Econ 15A Test Five - Probability and Statistics - December 2021 PDF

NAME: _________________________________ I.D.: _____________________________________ TEST FIVE PROBABILITY AND STATISTICS, ECON 15A...

NAME: _________________________________ I.D.: _____________________________________ TEST FIVE PROBABILITY AND STATISTICS, ECON 15A DECEMBER 3, 2021 You will need to use the following information for the next few questions. It will be repeated for each question. There is a population of four people: Amy, Brianna, Carlos, & Darius. [Note: this is a population, not a sample.] They are asked how many movies they saw last month. Amy saw 4 films, Brianna saw 10 films, Carlos saw 2 films, and Darius saw 4 film. We are too lazy to poll the entire population, so we take a sample of two (with replacement and order matters). Unlike some practice tests, you will not be asked to draw the sampling distribution in any question, but it will prove helpful if you do it for yourself in order to answer the questions. At no point am I asking you to post an image of the distribution. 1. What is the probability that the mean of a randomly selected sample equals the mean of the population?. 2. What is the probability that your error (the distance between what you want to know and your estimate of what you want to know) is two films or less? Explain how you know this. Details are welcome. 3. If we increased the sample size to nine (instead of two), what would the standard error of the sampling distribution of the mean equal? Explain your work and how you got your answer. 4. In 2016, 3 sets of triplets were born in Orange County hospitals. In 2017, 0 sets of triplets were born. In 2018, 6 sets of triplets were born. In 2019, 2 sets of triplets were born. In 2020, 4 sets of triplets were born. Assuming it's a Poisson distribution, what is the probability that Orange County will have 7 sets of triplets born in 2021? 5. Peter wants to know the average height of male freshmen at UCI. Amy wants to know the height of all freshmen at UCI. They both randomly select students from their populations and record height until their sampling distributions become approximately normal. Who will need a larger sample before their distribution is approximately normal? Peter, Amy, or they’re the same? Please explain. [Hint: Think about the shape of the population distributions.] 6. A carnival ride can fit 36 people, and it malfunctions if the mean weight of the passengers is over 190 lbs. per passenger. The distribution of weight for park goers is normally distributed with a mean of 175 and a standard deviation of 30 lbs. If 36 park goers are randomly selected for a free ride, what’s the probability that the ride will malfunction from exceeding the weight limit? 7. True or False and explain: A small sample (less than 10 people) taken from a population that is normally distributed will not result in a sampling distribution that is normal, while a large sample taken from the same distribution will result in a sampling distribution that is normal. 8. In a town, there are four people: Amy, Bob, Carla, Dave. (This is a population.) You randomly pick one person and asks the number of homework assignments they did last week. Amy did 9, Bob did 7, Carla did 15, Dave did 9. The standard deviation of this population is 3. What is the probability that your pick will be within 2 homework assignments from the mean? 9. I spin a spinner 10 times. The spinner has two colors: white, and black. How many possible ways (combinations) are there for landing on black 4 times? 10. If the Poisson distribution is used to estimate the binomial distribution, we could just use the binomial distribution instead. When is there a distinct advantage to using the Poisson distribution, even if we can use a computer to calculate the probabilities? (Thus, efficiency and ease of use are not acceptable answers.) 11. True or false and explain: Z-tables can only be used for normal distributions whereas z-scores can be used for any distribution. 12. There is a test. It’s mean is 300 with a standard deviation of 50. What is the range of the mid 60% of the distribution? Explain the steps you take to get the answer. The z-scores are between +/- 0.84 and +/- 0.85. Using either is fine. 13. An archer hits the bull’s-eye an average of 9/10 times. She fires 10 arrows. What is the probability that she misses the bull’s-eye exactly two times? Explain the steps you take to get the answer. 14. There is a test. The mean of the test is 75 and the standard deviation is 25. What proportion of the people got between a 94 and a 107? Explain the steps you take to get the answer. 15. A bowl of fruit has four apples, two oranges, one banana, and three peaches. Kate reaches in and picks a piece of fruit. She then puts it back and picks another one. She does this until she has picked 150 pieces of fruit. What is the probability that she picked more than 51 apples? For the sake of the question, assume the distribution is continuous. Explain the steps you take to get the answer.