Question 6.7: The article “Geographically Weighted Regression-Based Method......
The article “Geographically Weighted Regression-Based Methods for Merging Satellite and Gauge Precipitation” (L. Chao, K. Zhang, et al., Journal of Hydrology, 2018:275–289) describes a method of measuring precipitation by combining satellite measurements with measurements made on the ground. Errors were recorded monthly for 16 months, and averaged 0.393 mm with a standard deviation of 0.368 mm. Can we conclude that the mean error is less than 0.6 mm?
Let 𝜇 denote the mean error. The null and alternate hypotheses are
H_{0} : \mu ≥ 0.6 versus H_{1} : \mu < 0.6
Under H_{0}, the test statistic
t = \frac{\overline{X}\ −\ 0.6}{s ∕ \sqrt{n}}
has a Student’s t distribution with 15 degrees of freedom. Substituting \overline{X} = 0.393, s = 0.368, and n = 16, the value of the test statistic is
t = \frac{0.393\ −\ 0.6}{0.368 ∕ \sqrt{16}} = −2.25
Consulting the t-table, we find that the value t = −2.131 cuts off an area of 0.025 in the left-hand tail, and the value t = −2.602 cuts off an area of 0.01 (see Figure 6.8). We conclude that the P-value is between 0.01 and 0.025. (Software yields P = 0.0199.)
