-statistic against a critical Z-value.In the mathematical formula for Z:\[ z = \frac{\hat{p} - p_0}{SE} \]- \( \hat{p} \) is the sample proportion,- \( p_0 \) is the hypothesized population proportion,- And \( SE \) is the standard error.In the original exercise, a Z-test was used to determine if more than half of U.S. adult Twitter users get news from Twitter with a significance level of \( \alpha = 0.01 \). The calculated Z-statistic was less than the critical Z-value, leading to the conclusion of failing to reject the null hypothesis.