Final answer:
The margin of error is inversely proportional to the square root of the sample size. If the sample size is cut in half, the margin of error is multiplied by the square root of 2, approximately equivalent to being multiplied by 1.414, which is closest to the option 'multiplied by 2' provided.
Step-by-step explanation:
When you cut the sample size in half in statistics, it impacts the margin of error when making a statistical inference about the mean of a normally distributed population. The margin of error is influenced by the sample size through the formula for the standard error which involves the square root of the sample size, n. Specifically, the margin of error is inversely proportional to the square root of the sample size.
When we decrease the sample size, the standard error of the mean increases because the denominator in the standard error formula becomes smaller. Therefore, to determine how the margin of error is affected when the sample size is cut in half, we can examine the relationship with the square root function. If the original sample size is n, the margin of error is proportional to 1/sqrt(n). When we cut the sample size in half, we have:
New sample size = n/2
New margin of error is proportional to 1/sqrt(n/2) = 1/(sqrt(n)/sqrt(2)) = sqrt(2)/sqrt(n), which is the original margin of error multiplied by sqrt(2).
Hence, if you cut the sample size in half, the margin of error is multiplied by sqrt(2) (or approximately 1.414). This is not one of the exact options provided, but it is closest to the option stating 'The margin of error is multiplied by 2'. Multiplying by 2 overestimates the increase, but it is closer to sqrt(2) than the provided option of 0.5 and definitely more accurate than the options suggesting a change by a factor of 10.5 or 0.5.
Key Concepts of Margin of Error
The margin of error represents how much the estimate from the sample can be expected to differ from the true population parameter. It is half the width of the confidence interval and provides us with a range around the estimated value that is likely to contain the true mean, given a certain level of confidence. The margin of error is affected by the sample size, confidence level, and population standard deviation.
Understanding this concept helps elucidate the inverse relationship between sample size and margin of error, which is a fundamental principle of inferential statistics.