Equivalence of Hypothesis Test and Confidence Interval Two different simple random samples are drawn from two different populations. The first sample consists of 20 people with 10 having a common attribute. The second sample consists of 2000 people with 1404 of them having the same common attribute. Compare the results from a hypothesis test of p1=p2 (with a 0.05 significance level) and a 95% confidence interval estimate of p1-p2

Verified step by step guidance

1. Step 1: Calculate the sample proportions for each population. For the first sample, the proportion is p1 = 10/20. For the second sample, the proportion is p2 = 1404/2000. These proportions represent the observed probabilities of the common attribute in each sample.
2. Step 2: Formulate the null and alternative hypotheses for the hypothesis test. The null hypothesis (H0) is that the proportions are equal: p1 = p2. The alternative hypothesis (H1) is that the proportions are not equal: p1 ≠ p2. This is a two-tailed test.
3. Step 3: Compute the standard error for the difference in proportions. The formula for the standard error is: SE = sqrt((p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2)), where n1 and n2 are the sample sizes for the first and second populations, respectively.
4. Step 4: Perform the hypothesis test. Calculate the test statistic using the formula: z = (p1 - p2) / SE. Compare the calculated z-value to the critical z-value for a 0.05 significance level (±1.96 for a two-tailed test). If the calculated z-value falls outside the range of -1.96 to 1.96, reject the null hypothesis; otherwise, fail to reject it.
5. Step 5: Construct the 95% confidence interval for the difference in proportions (p1 - p2). The formula for the confidence interval is: (p1 - p2) ± z*SE, where z* is the critical z-value for a 95% confidence level (1.96). Compare the confidence interval to the null hypothesis value (0). If the interval does not contain 0, it suggests a significant difference between the proportions; otherwise, it does not.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0. The significance level, often set at 0.05, indicates the probability of making a Type I error, which is rejecting a true null hypothesis.

Confidence Intervals

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically 95%. It provides an estimate of the uncertainty around the sample statistic. For comparing two proportions, the confidence interval for the difference (p1 - p2) can indicate whether the two population proportions are statistically significantly different.

Proportions and Sample Size

Proportions represent the fraction of a sample that possesses a certain attribute, calculated as the number of successes divided by the total sample size. The sample size affects the precision of the estimates; larger samples generally yield more reliable results. In the context of hypothesis testing and confidence intervals, understanding the proportions from each sample is crucial for making valid comparisons between the two populations.