Exercises 1.The following density function describes a random variable X. 1 if 1 < x < 5 8 a). Find the probability that X lies between 2 and 4. b). Find the probability that X is less than 3. f (x) = x 2. The following density function describes a random variable X. x if 0 < x < 2 2 a). Find the probability that X is greater than 1. b). Find the probability that X is less than .5. c). Find the probability that X is equal to 1.5. f (x) = 1 3. Find the z-score such that: (a) The area under the standard normal curve to its left is 0.5 z= (b) The area under the standard normal curve to its left is 0.9826 z= (c) The area under the standard normal curve to its right is 0.1423 z= (d) The area under the standard normal curve to its right is 0.9394 z= 4. Determine the following z-scores: (a) z0.1 = (b) z0.025 = 5. Assume that the readings on the thermometers are normally distributed with a mean of 0 and a standard deviation of 1 C. A thermometer is randomly selected and tested. Find the probability of each reading in degrees. (a) Between 0 and 0.66: (b) Between 1.61 and 0: (c) Between 1.82 and 1.8: (b) Less than 1.86: (c) Greater than 0.61: 6. Using diaries for many weeks, a study on the lifestyles of visually impaired students was conducted. The students kept track of many lifestyle variables including how many hours of sleep obtained on a typical day. Researchers found that visually impaired students averaged 9.08 hours of sleep, with a standard deviation of 2.72 hours. Assume that the number of hours of sleep for these visually impaired students is normally distributed. (a) What is the probability that a visually impaired student gets less than 6.9 hours of sleep? (b) What is the probability that a visually impaired student gets between 6.4 and 8.46 hours of sleep? (c) Twenty percent of students get less than how many hours of sleep on a typical day? 7. Assume that the readings on the thermometers are normally idstributed with a mean of 9 0 and a standard deviation of 1.00 C. Find P80 , the 80th percentile. (This is the temperature reading separating the bottom 80 % from the top 20 %. ) 8. The extract of a plant native to Taiwan has been tested as a possible treatment for Leukemia. One of the chemical compounds produced from the plant was analyzed for a particular collagen. The collagen amount was found to be normally distributed with a mean of 78 and standard deviation of 7.9 grams per mililiter. (a) What is the probability that the amount of collagen is greater than 64 grams per mililiter? (b) What is the probability that the amount of collagen is less than 70 grams per mililiter? (c) What percentage of compounds formed from the extract of this plant fall within 2 standard? 9. Because of the relatively high interest rates, most consumers attempt to pay o their credit card bills promptly. However, this is not always possible. An analysis of the amount of interest paid monthly by a bank?s Visa cardholders reveals that the amount is normally distributed with a mean of 29 dollars and a standard deviation of 7 dollars a). What proportion of the bank?s Visa cardholders pay more than 32 dollars in interest? b). What proportion of the bank?s Visa cardholders pay more than 41 dollars in interest? c). What proportion of the bank?s Visa cardholders pay less than 20 dollars in interest? d). What interest payment is exceeded by only 19% of the bank's Visa cardholders? 10. A new car that is a gas- and electric-powered hybrid has recently hit the market. The distance travelled on 1 gallon of fuel is normally distributed with a mean of 45 miles and a standard deviation of 5 miles. Find the probability of the following events: a).The car travels more than 49 miles per gallon. b). The car travels less than 40 miles per gallon. c). The car travels between 37 and 50 miles per gallon. 11. The lifetime of lightbulbs that are advertised to last for 5500 hours are normally distributed with a mean of 5700 hours and a standard deviation of 100 hours. What is the probability that a bulb lasts longer than the advertised figure? 12. The top-selling Red and Voss tire is rated 70000 miles, which means nothing. In fact, the distance the tires can run until wearout is a normally distributed random variable with a mean of 85000 miles and a standard deviation of 5000 miles. a). What is the probability that the tire wears out before 70000 miles? b). What is the probability that a tire lasts more than 91000 miles? 10