Question

Confidence intervals Several factors are involved in the creation of a confidence interval. Among them are food fraud. Suppose an official said, "That's only the sample size, the level of confidence, and the margin 190 packages out of the billions of pieces of seafood of error. Which statements are true? sold in a year. With the small number tested, I don't a) For a given sample size, higher confidence means a know that one would want to change one's buying smaller margin of error. habits." (An official was quoted similarly in a dif- b) For a specified confidence level, larger samples ferent but similar context). Is this argument valid? provide smaller margins of error. Explain. c) For a fixed margin of error, larger samples provide greater confidence. d) For a given confidence level, halving the margin of error requires a sample twice as large.

Answer 1

Confidence intervals capture the range for the true population parameter; their width is influenced by factors such as sample size and confidence level. Higher confidence levels increase the margin of error, larger samples reduce it, and increasing sample size multiples rather than doubles reduces the margin of error.

Step-by-step explanation:

Understanding confidence intervals is important in statistics because they capture the range within which the true population parameter is likely to fall. Certain factors will affect the width of this interval, including sample size, confidence level, and margin of error. Let's explore the statements provided:

• a) For a given sample size, higher confidence does not mean a smaller margin of error. In fact, higher confidence levels typically lead to larger margins of error because to be more confident that the interval contains the true population parameter, the interval must be widened.
• b) For a specified confidence level, larger samples do indeed provide smaller margins of error. A larger sample size reduces the variability of the point estimate, which in turn reduces the margin of error.
• c) For a fixed margin of error, larger samples do not necessarily provide greater confidence; instead, they provide more precise point estimates. Increasing the sample size while keeping the margin of error constant will typically result in a narrower confidence interval but not necessarily a higher level of confidence.
• d) For a given confidence level, halving the margin of error requires increasing the sample size by a factor of four, not just doubling it. This is because the margin of error is inversely proportional to the square root of the sample size.

Regarding the official's argument about the sample size for testing seafood for fraud, the validity of the argument would depend on the context. If the sample size is too small, it may not be representative of the billions of pieces of seafood sold. Thus, the confidence interval derived from such a small sample may not be reliable, and might not warrant changes in buying habits. However, if that small sample is properly randomized and representative, then even a small sample can provide meaningful insights into the quality of the larger batch.