Calculate standard deviations to four decimal places and z-values to two decimal places. Determine whether each problem deals with a continuous or discrete variable before you start.
1. A coin is tossed
100 times. Use the normal curve approximation to find the probability of
obtaining
a) exactly 50 heads, and compare the result with the value 0.0796
obtained from the binomial distribution;
b) 60 or more heads. The actual value from the binomial distribution
is 0.028.
2. Use the normal curve approximation to find the probability of obtaining exactly 16 sixes in 96 tosses of a die. Compare the result with the value 0.110 obtained from the binomial distribution.
3. Assume that one-half
the people in a certain community are regular views of television. Of 100
investigators, each interviewing 10 individuals regarding their viewing
habits, how many would you expect to report that 3 people or fewer were
regular television viewers?
a) Calculate by an exact method (using binomial distribution).
b) Calculate by an approximate method (using normal distribution).
4. For a binomial frequency distribution for which p = 1/4, find the probability of obtaining 25 or more successes in 80 trials.
5. A manufacturer of light bulbs finds that on average 2% are defective. What is the probability that out of 1000 such bulbs selected at random 15 or more are defective?
6. A pair of unbiased dice is rolled 45 times. What is the probability that a sum of 6, 7, or 8 will appear 25 or more times?
7. According to American Experience Mortality Tables the probability of a man of age 41 dying within a year is 0.01. If an insurance company has 50,000 policies in force on men of this age, estimate by use of a normal curve the probability that the company will have to pay more than 525 death claims on this group of men within a year.
8. In a certain building trade the average wage is $3.60 per hour and the standard deviation is 45 cents. If the wages are assumed to follow a normal distribution, what percent of the workers receive wages between $3.00 and $3.50 per hour?
9. If the average life of a certain make of storage battery is 30 months with a standard deviation of 6 months, what percent of these batteries can be expected to last from 24 months to 36 months? Assume that the lifetimes follow a normal distribution.
10. A manufacturer knows that on the average 2% of his products are defective. Using the normal curve approximation, what is the probability that a lot of 100 pieces will contain exactly 5 defectives?
11. By use of the normal curve find how many throws with one die will be required in order that the probability of getting at least 5 sixes will have the value 1/2.
12. If the heights
of 10,000 college men closely follow a normal distribution with a mean
of 69.0 inches and a standard deviation of 2.5 inches,
a) how many of these men would you expect to be at least 6 feet in
height;
b) what range of heights would you expect to include the middle 75%
of the men in this group?
13. Three students take different tests. A got a score of 72, B one of 85, and C one of 17. All students taking the test with A had an average grade of 85, with B an average of 90, and with C an average of 25. The three standard deviations are 7, 3, and 7, respectively. Arrange the three students in order of excellence as you would judge them by these results, indicating your reasons.
14. In grading Satsuma plums whose weights are normally distributed, 20% are called small, 55% medium, 15% large, and 10% very large. If the mean weight of all Satsuma plums is 4.83 ounces with a standard deviation of 1.20 ounces, what are the lower and upper bounds for the weight of medium Satsuma plums?
15. On an examination the average grade was 70.0, and the standard deviation was 10.0. The instructor gave all students with grades from 61.0 to 79.0 the grade of C. There were 24 students who received a C grade. If the grades are assumed to satisfy a normal distribution, how many students took the examination?
16. The diameters of ball bearings are normally distributed with standard deviation of 2 mm. If it is known that 4% of the bearings have diameters larger than 23.5 mm, what is the mean of the distribution?
17. In a restaurant on a particular morning the amounts spent for breakfast by all patrons follow a normal distribution with mean 87.2 cents and standard deviation of 12.0 cents. On that morning if 420 people spend 85 cents or more for breakfast, what is the total number of people served?
18. In a certain cross of two varieties of peas, genetic theory indicates that half the seeds obtained will be smooth and half will be wrinkled. How many seeds have to be examined in order that at least 100 smooth seed are obtained with a probability of 90% ?
19. According to
Mendelian inheritance theory, certain crosses of peas give yellow and green
peas in the ratio of 3:1. How many seeds from this cross have to be planted
in order that there is a 95% probability that at least 50 green peas are
obtained?