[Solved] Assume that the readings on the thermometers are normally distributed with a mean of 0 and a standard deviation of 1C.

1.   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 0.65:

b.     Between −0.6 and 0:

c.     Between −1.01 and 0.56:

d.     Less than 2.05:

e.     Greater than −0.64:

2.   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 were obtained on a typical day. Researchers found that visually impaired students averaged 9.87 hours of sleep, with a standard deviation of 1.1 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 6.1 hours of sleep?

b.     What is the probability that a visually impaired student gets between 6.5 and 7.89 hours of sleep?

3.   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 61 and standard deviation of 9.8 grams per mililiter.

a.     What is the probability that the amount of collagen is greater than 60 grams per mililiter?

b.     What is the probability that the amount of collagen is less than 90 grams per mililiter?      

c.     What percentage of compounds formed from the extract of this plant fall within 1 standard deviation of the mean?                     

4.   Determine the following z-scores:                                                  

a.     z0.1 =                                               

b.     z0.025 =

5.   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 = ____

6.   The lifetime of lightbulbs that are advertised to last for 4100 hours are normally distributed with a mean of 4400 hours and a standard deviation of 300 hours. What is the probability that a bulb lasts longer than the advertised figure? 

7.   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 65 miles and a standard deviation of 7 miles. Find the probability of the following events:

a.     The car travels more than 69 miles per gallon. Probability = ________

b.     The car travels less than 58 miles per gallon. Probability = ________

c.     The car travels between 61 and 71 miles per gallon. Probability =________

8.   Select True or False                                    

1. A random variable X is normally distributed with a mean of 150 and a variance of

                               36. Given that X = 120, its corresponding z− score is 5.0    __________      

2.   Let z1 be a z−score that is unknown but identifiable by position and area. If the symmetrical area between −z1 and +z1 is 0.9544, the value of z1 is 2.0________

3.   The mean and standard deviation of a normally distributed random variable which has been standardized are one and zero, respectively._________

4.   Using the standard normal curve, the z−score representing the 90th percentile is

1.28._________

9.   Because of the relatively high interest rates, most consumers attempt to pay off 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 27 dollars and a standard deviation of 8 dollars.

a.     What proportion of the bank’s Visa cardholders pay more than 31 dollars in interest? Proportion = ________

b.     What proportion of the bank’s Visa cardholders pay more than 36 dollars in interest? Proportion = ________

c.     What proportion of the bank’s Visa cardholders pay less than 16 dollars in interest? Proportion =________

d.     What interest payment is exceeded by only 21% of the bank’s Visa cardholders?  Interest Payment =_________

10. The following density function describes a random variable X.        f(x) = 1 − (x /2) if 

0<x<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. _________

11. Compute the following probabilities for the standard normal distribution Z.

a.     P(0 < Z < 1.95)= ________

b.     P(−1.9 < Z < 0.9)= ________

c.     P(Z > −1.95)=________

12. The top-selling Red and Voss tire is rated 60000 miles, which means nothing. In fact, the distance the tires can run until wear out is a normally distributed random variable with a mean of 71000 miles and a standard deviation of 5000 miles.

a.     What is the probability that the tire wears out before 60000 miles? ________

b.     What is the probability that a tire lasts more than 81000 miles? ________

13. The following density function describes a random variable X. f (x) = (x – 1)/8 if 1 < x < 5

a.     Find the probability that X lies between 2 and 4. ________

b.     Find the probability that X is less than 3. ________

14. College students average 8.6 hours of sleep per night with a standard deviation of 35 minutes. If the amount of sleep is normally distributed, what proportion of college students sleep for more than 9.6 hours?            ________

15. The following density function describes a random variable X. f(x) = (x/81) if 0<x<9 and f(x)= (18-x)/81 if 9<X<18.

a.     Draw a graph of the density function and then use it to find the probabilities below:

i.          Find the probability that X lies between 1 and 8. ________

ii.         Find the probability that X lies between 8 and 11. ________

iii.        Find the probability that X is less than 10. ________ iv.      Find the probability that X is greater than 7. ________

16. 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 were obtained on a typical day. Researchers found that visually impaired students averaged 8.95 hours of sleep, with a standard deviation of 2.11 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 6.2 and 10.5 hours of sleep?________

c.     Thirty percent of students get less than how many hours of sleep on a typical day?________

17. For a normal distribution, find the percentage of data that are 

a.     Within 1 standard deviation of the mean.________

b.     To the right of 1.5 standard deviations below the mean________

c.     More than 0.5 standard deviations away from the mean________

18. Suppose the random variable x is best described by a normal distribution with µ = 30 and σ = 7.9. Find the z-score that corresponds to each of the following x values.

a.     X=36, z=________

b.     X=35, z=________

c.     X=22, z=________

d.     X=30, z=________

e.     X=19, z=________

f.      X=36, z=________

19. Assume that the readings on the thermometers are normally idistributed with a mean of 0◦ and a standard deviation of 1.00◦C. Find P25, the 25th percentile. This is the temperature reading separating the bottom 25 % from the top 75 %.________

20. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. Mensa is an international society that has one and only one qualification for membership: a score in the top 2% on an IQ test.

a.     What IQ score should one have in order to be eligible for Mensa?________

b.     In a typical region of 115,000 people, how many are eligible for Mensa?________

21. Find the following probabilities for the standard normal random variable z:

a.     P(−1.36 ≤ z ≤ 1.38)= ________

b.     P(−2.27 ≤ z ≤ 1.64)= ________

c.     P(z ≤ 1.33)=________

d.     P(z > −0.68)=________

22. Suppose x is a normally distributed random variable with µ = 10.1 and σ = 3.4. Find each of the following probabilities:

a.     P(5.7 ≤ X ≤ 16.2)= ________

b.     P(5.6 ≤ X ≤ 15.6)= ________

c.     P(11.5 ≤ X ≤ 14.1)= ________

d.     P(X ≥ 10.7)=________

e.     P(X ≤ 14.4)=________

23. Women’s weights are normally distributed with a mean given by µ = 143 lb and a standard deviation given by σ = 29 lb. Find the third decile, D3, which separates the bottom 30% from the top 70%.________

Mean= 143

S.D = 29

P[Z<((x-143)/29))]=0.3

(x-143)/29=0.52

X=127.79

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