A building code's requirements are that a particular type of foundation can be used only if a particular type of soil exists below the site. The soil is identified by a mean "index" value of 10. At higher or lower values some other foundation type must be considered.
(a) Can the engineer accept at the 10 percent significance level the hypothesis that this mean value is 10 if he observes four soil specimens at different points around the site with index values of 6, 9, 10, and 7? Assume that these values are observations of independent identically distributed normal random variables with standard deviation 2 and mean equal to the (unknown) mean index.
(b) Suppose, instead, that the soil specimens whose index values are listed above were taken quite close together on the site, so that they are not independent samples but rather have a common correlation
coefficient, 0.5, between all pairs. What moment of the sample statistic changes from the problem in part (a)? Assuming that the sample mean is normally distributed, can the engineer accept the hypothesis that the (unknown) mean is 10 (at the 10 percent significance level)?
(c) Reconsider the situation in case (a) if the code states that the foundation can be used if the soil has an index of 10 or more.