The Normal Distribution - Worksheets, Questions and Revision

16 original exam-style questions - 11 pages of questions with a full mark scheme - free printable PDF.

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H06 The Normal Distribution

EDEXCEL 1ST0 · Calculator allowed · about 80 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

For each statement about the normal distribution, write down whether it is True or False.

(a)The normal distribution curve is symmetrical about the mean.(1)

(b)For a normal distribution, the mean, median and mode are all equal.(1)

(c)The tails of a normal distribution curve eventually meet the horizontal axis.(1)

(d)A normal distribution curve can have two separate peaks.(1)

(Total for Question 1 is 4 marks)

The diagrams show four sketched distribution curves, labelled A to D.

Write down the letter of the curve that shows a normal distribution.

(Total for Question 2 is 1 mark)

A machine packs bags of sugar. The masses of the bags are normally distributed with a mean of 1000 g and a standard deviation of 15 g.

On the axes below, sketch a normal distribution curve to show the distribution of the masses. Label the mean, and the values 1 and 2 standard deviations above and below the mean, on the horizontal axis.

(Total for Question 3 is 4 marks)

For any normal distribution, complete each sentence below using one of the values 68%, 95% or 99.7%.

(a)Approximately ____ of the data values lie within 1 standard deviation of the mean.(1)

(b)Approximately ____ of the data values lie within 2 standard deviations of the mean.(1)

(c)Approximately ____ of the data values lie within 3 standard deviations of the mean.(1)

(Total for Question 4 is 3 marks)

The diagram shows a normal distribution curve. Dashed lines are drawn at the mean and at 1, 2 and 3 standard deviations above and below the mean, splitting the curve into 8 regions, labelled P to W.

Using the 68-95-99.7 rule, write down the approximate percentage of the distribution lying in each region described below.

(a)Regions S and T combined (within 1 standard deviation of the mean).(1)

(b)Region R (between 2 and 1 standard deviations below the mean).(1)

(c)Region Q (between 3 and 2 standard deviations below the mean).(1)

(d)Region P (more than 3 standard deviations below the mean).(1)

(Total for Question 5 is 4 marks)

The lengths of steel rods produced by a machine are normally distributed with a mean of 250 cm and a standard deviation of 3 cm.

(a)Work out the interval within which approximately 68% of the rods' lengths lie.(2)

(b)Work out the interval within which approximately 95% of the rods' lengths lie.(2)

(Total for Question 6 is 4 marks)

The resting heart rates of adults in a large population are normally distributed with a mean of 72 beats per minute (bpm) and a standard deviation of 6 bpm.

(a)Write down the percentage of adults with a resting heart rate between 60 bpm and 84 bpm.(2)

(b)Work out the percentage of adults with a resting heart rate above 90 bpm.(3)

(Total for Question 7 is 5 marks)

The marks scored by 240 candidates in a statistics exam are normally distributed with a mean of 65 and a standard deviation of 8.

Work out an estimate for the number of candidates who scored more than 81 marks.

(Total for Question 8 is 3 marks)

The annual salaries of 500 employees at a company are normally distributed with a mean of £32,000 and a standard deviation of £4,000. For each statement, write down whether it is True or False.

(a)Since about 95% of employees earn between £24,000 and £40,000, we can be certain that no employee earns less than £24,000.(1)

(b)About 68% of employees earn between £28,000 and £36,000.(1)

(c)It is impossible for an employee to earn more than £44,000.(1)

(Total for Question 9 is 3 marks)

In a national test, a pupil's raw score is converted into a standardised score (z-score), where z = 0 represents the mean and z = 1 represents 1 standard deviation above the mean, and so on. The raw test scores for all pupils are normally distributed with a mean of 100 and a standard deviation of 15. A particular pupil's result has a standardised score of z = 2.

(a)Work out the pupil's raw test score.(2)

(b)Using the empirical rule, estimate the percentage of pupils who scored higher than this pupil.(2)

(Total for Question 10 is 4 marks)

A machine fills bottles of sauce. The volumes are normally distributed with a mean of 500 ml and a standard deviation of 4 ml. A bottle is rejected by quality control if its volume is more than 2 standard deviations from the mean.

