- Calculate the mean of the control group: \(\bar{x}_{\text{Control}} = \frac{1960}{8} = 245.0\).
- Calculate the mean of the treatment group: \(\bar{x}_{\text{Treatment}} = \frac{1810}{8} = 226.25\).
- Find the difference: \(226.25 - 245.0 = -18.75\).
- Round the difference to the nearest tenth: \(\boxed{-18.8}\).
Explanation
- Analyze the given data
We begin by organizing the provided cholesterol level data. The control group consists of 8 recorded values: \(241, 243, 242, 245, 250, 248, 246, 245\). The treatment group, which received the cholesterol-lowering medication, also has 8 recorded values: \(230, 225, 220, 218, 224, 240, 232, 221\). Although the initial description mentions 10 patients in each group, we will perform our analysis using the provided set of 8 values per group. Our goal is to determine the mean of each group and find the difference between the treatment mean and the control mean.
- Calculate group means
To find the mean of the control group (\(\bar{x}_{\text{Control}}\)), we sum the values and divide by the number of observations (\(n = 8\)):
\[S_{\text{Control}} = 241 + 243 + 242 + 245 + 250 + 248 + 246 + 245 = 1960\]
\[\bar{x}_{\text{Control}} = \frac{1960}{8} = 245.0 \text{ mg/dL}\]
Similarly, for the treatment group (\(\bar{x}_{\text{Treatment}}\)):
\[S_{\text{Treatment}} = 230 + 225 + 220 + 218 + 224 + 240 + 232 + 221 = 1810\]
\[\bar{x}_{\text{Treatment}} = \frac{1810}{8} = 226.25 \text{ mg/dL}\]
- Calculate the difference and round the result
Now, we calculate the difference between the mean of the treatment group and the mean of the control group:
\[\text{Difference} = \bar{x}_{\text{Treatment}} - \bar{x}_{\text{Control}}\]
\[\text{Difference} = 226.25 - 245.0 = -18.75\]
Finally, rounding the result to the nearest tenth, we get:
\[-18.75 \approx -18.8\]
Thus, the treatment group's mean cholesterol level is \(18.8\) mg/dL lower than the control group's mean.
Examples
In clinical research, this type of analysis is crucial to evaluate the efficacy of a drug. For instance, if you were comparing the average recovery time of two groups of patients undergoing different treatments, you would use similar statistical methods to see if one treatment results in a statistically significant improvement over the other. Calculations like this help doctors and researchers make data-driven decisions about patient care. If \(\bar{x}_1\) and \(\bar{x}_2\) are the means of two treatment groups, the difference \(D = \bar{x}_1 - \bar{x}_2\) provides a quick summary of how much the average outcome differs.