Question

Assume that a scientist is able to measure the average weight of lobsters within a 50-mile radius of an island with a confidence level of \(90 \%\) by collecting data from 100 random spots around the island. If he wishes to increase the confidence level in his results to \(95 \%\), what would best help him achieve his goal? (A) Compare the results to those from another island 300 miles away. (B) Expand the radius of sampling to 100 miles and redistribute his 100 random spots within the larger range. (C) Increase the number of data samples. (D) Use a scale with \(5 \%\) more accuracy.

Short answer

Increase the number of data samples by choosing Option C.

Step-by-step solution

Understand Confidence Levels

Confidence level is the probability that a confidence interval contains the true population parameter. Increasing the confidence level means you are more certain that the parameter lies within the interval, but it typically requires more data or a larger sample size to achieve this.

Review Options for Increasing Confidence

Option A suggests comparing results with another location, but this doesn't address the current dataset's confidence level. Option B suggests expanding the geographical area which redistributes existing data but doesn't inherently increase sample size richness. Option C suggests increasing the number of data samples, directly affecting confidence intervals by reducing their size. Option D suggests using a more accurate scale; however, this doesn't increase the confidence level directly.

Determine the Best Option

To increase the confidence level from 90% to 95%, the most effective method is to increase the sample size. This reduces the margin of error, tightening the confidence interval, and thereby increasing the confidence level. Thus, Option C is the best choice.

Key concepts

Understanding Confidence Levels

When conducting research or scientific studies, the term 'confidence level' often comes into play. It's a way to express how confident we are that a given range (confidence interval) actually includes the true value we seek. For example, a 90% confidence level means that if the study were repeated multiple times, approximately 90% of the intervals calculated from those studies would contain the true population parameter.

Higher confidence levels are typically preferable because they suggest greater reliability in the results. However, a higher confidence level also often necessitates a broader confidence interval or a larger sample size. This is because as your confidence increases, the range of values that could potentially include the true parameter also becomes wider unless more data is gathered to narrow it back down.

Significance of Sample Size in Studies

Sample size plays a crucial role in determining the accuracy and reliability of statistical analysis. When the sample size is increased, the true population parameter is estimated more accurately. This happens because a larger dataset provides more information, reducing random sampling error.

• Increasing the sample size helps narrow the confidence interval.
• It improves the precision of the estimate of the population parameter.
• More samples mean that variations due to chance are minimized, leading to more robust conclusions.

For instance, in the original exercise, increasing the number of sampling spots around the island would directly contribute to a higher confidence level by minimizing the range of error and making the results more reliable.

Exploring Margin of Error

The margin of error is an important concept in statistics, especially when discussing confidence intervals. It represents the amount of error that one can tolerate in the estimation of a population parameter. Essentially, it's an expression of the amount of random sampling error in a survey's results.

• A smaller margin of error indicates more precise results.
• A larger sample size generally reduces the margin of error.
• When reducing the margin of error, the confidence interval becomes narrower, meaning more specific estimates of the population parameter.

In the context of the exercise, if the scientist wants to decrease the margin of error while increasing his confidence level, he would need to increase his sample size. This is because a larger sample size provides a better approximation of the population parameter, hence a smaller margin of error and, subsequently, more confidence in the results.