Final answer:
In statistics, X usually represents the number of successes in a sample and P' the sample proportion. To construct a 95 percent confidence interval for a population proportion, we use the normal distribution, and there are specific formulas to calculate this interval. Surveys via email may encounter non-response and coverage biases.
Step-by-step explanation:
When we refer to the random variable X, we're typically talking about the number of successes or particular outcomes in a sample. The random variable ’ (often denoted as P’ or p-hat) represents the sample proportion, which is the ratio of the number of successes to the total number of trials in the sample. For example, if a poll asks about the most significant issue in an election and 65% of the respondents say „the economy,” then X could represent the number of people who say the economy is the most significant issue, and P’ would be 0.65 if the sample size is the same as the total number of respondents.
To construct a 95 percent confidence interval for a population proportion, one would typically use the normal distribution assumption if the sample size is large enough (using the Central Limit Theorem) and if np and n(1-p) are both greater than 5. The confidence interval can be calculated using the formula for a proportion: CI = P’ ± Z*(sqrt((P’(1-P’))/n)), where Z* is the Z-value corresponding to the desired level of confidence and n is the sample size.
Difficulties in obtaining random results in a survey, such as one conducted by email, may include non-response bias, where certain segments of the population may not respond to emails as readily, and coverage bias, where the sample may not adequately represent the entire population due to some people not having access to email.