# Statistics Problem Set: Probabilities and Distribution

1. The table below shows the total number of man-days lost to sickness during one week's operation of a small chemical plant. Showing all of your work, calculate the arithmetic mean and standard deviation of the number of lost days.

Days Lost 1-3 4-6 7-9 10-12 13-15

Frequency 8 7 10 9 6

2. The school of Science and Engineering at a local university regularly purchases a particular type of electrical component. 60% are purchased from company A and 40% are from Company B. 2% from Company A and 1% from Company B are known to be defective. The components are identical and thoroughly mixed on receipt.

a. Draw a tree diagram to represent the possible outcomes when a single component is selected at random.

b. A component is selected at random.

i. What is the probability that this component was supplied by Company A and was defective?

ii. Calculate the probability that the component was defective.

iii. Given that the component was defective, what is the probability that it was supplied by Company A?

3. Length of metal strips produced by a machine are normally distributed with mean length of 150 cm and a standard deviation of 10 cm. Find the probability that the length of a randomly selected strip is:

a. Shorter than 165 cm

b. Longer than 170 cm

c. Between 145 and 155 cm

4. The torque, T Nm, required to rotate shafts of different diameters, D mm, on a machine has been tested and recorded below.

a. Plot a scatter diagram of the results with diameter of the machine as the independent variable.

b. Use the method of least squares to determine the linear regression equation that relates the torque to the diameter.

c. Estimate the value of the torque required when the diameter is 16 mm.

d. Calculate the coefficient of correlation between these variables and comment on your prediction obtained in (c).

D (mm) 6 10 14 18 21 25

T (Nm) 5.5 7.0 9.5 12.5 13.5 16.5

#### Solution Preview

Please see the attached documents for full solutions.

1.

Days lost Mid value, X Frequency, F X^2 X*F X^2*F

1 -- 3 2 8 4 16 32

4 -- 6 5 7 25 35 175

7 -- 9 8 10 64 80 640

10 -- 12 11 9 121 99 1089

13 -- 15 14 6 196 84 1176

Total ...