(a)Work out the tolerance limits between which a bottle's volume must lie to avoid rejection.(2)

(b)The factory fills 5000 bottles per day. Work out an estimate for the number of bottles rejected each day.(3)

(Total for Question 11 is 5 marks)

The heights of a large group of adult cyclists are normally distributed with a mean of 170 cm and a standard deviation of 6 cm. The diagram shows the distribution, with several height values marked on the horizontal axis.

(a)State how many standard deviations the height 158 cm is from the mean, and in which direction.(1)

(b)State how many standard deviations the height 176 cm is from the mean, and in which direction.(1)

(c)State how many standard deviations the height 188 cm is from the mean, and in which direction.(1)

(d)Using the empirical rule, write down the percentage of cyclists with a height between 164 cm and 188 cm.(1)

(Total for Question 12 is 4 marks)

The diagrams show the distributions of the annual bonuses paid to staff at two companies, Company A and Company B.

(a)Which company's bonus distribution, A or B, could be modelled by a normal distribution? Give a reason for your answer.(2)

(b)For Company B's distribution, state whether the mean is greater than, less than, or equal to the median. Give a reason for your answer.(2)

(Total for Question 13 is 4 marks)

A factory produces bags of flour with a labelled mass of 1 kg (1000 g). Before setting tolerance limits, a quality inspector wants to check whether the masses of the bags are approximately normally distributed. This investigation follows the statistical enquiry cycle: plan, collect, process and interpret.

(a)PLAN: Describe a suitable method the inspector could use to select a sample of 50 bags from a day's production for weighing, and explain how this method helps to avoid bias.(2)

(b)COLLECT/PROCESS: The inspector weighs a small sample of 8 bags. Their masses, in grams, are shown in the table.

Bag	1	2	3	4	5	6	7	8
Mass (g)	994	998	1001	1002	999	1005	997	1000

Work out the mean mass of this sample.(2)

(c)PROCESS: Work out the median mass of the sample of 8 bags in part (b).(1)

(d)INTERPRET: A much larger sample confirms that the masses of all bags produced are normally distributed with a mean of 1000 g and a standard deviation of 8 g. The factory sets tolerance limits of 984 g to 1016 g, and rejects any bag outside these limits. Use the empirical rule to estimate the percentage of bags that would be rejected, and comment on whether the tolerance limits seem reasonable.(3)

(Total for Question 14 is 8 marks)

The times taken by runners to complete a marathon are normally distributed with a mean of 240 minutes and a standard deviation of 20 minutes.

(a)210 minutes is 1.5 standard deviations below the mean, and 270 minutes is 1.5 standard deviations above the mean. Since 1.5 standard deviations is halfway between 1 and 2 standard deviations, use linear interpolation between the known percentages for 1 standard deviation (68%) and 2 standard deviations (95%) to estimate the percentage of runners with a time between 210 and 270 minutes.(3)

(b)Out of 3000 entrants, estimate the number of runners whose time is outside the interval 210 to 270 minutes.(2)

(Total for Question 15 is 5 marks)

The wingspans of a large population of adult swans are normally distributed with a mean of 218 cm and a standard deviation of 6 cm. The diagram shows the distribution, with several wingspan values marked on the horizontal axis.

(a)Using the diagram, write down the interval of wingspans within which approximately 95% of the swans lie.(2)

(b)A swan has a wingspan of 227 cm. This corresponds to a standardised score of z = 1.5. Show that this value of z is correct. Then, using linear interpolation between the known percentages for 1 standard deviation (68%) and 2 standard deviations (95%), estimate the percentage of swans with a wingspan greater than 227 cm.(3)

(c)Give one reason why the normal distribution may not be a perfectly accurate model for the wingspans of swans.(1)

(Total for Question 16 is 6 marks)

Mark scheme · H06 The Normal Distribution

Question 1

(a) B1 True cao
(a) Answer: True
(b) B1 True cao
(b) Answer: True
(c) B1 False cao
(c) Answer: False
(d) B1 False cao
(d) Answer: False

Question 2

B1 A cao
Answer: A

Question 3

B1 symmetric, single-peaked bell-shaped curve drawn, centred on the middle tick
B1 the middle tick correctly labelled as the mean, 1000 (g)
B1 985 and 1015 correctly labelled at 1 standard deviation below/above the mean
B1 970 and 1030 correctly labelled at 2 standard deviations below/above the mean
Answer: A symmetric bell-shaped curve peaking at 1000 g, with 985 g and 1015 g marked at ±1 sd, and 970 g and 1030 g marked at ±2 sd.

