25
MEI Conference 2017 Statistics: Introducing the Normal distribution using graphing technology Nick Thorpe

# Statistics: Introducing the Normal distribution using ... · THE NORMAL DISTRIBUTION The normal curve − • probability Useful rules of thumb The Standard Normal Distribution For

others

• View
41

0

Embed Size (px)

Citation preview

MEI Conference 2017

Statistics: Introducing

the Normal distribution

using graphing

technology

Nick Thorpe [email protected]

Data on the number of mobile phone subscribers per 1000 (MEI LDS)

The data below shows the number of mobile phone subscribers per 1000 in the 217 countries from

the MEI LDS where data is available.

1. Find the value of

(a) mean + 1 S.D = (b) mean – 1 S.D =

(c) Work out the percentage of the 217 countries in the interval mean ± 1 S.D.

2. Find the value of

(a) mean + 2 S.D = (b) the mean – 2 S.D =

(c) Work out the percentage of the 217 countries in the interval mean ± 2 S.D.

3. Find the percentage of the 217 countries in the interval mean ± 3 S.D.

Rank MPS/1000 Rank MPS/1000 Rank MPS/1000 Rank MPS/1000 Rank MPS/1000 Rank MPS/1000 Rank MPS/1000

1 47.8 32 513.4 63 760.6 94 915.9 125 1099.5 156 1219.9 187 1448.3

2 53.5 33 516.8 64 763.3 95 918.6 126 1107.6 157 1220.7 188 1462.1

3 63.1 34 526.1 65 764.3 96 939.9 127 1111.0 158 1221.3 189 1488.1

4 68.4 35 533.5 66 767.0 97 946.7 128 1111.3 159 1224.6 190 1490.2

5 97.6 36 543.4 67 769.7 98 950.8 129 1111.7 160 1225.3 191 1500.2

6 152.3 37 548.4 68 770.9 99 952.0 130 1111.8 161 1232.6 192 1509.0

7 153.1 38 565.6 69 779.5 100 954.2 131 1113.5 162 1235.6 193 1535.8

8 173.0 39 566.3 70 789.2 101 956.6 132 1113.5 163 1239.3 194 1539.3

9 202.7 40 567.1 71 789.7 102 971.1 133 1114.0 164 1241.1 195 1556.7

10 212.4 41 584.4 72 792.5 103 972.1 134 1115.6 165 1260.9 196 1576.3

11 216.1 42 585.0 73 807.1 104 975.2 135 1118.1 166 1267.4 197 1600.7

12 251.7 43 585.3 74 809.5 105 994.6 136 1119.0 167 1288.1 198 1616.3

13 254.4 44 608.7 75 810.8 106 1012.8 137 1119.7 168 1293.5 199 1617.1

14 258.0 45 609.7 76 811.4 107 1015.2 138 1129.6 169 1297.5 200 1629.9

15 259.7 46 636.6 77 821.2 108 1018.4 139 1132.8 170 1303.0 201 1639.2

16 261.2 47 637.2 78 827.5 109 1029.9 140 1138.5 171 1307.4 202 1645.6

17 302.3 48 640.7 79 829.2 110 1044.6 141 1138.6 172 1316.6 203 1652.7

18 309.2 49 649.6 80 837.9 111 1045.3 142 1142.8 173 1325.8 204 1674.2

19 326.0 50 665.2 81 841.2 112 1051.3 143 1144.2 174 1329.7 205 1704.1

20 328.4 51 675.6 82 845.1 113 1053.1 144 1158.9 175 1339.9 206 1732.2

21 368.0 52 680.0 83 851.4 114 1057.2 145 1162.0 176 1362.0 207 1751.8

22 369.1 53 682.8 84 857.1 115 1061.0 146 1163.9 177 1374.1 208 1753.7

23 384.8 54 693.7 85 862.5 116 1061.3 147 1167.4 178 1389.6 209 1768.9

24 413.4 55 700.8 86 865.3 117 1066.9 148 1173.9 179 1389.7 210 1838.3

25 416.7 56 716.7 87 867.7 118 1068.5 149 1179.4 180 1412.1 211 1876.2

26 455.3 57 719.0 88 873.3 119 1084.1 150 1182.8 181 1413.9 212 1938.1

27 461.2 58 719.4 89 888.0 120 1088.2 151 1185.1 182 1419.2 213 1969.4

28 478.5 59 720.2 90 893.0 121 1088.6 152 1190.7 183 1422.8 214 2014.8

29 481.7 60 723.0 91 893.3 122 1090.3 153 1197.4 184 1426.2 215 2306.2

30 495.3 61 743.7 92 909.5 123 1093.5 154 1197.9 185 1429.6 216 2447.2

31 513.2 62 753.9 93 914.1 124 1098.7 155 1213.4 186 1435.1 217 2743.6

THE NORMAL DISTRIBUTION

The normal curve

Useful rules of thumb

The Standard Normal Distribution

For the standard normal distribution

𝑍~N(0,1)

mean = 0; standard deviation =1

• Symmetrical

• Bell shaped

• Horizontal axis is our random variable

• Area under curve represents frequency

• Usually we scale total area to be 1 so then area represents

probability

• We write 𝑋~N(𝜇, 𝜎2)

