Apple claims that with a fully charged battery, its newest iPhone X allows for up to 12 hours of
internet use. The minimum battery life is 9.5 hours. Assume that for a randomly selected iPhone
X, battery life is uniformly distributed between 9.5 and 12 hours. The density function of this
uniform probability distribution is given in the figure below. For values of battery life between 9.5 and 12 hours, the density is 0.4. For all other values of
battery life, the density is 0.
1. For a randomly selected iPhone X, what is the probability that the battery life is larger than
11 hours? Indicate this probability in the figure above, and calculate it. 2. Calculate the probabilities that (i) the battery life lies between 9.5 and 10 hours; and (ii) the
battery life is less than 10 hours. Are these two probabilities different? Explain why or why
not. 3. What do you think the expected value of the battery life is? According to AAA, the average price for a gallon of regular gas in North Carolina is $2.63.
Assume that the gas price is normally distributed with a standard deviation of $0.17. 4. Sketch the density function of the gas price. In your graph, indicate the probability that the
price is greater than $2.75 (no calculations necessary). 5. Calculate the probability that the price is greater than $2.75. 6. What is the probability that the price is greater than or equal to $2.75? Compare your
answers to questions 5 and 6: are they different? Why or why not? 7. How would your answer to question 6 change if the standard deviation was $0.25 instead of
$0.17? (You do not have to do any calculations, just provide an explanation in words) 8. Calculate the probability that the price lies within 1.5 standard deviations of the mean.
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