(Solved): 1. A variable \( X \) is uniformly distributed between 0 and 12 . What is its mean? 2. A variabl ...

1. A variable \( X \) is uniformly distributed between 0 and 12 . What is its mean? 2. A variable \( X \) is uniformly distributed between 100 and 300 . What is the \( 40^{\text {th }} \) percentile? 3. If \( X \sim \operatorname{Exp}(0.25) \), what is the probability that \( X \) is at least 7 ? 4. In Middletown, \( 53 \% \) of people support the governor's reelection campaign. Ten Middletown residents are surveyed, and we are interested in the number of these ten people who support the governor. a. If many of these ten-person surveys are conducted, what is the distribution of the survey results? b. What are the mean and standard deviation of these results? 5. In Westport, the governor is far more unpopular, and only has \( 10 \% \) support. A pollster decides to survey people in Westport until he finds one who supports the governor. How many people should he expect to talk to? A researcher wants to test a new fertilizer compound on a corn crop. Under typical growing conditions, the height of the corn is normally distributed with mean 8 feet and standard deviation 1 foot. The researcher believes that her fertilizer will help the corn grow taller. She measures a sample of 20 plants that have been treated with the new fertilizer and finds that they have an average height of \( 8.6 \) feet. 6. Under typical growing conditions, what is the probability that the corn is at least 10 feet tall? 7. What should the researcher use as her null and aiternative hypotheses? 8. What is the \( p \)-value of this hypothesis test? 9. If the significance level of the test is \( \alpha=0.05 \), what is the conclusion?