ACCA MA · Guided practice (Show me how)
Chapter 23: The Normal distribution
The normal distribution - from a frequency table to a z score
Part (a). The following table shows the annual salaries earned by 150 workers.
| Salary | Frequency |
|---|---|
| $0 - $1,000 | 25 |
| $1,000 - $2,000 | 35 |
| $2,000 - $3,000 | 40 |
| $3,000 - $5,000 | 50 |
| 150 |
The salary bands share their end points, so read each band as running up to but not including its upper figure, which is how the printed answer reads them.
Parts (b) and (c). A company produces units with an average length of 10 cm and a standard deviation of 0.2 cm. Lengths are normally distributed.
The normal distribution tables give the area under the curve BETWEEN THE MEAN AND A POINT z standard deviations away from it. The whole curve has an area of 1, and it is symmetrical, so the area on each side of the mean is 0.5. Table values you will need:
| z | Area between the mean and z |
|---|---|
| 1.00 | 0.3413 |
| 1.64 | 0.4495 |
| 2.00 | 0.4772 |
Work probabilities to 4 decimal places where the tables allow it, and to 2 decimal places in part (a).
(a) Probability from a frequency table
Calculate the probability that a worker chosen at random from the salary table earns less than $1,000, between $1,000 and $2,000, and more than $2,000.
(b) and (c) The normal distribution
Calculate the proportion of units longer than 10 cm, between 10 cm and 10.4 cm, and shorter than 9.8 cm, and the length X cm such that there is a 0.95 probability that a unit is longer than X cm.
Opening this exercise…