Higher Engineering Mathematics

Assign-17-H8152.tex 23/6/2006 15: 16 Page 625

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Statistics and probability

Assignment 17

This assignment covers the material contained

in Chapters 61 to 63.

The marks for each question are shown in

brackets at the end of each question.

1. 1200 metal bolts have a mean mass of 7.2 g
   and a standard deviation of 0.3 g. Determine the
   standard error of the means. Calculate also the
   probability that a sample of 60 bolts chosen at
   random, without replacement, will have a mass
   of (a) between 7.1 g and 7.25 g, and (b) more
   than 7.3 g. (12)


2. A sample of 10 measurements of the length of
   a component are made and the mean of the
   sample is 3.650 cm. The standard deviation of
   the samples is 0.030 cm. Determine (a) the 99%
   confidence limits, and (b) the 90% confidence
   limits for an estimate of the actual length of the
   component. (10)


3. An automated machine produces metal screws
   and over a period of time it is found that 8%
   are defective. Random samples of 75 screws are
   drawn periodically.

(a) If a decision is made that production con-

tinues until a sample contains more than

8 defective screws, determine the type I

error based on this decision for a defect

rate of 8%.

(b) Determine the magnitude of the type II error

when the defect rate has risen to 12%.

The above sample size is now reduced to

55 screws. The decision now is to stop the

machine for adjustment if a sample contains

4 or more defective screws.

(c) Determine the type I error if the defect rate

remains at 8%.

(d) Determine the type II error when the defect

rate rises to 9%. (22)

4. In a random sample of 40 similar light bulbs
   drawn from a batch of 400 the mean lifetime is
   found to be 252 hours. The standard deviation of
   the lifetime of the sample is 25 hours. The batch is
   classed as inferior if the mean lifetime of the batch
   is less than the population mean of 260 hours. As
   a result of the sample data, determine whether
   the batch is considered to be inferior at a level of
   significance of (a) 0.05, and (b) 0.01. (9)


5. The lengths of two products are being compared.

Product 1: sample size=50, mean value of

sample= 6 .5 cm, standard devia-

tion of whole of batch= 0 .40 cm.

Product 2: sample size=60, mean value of

sample= 6 .65 cm, standard devia-

tion of whole of batch= 0 .35 cm.

Determine if there is any significant difference

between the two products at a level of significance

of (a) 0.05, and (b) 0.01. (7)

6. The resistance of a sample of 400 resistors pro-
   duced by an automatic process have the following
   resistance distribution.

Resistance Frequency

()

50.11 9

50.15 35

50.19 61

50.23 102

50.27 89

50.31 83

50.35 21

Calculate for the sample: (a) the mean, and (b) the

standard deviation. (c) Test the null hypothesis

that the resistance of the resistors are normally

distributed at a level of significance of 0.05, and