Posted: October 15th, 2022

A biased weather forecaster makes daily predictions about the temperature, in degrees Fahren- heit, at a station at noon each day. Let random variable W represent the forecast error for any given day. For example, if the forecast is 89.1 degrees and the actual temperature is 90.1 degrees, then we would have W = -1.0. Assume W follows a normal distribution with mean 1.5 and standard deviation 6.2. (a) (1 point) Explain in one or two sentences why this forecaster is considered “biased.” (b) (2 points) Calculate the probability that the next forecast will miss on the low side, i.e., P(W<0). Set up the relevant integral and then use R to make the calculation. Write the R code necessary to get the answer. (c) (4 points) A different forecaster of temperatures at the same station has error distribution U ~ N(1.9, sd=3.0). This forecaster’s predictions are further off, on average, but they are less variable. Suppose the criterion for judging a forecaster is the probability of getting a forecast within five degrees of the truth (i.e., P(Jerror] < 5)). Using this criterion, determine which of the two forecasters is preferred. Include your R code. (d) (2 points EXTRA CREDIT) Calculate the probability that the first forecaster (with daily error W) will miss at least three of their next five forecasts on the low side. Assume each forecast error independently follows the same normal distribution with mean 1.5 and standard deviation 6.2.

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