Question 4

(a) B1 68% cao
(a) Answer: 68%
(b) B1 95% cao
(b) Answer: 95%
(c) B1 99.7% cao
(c) Answer: 99.7%

Question 5

(a) B1 68% cao
(a) Answer: 68%
(b) B1 13.5% cao
(b) Answer: 13.5%
(c) B1 2.35% cao
(c) Answer: 2.35%
(d) B1 0.15% cao
(d) Answer: 0.15%

Question 6

(a) M1 250-3 and 250+3, oe
(a) A1 247 cm to 253 cm cao
(a) Answer: 247 cm to 253 cm
(b) M1 250-(2x3) and 250+(2x3), oe
(b) A1 244 cm to 256 cm cao
(b) Answer: 244 cm to 256 cm

Question 7

(a) M1 identifies 60 and 84 as 2 standard deviations from the mean, oe
(a) A1 95% cao
(a) Answer: 95%
(b) M1 identifies 90 as 3 standard deviations above the mean, oe
(b) M1 (100-99.7)/2, oe
(b) A1 0.15% cao
(b) Answer: 0.15%

Question 8

M1 identifies 81 as 2 standard deviations above the mean, oe
M1 (100-95)/2, oe
A1 6 cao
Answer: 6 candidates

Question 9

(a) B1 False cao
(a) Answer: False
(b) B1 True cao
(b) Answer: True
(c) B1 False cao
(c) Answer: False

Question 10

(a) M1 100+2x15, oe
(a) A1 130 cao
(a) Answer: 130
(b) M1 (100-95)/2, oe
(b) A1 2.5% cao
(b) Answer: 2.5%

Question 11

(a) M1 500-2(4) and 500+2(4), oe
(a) A1 492 ml to 508 ml cao
(a) Answer: 492 ml to 508 ml
(b) M1 100-95 (=5%), oe
(b) M1 5% x 5000, oe
(b) A1 250 cao
(b) Answer: 250 bottles

Question 12

(a) B1 2 standard deviations below the mean, cao
(a) Answer: 2 standard deviations below the mean
(b) B1 1 standard deviation above the mean, cao
(b) Answer: 1 standard deviation above the mean
(c) B1 3 standard deviations above the mean, cao
(c) Answer: 3 standard deviations above the mean
(d) B1 83.85% cao
(d) Answer: 83.85%

Question 13

(a) B1 A cao
(a) B1 valid reason referring to symmetry, e.g. Curve A is symmetrical (bell-shaped) with mean equal to median
(a) Answer: Company A - because its distribution is symmetrical, with the mean and median equal.
(b) B1 mean is greater than the median, cao
(b) B1 valid reason, e.g. the distribution is positively skewed, so the long tail of a few very large bonuses pulls the mean above the median
(b) Answer: The mean is greater than the median, because Company B's distribution is positively skewed - the long tail to the right, from a few very large bonuses, pulls the mean above the median.

Question 14

(a) B1 suitable sampling method described, e.g. a systematic sample, selecting every nth bag as they come off the production line
(a) B1 valid reason linked to avoiding bias, e.g. this spreads the sample across the whole day's production, rather than only bags made at one time, so it better represents the day's output
(a) Answer: e.g. Take a systematic sample, selecting every nth bag from the production line throughout the day; this avoids bias because bags from all points in the day's production are included, not just bags made at one time.
(b) M1 994+998+1001+1002+999+1005+997+1000 (=7996), oe
(b) A1 999.5 g cao
(b) Answer: 999.5 g
(c) B1 999.5 g cao
(c) Answer: 999.5 g
(d) M1 identifies 984 and 1016 as 2 standard deviations from the mean, oe
(d) A1 5% cao (100-95)
(d) B1 valid comment, e.g. the limits seem reasonable since only a small proportion (about 5%) of bags would fall outside them
(d) Answer: About 5% of bags would be rejected; the tolerance limits seem reasonable since only a small proportion of bags fall outside them.

Question 15

(a) M1 recognises the required percentage is halfway between 68% and 95%, oe
(a) M1 68+(95-68)/2, oe
(a) A1 81.5% cao
(a) Answer: 81.5%
(b) M1 100-81.5 (=18.5%), oe
(b) A1 555 cao
(b) Answer: 555