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4. (10 marks) (Do this question by hand, showing all your working clearly and stating all used formulas) A manufacturer of outdoor lights is investigating the on/off remote- control unit sold with their lights. This plays an important role in their product as Canadian winters are very cold and homeowners want a reliable remote to switch the outdoors lights on and off while they themselves are indoors. The company has determined that activation distances are normally distributed with an average activation distance of 2 meters and a standard deviation of 0.5 meters. a) (2) If a remote's maximum range is 2.5 meters, what fraction of the time will users attempt to operate the remote outside of the maximum range? b) (2) Solve for the 70% percentile and describe in your own words what the value means: c) (2) A random sample of 20 observations is drawn from the population of remotes. What is the probability that the average activation distance is between 1.9 meters and 2 meters? d) (4) As part of the manufacturing process, samples of size 10 remotes are randomly selected out of the production line every hour and tested for their average activation distance. These samples are used to test whether the distribution of the activation distances has changed in some way, i.e this check is used to ensure that the production process maintains its accuracy. One such sample is listed below. Use this sample to test, at a 1% level of significance, whether the average activation distance has decreased i.e is now less than 2 meters. (The critical value should be used to make your conclusion) 1.42 1.85 1.79 2.13 1.28 1.86 2.42 1.45 2.27 1.49