which of the following statements is true? Justify your answer with an example.
i)When the margin of error is small, the confidence level is high.
ii)When the margin of error is small, the confidence level is low.
iii)A confidence interval is a type of point estimate.
iv) A population mean is an example of a point estimate.
A) I only
B) II only
C) III only
D) IV only
E) None of the above
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To determine which statement is true, let's go through each statement and justify our answer with an example:
i) "When the margin of error is small, the confidence level is high."
This statement is true. In statistical terms, the margin of error is the range within which we expect the true population parameter to fall. The confidence level is the degree of certainty we have that the population parameter falls within that range. So, when the margin of error is small, it means the range is narrow, and we can be more confident that the population parameter is close to our estimate.
For example, let's say we are conducting a survey to estimate the proportion of people in a city who prefer a specific brand of soda. With a large sample size and a small margin of error, such as +/- 2%, we can be highly confident that the estimated proportion falls within a narrow range around our result.
ii) "When the margin of error is small, the confidence level is low."
This statement is incorrect. As explained earlier, when the margin of error is small, it means we have a narrow range for our estimate, which results in a higher confidence level.
iii) "A confidence interval is a type of point estimate."
This statement is incorrect. A confidence interval is not a type of point estimate but rather an estimation of a range in which the true population parameter is expected to fall. It provides a level of uncertainty around the point estimate.
For example, if we have a confidence interval of 95% for the population mean, it means we are 95% confident that the true population mean falls within that interval.
iv) "A population mean is an example of a point estimate."
This statement is true. A point estimate refers to a single value used to estimate an unknown parameter of a population. The population mean is one such example of a point estimate, where we estimate the mean value of a population based on a sample.
For instance, if we collect a sample of student heights and calculate the average, that value would be an estimate of the population mean height.
Based on our analysis, the correct answer is:
D) IV only