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1. (a) Show that any open set U in R is a union of countably many disjoint open intervals, one or two of which may be unbounded (the line or half-lines). Hint: For each x ∈ U , find a largest open interval contain- ing x and included in U.
(b) Let F be a closed set in R whose complement is the union of disjoint open intervals (an, bn ) from part (a). Let f be a continuous real-valued function on F . Extend f to be linear on [an, bn ], or if an or bn is infinite, extend f to be constant on a half-line. Show that the resulting f is continuous on R. Note: Be aware that subsequences of the an may converge.
1. An experiment has four possible outcomes A, B, C, and D. Check whether the following assignments of probability are possible:
What percentage of population would have readings exceeding 250 and what percentage would have readings less than 150?
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Use the attached data set for the following questions (laborEcon.dta) . Feel free to resample the data set if you think it is appropriate. Consider the earning function in the form of a linear regression equation we discussed before:
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