Question

The earth's temperature can be measured using either ground-based sensors or infrared-sensing devices mounted in aircraft or space satellites. Ground-based sensoring is very accurate but tedious, while infrared-sensoring appears to introduce a bias into the temperature readings- that is, the average temperature reading may not be equal to the average obtained by ground-based sensoring. To determine the bias, readings were obtained at five different locations using both ground- and air-based temperature sensors. The readings (in degrees Celsius) are listed here: $$ \begin{array}{ccc} \text { Location } & \text { Ground } & \text { Air } \\ \hline 1 & 46.9 & 47.3 \\ 2 & 45.4 & 48.1 \\ 3 & 36.3 & 37.9 \\ 4 & 31.0 & 32.7 \\ 5 & 24.7 & 26.2 \end{array} $$ a. Do the data present sufficient evidence to indicate a bias in the air-based temperature readings? Explain. b. Estimate the difference in mean temperatures between ground- and air-based sensors using a \(95 \%\) confidence interval. c. How many paired observations are required to estimate the difference between mean temperatures for ground- versus air-based sensors correct to within \(.2^{\circ} \mathrm{C}\), with probability approximately equal to \(.95 ?\)

Short answer

Answer: The purpose of conducting a paired t-test in this exercise is to determine if there is a bias in the air-based temperature readings by testing the null hypothesis that there exists no difference between the mean ground and air-based temperature readings.

Step-by-step solution

Calculate the differences between ground and air-based readings

For each location, calculate the difference in temperature readings between ground and air-based sensors by subtracting the air temperature reading from the ground temperature reading (Ground - Air).

Calculate the mean and standard deviation of the differences

Using the calculated differences, find the mean and standard deviation of the differences. These values are necessary for conducting a paired t-test and calculating the confidence interval.

Conduct a paired t-test

The goal of this analysis is to determine if there is a bias in the air temperature readings. Therefore, conduct a paired t-test to test the null hypothesis that there exists no difference between the mean ground and air-based temperature readings. Calculate the t-statistic and find the corresponding p-value.

Interpret the result of the paired t-test

Compare the p-value to the significance level (usually \(\alpha=0.05\)). If the p-value is less than the significance level, reject the null hypothesis, and there exists sufficient evidence to indicate a bias in the air-based temperature readings. If the p-value is greater than the significance level, we cannot reject the null hypothesis; hence, there is insufficient evidence to indicate a bias.

Calculate the 95% confidence interval for the difference in means

Using the mean and standard deviation of the differences, along with the number of observations, calculate the 95% confidence interval for the difference in mean temperatures between ground and air-based sensors.

Determine the required number of paired observations

To estimate the difference in mean temperatures with an accuracy of 0.2°C and a probability of 0.95, utilize the formula for determining the required sample size for a paired t-test. For this step, you need the standard deviation of the differences, the desired precision, and the desired probability.

Interpret the required number of paired observations

The calculated number of paired observations is the minimum required to estimate the difference between mean temperatures for ground versus air-based sensors correct to within 0.2°C with a probability approximately equal to 0.95.

Key concepts

Bias

In statistics, bias refers to the systematic deviation from the true value or a lack of neutrality. Here, we're examining whether there is a bias in air-based temperature readings when compared to ground-based readings. If the air sensors consistently show higher or lower temperatures than the ground sensors, this suggests a bias. To assess this, we use the differences in measurements between the two sensor types at each location.
By running a paired t-test, we can assess whether these differences are statistically significant. If the mean of these differences is considerably different from zero, it indicates that one set of readings consistently diverges from the other. Thus, any such difference confirms the presence of bias.

Confidence interval

The confidence interval gives us a range of likely values for the difference in mean temperatures measured by ground versus air-based sensors. Specifically, here we're focusing on a 95% confidence interval, which means we're 95% confident that this interval contains the true mean difference.
To calculate the confidence interval, we use the mean and standard deviation of the differences in sensor readings along with the number of observations. This involves statistical calculations: the formula for the confidence interval is \[\left(\text{Mean Difference} - t_{\frac{\alpha}{2}, n-1} \times \frac{\text{SD}}{\sqrt{n}}, \ \text{Mean Difference} + t_{\frac{\alpha}{2}, n-1} \times \frac{\text{SD}}{\sqrt{n}}\right)\] Where \(t\) is the t-distribution value, \(SD\) is the standard deviation, and \(n\) is the sample size. This interval provides valuable insight into the reliability and precision of our estimate of the true mean difference.

Hypothesis testing

Hypothesis testing is a statistical method used to decide if there is enough evidence to reject a null hypothesis. In the context of our exercise, we are testing the null hypothesis that there is no difference between the mean temperatures recorded by the ground and air-based sensors.
The paired t-test is used to perform this hypothesis test by comparing the mean difference of the paired observations. If the calculated t-statistic leads to a p-value less than our significance level (often 0.05), we reject the null hypothesis. This rejection indicates sufficient evidence of a bias in the air-based readings. If the p-value is higher, we lack evidence to conclude a bias.

Sample size determination

Determining an adequate sample size is crucial to ensure the accuracy and reliability of our results. For estimating the mean difference between ground and air-based sensors within a desired precision, we utilize the sample size formula for a paired t-test.
This formula is \[n = \left(\frac{t_{\frac{\alpha}{2}, n-1} \times SD}{E}\right)^2\] Where \(E\) is the precision (e.g., 0.2°C), \(SD\) is the standard deviation of differences, and \(t\) is derived from the t-distribution.
By performing this computation, we ensure our sample size is large enough to achieve our goal with the desired level of confidence (e.g., 95%), providing a robust conclusion about the bias in temperature readings.