\(\cos\theta=(\frac{\sqrt2}{2})\to\theta=\pi/4\ and\ \theta=\frac{7\pi}{4}\)

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Geographical Analysis (Oct. 2006) published a study of a new method for analyzing remote-sensing data from satellite pixels in order to identify urban land cover. The method uses a numerical measure of the distribution of gaps, or the sizes of holes, in the pixel, called lacunarity. Summary statistics for the lacunarity measurements in a sample of 100 grassland pixels are \(\bar{x}=225\ and\ s=20s=20\). It is known that the mean lacunarity measurement for all grassland pixels is 220. The method will be effective in identifying land cover if the standard deviation of the measurements is 10% (or less) of the true mean (i.e., if the standard deviation is less than 22).

a. Give the null and alternative hypotheses for a test to determine whether, in fact, the standard deviation of all grassland pixels is less than 22.

b. A MINITAB analysis of the data is provided below. Locate and interpret the p-value of the test. Use \(\alpha=0.10\). Test for One Standard Deviation Method Null hypothesis \(\Sigma = 22\) Method Alternative hypothesis \(\Sigma = < 22\) The standard method is only for the normal distribution. Statistics NStDevVariance 10020.0400 Tests