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 by comparing standard deviations; the smallest standard deviation indicates the most reliable thermometer.
Determine validity by comparing the means to the true boiling point ( 100. 0 ∘ C ); the closest mean indicates the highest validity.
Conclude that Thermometer Z is the most reliable based on its lowest standard deviation. A
Explanation
Understanding the Problem We are given temperature readings from four different thermometers (W, X, Y, and Z) measuring the temperature of impure boiling water. We know that the actual boiling point of pure water is 100. 0 ∘ C . We need to determine which statement about the reliability and validity of the thermometers is best supported by the data.
Defining Reliability and Validity Reliability refers to the consistency of the measurements. A thermometer is considered reliable if its readings are close to each other. This can be quantified by a small standard deviation. Validity refers to how close the measurements are to the true value ( 100. 0 ∘ C ). A thermometer is considered valid if its mean reading is close to 100. 0 ∘ C .
Gathering the Data From the calculations, we have the following means and standard deviations for each thermometer:
Thermometer W: Mean = 98.9 7 ∘ C , StDev = 1.7 9 ∘ C Thermometer X: Mean = 101.3 7 ∘ C , StDev = 0.9 5 ∘ C Thermometer Y: Mean = 94.6 0 ∘ C , StDev = 4.3 3 ∘ C Thermometer Z: Mean = 90.7 0 ∘ C , StDev = 0.1 0 ∘ C
Analyzing Reliability Now, let's analyze the reliability of each thermometer. The thermometer with the smallest standard deviation is the most reliable. Comparing the standard deviations, we see that Thermometer Z has the smallest standard deviation ( 0.1 0 ∘ C ). Therefore, Thermometer Z is the most reliable.
Analyzing Validity Next, let's analyze the validity of each thermometer. The thermometer with the mean closest to 100. 0 ∘ C has the highest validity. The differences between the means and 100. 0 ∘ C are:
Thermometer W: ∣98.97 − 100.0∣ = 1.0 3 ∘ C Thermometer X: ∣101.37 − 100.0∣ = 1.3 7 ∘ C Thermometer Y: ∣94.60 − 100.0∣ = 5.4 0 ∘ C Thermometer Z: ∣90.70 − 100.0∣ = 9.3 0 ∘ C
Thermometer W has the mean closest to 100. 0 ∘ C , so it has the highest validity.
Conclusion Based on our analysis, Thermometer Z is the most reliable, and Thermometer W has the highest validity. Now, let's evaluate the given options:
A. Thermometer Z is the most reliable. - This is consistent with our analysis. B. Thermometer X has the highest validity. - This is not consistent with our analysis. C. Thermometer W is the most reliable. - This is not consistent with our analysis. D. Thermometer Y has the highest validity. - This is not consistent with our analysis.
Therefore, the best answer is A.
Examples
Understanding reliability and validity is crucial in many real-world applications, such as medical testing. For example, when testing for a disease, a reliable test will consistently give similar results when repeated on the same sample. A valid test will accurately identify those who have the disease (true positive) and correctly identify those who do not (true negative). Balancing both reliability and validity ensures accurate and trustworthy results, which is essential for proper diagnosis and treatment.
