Four different thermometers were used to measure the temperature of a sample of impure boiling water. The measurements are recorded in the table below.
| Thermometer | Temperature Reading 1 | Temperature Reading 2 | Temperature Reading 3 |
| :---------- | :-------------------- | :-------------------- | :-------------------- |
| W | $100.1^{\circ} C$ | $99.9^{\circ} C$ | $96.9^{\circ} C$ |
| X | $100.4^{\circ} C$ | $102.3^{\circ} C$ | $101.4^{\circ} C$ |
| Y | $90.0^{\circ} C$ | $95.2^{\circ} C$ | $98.6^{\circ} C$ |
| Z | $90.8^{\circ} C$ | $90.6^{\circ} C$ | $90.7^{\circ} C$ |
The actual boiling point of pure water is $100.0^{\circ} C$.
What can be concluded from the data about the reliability and validity of the thermometers?
A. Thermometer $Z$ is the most reliable.
B. Thermometer $X$ has the highest validity.
C. Thermometer $W$ is the most reliable.
D. Thermometer $Y$ has the highest validity.
Answer (1)
Calculate the mean and standard deviation for each thermometer's readings.
Determine reliability based on the lowest standard deviation.
Determine validity based on the mean closest to the true value (100.0°C).
Conclude that Thermometer Z is the most reliable: A .
Explanation
Understanding Reliability and Validity We are given temperature readings from four different thermometers (W, X, Y, and Z) measuring the boiling point of impure water. We need to determine which statement about the reliability and validity of the thermometers can be concluded from the data. Reliability refers to the consistency of the measurements (low standard deviation), and validity refers to how close the measurements are to the true value (100.0°C).
Calculating Mean and Standard Deviation First, we calculate the mean and standard deviation for each thermometer's readings. The calculations are as follows:
Thermometer W: Mean = 98.97, Standard Deviation = 1.79 Thermometer X: Mean = 101.37, Standard Deviation = 0.95 Thermometer Y: Mean = 94.60, Standard Deviation = 4.33 Thermometer Z: Mean = 90.70, Standard Deviation = 0.10
Analyzing Reliability and Validity Now, we analyze the reliability of each thermometer based on the standard deviation. The thermometer with the lowest standard deviation is the most reliable. From the calculated values, Thermometer Z has the lowest standard deviation (0.10). Therefore, Thermometer Z is the most reliable.
Next, we analyze the validity of each thermometer based on how close its mean is to the true boiling point of pure water (100.0°C). We calculate the absolute difference between each thermometer's mean and 100.0°C:
Thermometer W: |98.97 - 100.0| = 1.03 Thermometer X: |101.37 - 100.0| = 1.37 Thermometer Y: |94.60 - 100.0| = 5.40 Thermometer Z: |90.70 - 100.0| = 9.30
Thermometer W has the mean closest to 100.0°C. Therefore, Thermometer W has the highest validity.
Evaluating the Statements Based on our analysis, Thermometer Z is the most reliable, and Thermometer W has the highest validity. Now, we evaluate the given statements:
A. Thermometer Z is the most reliable. - This statement is correct. B. Thermometer X has the highest validity. - This statement is incorrect. C. Thermometer W is the most reliable. - This statement is incorrect. D. Thermometer Y has the highest validity. - This statement is incorrect.
Final Answer Therefore, the correct conclusion is that Thermometer Z is the most reliable.
Examples
Understanding the reliability and validity of measurement tools is crucial in various real-life scenarios. For example, in medical diagnostics, the reliability of a thermometer ensures consistent readings, while its validity ensures accurate temperature measurements for proper diagnosis. Similarly, in scientific experiments, reliable and valid instruments are essential for obtaining accurate and reproducible results, leading to sound conclusions and advancements in knowledge. In manufacturing, consistent and accurate measurements ensure product quality and adherence to specifications.
