Review Exercises (week 7: Normal Probability Distributions)
Review Exercises ( week 7: Normal Probability Distributions)
In Exercises 1-4, use the following information and the Empirical Rule to answer the questions. A certain light bulb’s lifespan is normally distributed, with a mean of 670 hours and a standard deviation of 65 hours.
Between what two lifespans will about 95% of these bulbs fall?
Between what two lifespans will about 99.7% of these bulbs fall?
Estimate the probability that a randomly selected bulb will last between 605 hours and 735 hours.
Estimate the probability that a randomly selected bulb will last more than 865 hours.
In Exercises 5 and 6, use the following information and standard scores to investigate observations about a normal population. A batch of 2500 resistors is normally distributed, with a mean resistance of 1.5 ohms and a standard deviation of 0.08 ohm. Four resistors are randomly selected and tested. Their resistances were measured at 1.32, 1.54, 1.66, and 1.78 ohms.
How many standard deviations from the mean are these observations?
Do any of these observations seem more or less likely than others?
In Exercises 7-18, use the Standard Normal Table to find areas under the standard normal curve.
7. Find the area to the left of z = - 0.84.
8. Find the area to the left of z = 2.55
9. Find the area to the left of z = - 0.27.
10. Find the area to the left of z = 1.26.
11. Find the area to the right of z = 1.68.
12. Find the area to the right of z = - 0.83.
13. Find the area between z = - 1.64 and the mean.
14. Find the area between z = - 1.22 and z = - 0.43.
15. Find the area between z = 0.15 and z = 1.35.
16. Find the area between z = - 1.96 and z = 1.96.
17. Find the area to the left of z = - 1.5 and to the right of z = 1.5.
18. Find the area to the left of z = 0.12 and to the right of z = 1.72.
Find the indicated probabilities;
P(z < 1.28)
P(Z > -0.74)
P(-2.15 < z < 1.55)
P(0.42 < z < 3.15)
P(z < - 2.50 or z > 2.50)
P(z < 0 or z > 1.68)
Use the standard normal distribution to compare two other normal distributions.
Blood pressure is described by two numbers: systolic pressure and diastolic pressure (measured in mmHg, or millimeters of mercury). For example, a blood pressure of 120/80 denotes a systolic pressure of 120 mmHg and a diastolic pressure of 80 mmHg. A study of 625 persons ages 26 to 45 finds that their blood pressures are normally distributed. Their systolic pressure had a mean of 122 and a standard deviation of 14, and their diastolic pressure had a mean of 83 and a standard deviation of 9. Two study participants are randomly selected and their blood pressures are measured again. The first participant had a systolic reading of 99 mmHg, while the second had a diastolic reading of 70 mmHg. Which participant had the lower reading with regard to the study population?
Health experts often consider blood pressure to be elevated if systolic pressure is 140 or greater or the diastolic pressure is 90 or greater. (a) How many participants in the study in Exercise 25 would you expect to have elevated systolic blood pressure? (b) How many would you expect to have elevated diastolic blood pressure?
Find the indicated probabilities:
The green turtle migrates across the Southern Atlantic in the winter, swimming great distances. A study found that the mean migration distance was 2200 kilometers and the standard deviation was 625 kilometers. Assuming that the distances are normally distributed, find the probability that a randomly selected green turtle migrates a distance (a) less than 1900 kilometers, (b) between 200 kilometers and 2500 kilometers, and (c) greater than 2450 kilometers.
The world’s smallest mammal is the Kitti’s hog-nosed bat, with a mean weight of 1.5 grams and a standard deviation of 0.25 gram. Assuming that the weights are normally distributed, find the probability of randomly selecting a bat that weighs (a) between 1.0 gram and 2.0 grams, (b) between 1.6 grams and 2.2 grams, and (c) more than 2.2 grams.
