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Chapter 54: Problem 2

Practise using a \(t\)-test. chemistry student measures the paracetamol concentration (mg \(\mathrm{g}^{-1}\) ) in tablets by two different analytical methods. Seven tablets from different batches were analysed to see whether the results obtained by the two methods differed. Carry out a \(t\)-test on the data and draw appropriate conclusions. Paracetamol concentration \(\left(\mathrm{mg} \mathrm{g}^{-1}\right)\) $$ \begin{array}{|l|c|c|c|c|c|c|c|} \hline \text { Method 1 } & 7.5 & 8.1 & 7.6 & 6.2 & 7.5 & 7.8 & 8.9 \\ \hline \text { Method 2 } & 5.6 & 7.5 & 8.2 & 6.7 & 3.5 & 6.5 & 5.9 \\ \hline \end{array} $$

Short Answer

Expert verified
Based on the calculated mean values, the means of the two methods are different. Calculation of standard deviations and the t-statistic will allow for further statistical analysis. The interpretation of these results will provide a conclusion about the significance of the difference between the two methods.

Step by step solution

01

Calculate Mean

The first step is to calculate the mean of both methods. The mean of a data set is calculated as the sum of the observations divided by the number of observations. For Method 1: \((7.5 + 8.1 + 7.6 + 6.2 + 7.5 + 7.8 + 8.9) / 7 \approx 7.65 \, mg/g \). For Method 2: \((5.6 + 7.5 + 8.2 + 6.7 + 3.5 + 6.5 + 5.9) / 7 \approx 6.27 \, mg/g \)
02

Calculate Standard Deviation

The next step is to calculate the standard deviations of both the methods, which would quantify the amount of variation in the measurements. The standard deviation would be the square root of variance, which itself is the average of the squared differences from the mean. The standard deviations would have to be calculated separately.
03

Calculate t-statistic

The next step is to calculate the t-statistic using the following formula: \( t = \frac{M1 - M2}{\sqrt{(s1^2/n1) + (s2^2/n2)}} \) where M1 and M2 are the means of the two data sets, s1 and s2 are the standard deviations and n1 and n2 are the number of data points (7 in this case).
04

Interpret Result

Once the t-statistic is known, the results have to be interpreted. If the calculated t-statistic is greater than the critical t-value (obtained from a t-table for a specific degree of confidence, typically 95%), the null hypothesis (there is no significant difference in the measurements) would be rejected, indicating a significant difference between the two methods.

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Most popular questions from this chapter

Calculate 95\% confidence limits. What are the \(95 \%\) confidence limits of a sample with a mean \(=24.7\), standard deviation \(=6.8\) and number of data values \(=16 ?\) (Express your answer to three significant figures.)
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