𝑦 =1

𝜎√2𝜋𝑒

− (𝑥−𝜇)2

2𝜎2

If 𝑋~N(𝜇, 𝜎2) then 𝑍 = ~N(0,1)

Using the symmetry of the Normal distribution

The shaded area under the Normal curve below is 𝑝. Find the areas of other regions shown

𝒑

Calculating probabilities from the normal distribution using the CG-20

If 𝑋~𝑁(71,192) calculate 𝑃(𝑋 > 100).

(1) Choose option 2 on the MENU.

KEY PRESSES p2

(2) Choose the distribution menu (DIST). Any entries in the lists can be ignored.

KEY PRESSES y

(3) Choose Normal Distribution (NORM).

KEY PRESSES q

(4) Choose Normal Cumulative distribution (Ncd)*.

KEY PRESSES w

(5) Enter the parameters, ensuring the lower/upper bound is a large number of SDs from the mean.

KEY PRESSES wBNl

KEY PRESSES d

(7) By scrolling down to the bottom we have the opportunity to visualise the probability and the normal curve.

KEY PRESSES u

(8) Just as above, we can read off the required probabilities and return by pressing EXIT.

KEY PRESSES d

*OTHER OPTIONS:

Npd gives the height of the normal curve – this is rarely useful when calculating probabilities.

InvN returns the limits for given probabilities.

OCR A level Specimen Paper 2 Q7

(i) The heights of English men aged 25 to 34 are normally distributed with mean 178cm and standard deviation 8cm. Three English men aged 25 to 34 are chosen at random. Find the probability that all three of them have a height less than 194cm. [3] (ii) The diagram shows the distribution of heights of Scottish women aged 25 to 34.

It is given that the distribution is approximately normal. Use the diagram to estimate the standard

deviation of these heights, explaining your method. [3]

AQA A level Specimen Paper 3 Q13

In the South West region of England, 100 households were randomly selected and, for each household,

the weekly expenditure, £X, per person on food and drink was recorded.

The maximum amount recorded was £40.48 and the minimum amount recorded was £22.00. The results

are summarised below, where �̅� denotes the sample mean.

∑ 𝑥 = 3046.14 ∑(𝑥 − �̅�)2 = 1746.29

(a) (i) Find the mean and the standard deviation of X. [2]

(a) (ii)Using your results from part (a)(i) and other information given, explain why the normal distribution

can be used to model X. [2]

(a) (iii)Find the probability that a household in the South West spends less than £25.00 on food and drink

per person per week. [2]

(b) For households in the North West of England, the weekly expenditure, £Y, per person on food and

drink can be modelled by a normal distribution with mean £29.55

It is known that P(Y < 30) = 0.55

Find the standard deviation of Y, giving your answer to one decimal place. [3]

Statistics:

Introducing the

Normal

distribution

using graphing

technology

The Normal Distribution

This probability model fits many real world situations.

The data on page 2 is taken from the MEI LDS and gives

the number of mobile phone subscribers (per 1000 of

population).

Use the data to answer the questions at the bottom on

page 2.

The Normal Distribution

What do you think the histogram looks

like?

What do you think the histogram looks like?

‘The average man’

The chest circumferences of 5738

Scottish soldiers. Quetelet,1846

The normal curve

• Symmetrical

• Bell shaped

• A continuous random variable plotted on horizontal axis

• Area under curve represents frequency

• Usually we scale total area under curve to be 1 so then area represents probability

Use the CG-20 to check these

proportions

REMINDER: OUTLIERS

What is ‘too’ big or ‘too’ small?

Values which are more than 1.5 x interquartile range

above upper quartile or below lower quartile

OR

Values which are more than 2 standard deviations away

from the mean (if distribution is approx symmetric)

Using the symmetry of the

standard normal distribution

Normal?

The results of a national exam have a mean of

71% and a standard deviation of 19%.

Do you expect a normal distribution to be a good

model of the scores of this exam?

The Standard Normal Distribution

Converting to standard normal

1 7

Normal Distribution Tables

Normal Distribution Tables

• Registered charity committed to improving

mathematics education

• Independent UK curriculum development body

• We offer continuing professional development

courses, provide specialist tuition for students

and work with employers to enhance

mathematical skills in the workplace

• We also pioneer the development of innovative

teaching and learning resources