Q.1.1 - The Practice of Statistics for AP

Question

To assess the accuracy of a laboratory scale, a standard weight known to weigh $10$ grams is weighed repeatedly. The scale readings are Normally distributed with unknown mean (this mean is $10$ grams if the scale has no bias). In previous studies, the standard deviation of the scale readings has been about $0.0002$ gram. How many measurements must be averaged to get a margin of error of $0.0001$ with $98\%$ confidence? Show your work.

Step-by-Step Solution

Answer

The required number is $22$ .

Step 1: Given information

Given in the question that, To assess the accuracy of a laboratory scale, a standard weight known to weigh $10$ grams is weighed repeatedly. The scale readings are Normally distributed with unknown mean (this mean is$10$ grams if the scale has no bias). In previous studies, the standard deviation of the scale readings has been about $0.0002$ gram . We need to find that how many measurements must be averaged to get a margin of error of $0.0001$ with $98\%$ confidence.

Step 2: Explanation

Population standard deviation $\left(\sigma \right)=0.0002$

Margin of error $\left(E\right)=0.0001$

Confidence level $=98\%$

The formula to compute the sample size is:

$n={\left(\frac{z_{\alpha /2}\times \sigma }{E}\right)}^2$

 The z-score at $99\%$confidence level is calculated from the standard normal table like this,

$z_{a/2}=z_{0.02/2}$

$=2.33$

The number of measurements is calculated as:

$n={\left(\frac{z_{\alpha /2}\times \sigma }{E}\right)}^2$

  $={\left(\frac{2.33\times 0.0002}{0.0001}\right)}^2$

  $=21.648$

  $\approx 22$