Question

Suppose that a survey about voting was conducted of 45 randomly selected individuals from Gainesville. The respondents were asked if they planned on voting in the next election. Thirty- five people said that they did plan on voting. Which of the following statements correctly describes how the confidence interval for the population proportion of people that plan on voting in the next election? There are not 15 successes and 15 failures. A confidence interval can not be done. There are not 15 successes and 15 failures. A confidence interval can be computed by adding 2 successes and 2 failures. There are not 15 successes and 15 failures. A confidence interval can be computed by adding 4 successes and 5 failures. There are at least 15 successes and 15 failures. A large sample confidence interval for the population proportion can be computed (phat +/- 2 sqrt{p' (1-p)/n) with no additional values added.

Answer 1

According to best practices in statistics, the plus-four method should be used since there are not enough failures to compute the confidence interval directly. By adding 2 successes and 2 failures, a confidence interval for the Gainesville voting proportion can be calculated accurately.

Step-by-step explanation:

The question concerns computing a confidence interval for a population proportion, which in this case is the proportion of individuals in Gainesville who plan to vote in the next election. The survey results show that 35 out of 45 people plan to vote, and the objective is to determine if a confidence interval can be calculated with these results. According to statistical practices for confidence intervals of proportions, one should observe at least 15 successes and 15 failures to use the normal approximation. Since we have 35 successes (people who plan to vote) and 10 failures (people who do not plan to vote), we do not initially have the recommended number of failures.

However, a method known as the plus-four method can be used when the number of successes or failures is less than 15. This method involves adding 2 successes and 2 failures to the data which adjusts the sample size and number of successes to better meet the normal approximation requirements. Hence, the correct statement from the given options is: "There are not 15 successes and 15 failures.

Answer 2

A confidence interval can be computed by adding 4 successes and 5 failures. The confidence interval for the population proportion of people planning on voting in the next election is (0.67, 0.92) with 95% confidence.

Step-by-step explanation:

A confidence interval can be computed by adding 4 successes and 5 failures. In this case, the survey had 35 people who said they planned on voting and 45 people were surveyed in total. Therefore, there were 35 successes and 10 failures. The formula to calculate the confidence interval for a population proportion is p ± 1.96 *
`\sqrt`(1 - p))/n), where p is the sample proportion, n is the sample size, and 1.96 is the critical value for a 95% confidence level. By substituting the values, the confidence interval for the population proportion of people planning on voting is (0.67, 0.92). This means that with 95% confidence, the true proportion of people planning on voting in the next election is estimated to be between 67% and 92%.