HKU/SON/2018 1

STATISTICS (NURS4302) School of Nursing, The University of Hong Kong

Formula Sheet

1. If ),(~ pnBinX , then mean = np and standard deviation = )1( pnp − .

2. If ),(~ 2σµNX , then mean = µ and standard deviation =σ .

3. A (1-α)100% confidence interval for a population mean is

))2/,(,)2/,(( SEMdftXSEMdftX αα +− where X = sample mean, SEM = sample standard error for the mean (= nSD / ), )2/,( αdft = t critical value from a t distribution, df = degrees of freedom = n-1, α = level of significance.

4. A (1-α)100% confidence interval for a population proportion is

))2/(,)2/(( SEPzpSEPzp αα +− where p = sample proportion

SEP = sample standard error for the proportion, i.e. npp /)1( − ,

)2/(αz = z value from the standard normal distribution, α is called the level of significance.

5. A (1-α)100% confidence interval for the difference between two population means is

))2/,(,)2/,((2121 2121 XXXX sedftXXsedftXX

−−+−−− αα

where ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

−2

22

1

21

21 nSD

nSD

se XX and ( )

( ) ( )1/

1/

2

22

22

1

21

21

2

2

22

1

21

−+

−

+=

nnSD

nnSD

df nSD

nSD

.

6. A (1-α)100% confidence interval for the difference between two population proportions is

( 1p - 2p - )2/(αz2

22

1

11 )1()1(npp

npp −

+−

, 1p - 2p + )2/(αz2

22

1

11 )1()1(npp

npp −

+−

)

where )2/(αz is a critical value from the standard Normal distribution, and 1n and

2n are the sample sizes of groups 1 and 2 respectively.

7. When testing 00 : µµ =H , the test statistics is

nSDX

T/

0µ−=

which follows a t-distribution with degrees of freedom = n – 1.

HKU/SON/2018 2

8. For testing 210 : µµ =H , the test statistics is

21

21

XXseXXT−

−=

where ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

−2

22

1

21

21 nSD

nSD

se XX. Under 0H , T follows a t-distribution with degrees

of freedom equal ( )( ) ( )

1/

1/

2

22

22

1

21

21

2

2

22

1

21

−+

−

+=

nnSD

nnSD

df nSD

nSD

.

9. The test statistics for a 2χ test is

∑−

=expected

)expectedobserved( 22X

which follows a chi-square distribution with degrees of freedom equals to (no. of rows – 1)(no. of columns – 1).

10. Cramer’s V index = )1(

2

−=

knXV where 2X is the test statistics of the 2χ test

and k is the minimum of the numbers of rows and columns.

11. The Pearson correlation coefficient is

∑∑∑

==

=

−−

−−=

n

i in

i i

in

i i

YYXX

YYXXr

122

1

1

)()(

))(( =

))((1

221

22

1

∑∑∑

==

=

−−

−n

i in

i i

in

i i

YnYXnX

YXnYX.

12. When testing 0:0 =ρH , the test statistics is

212r

nrt−

−=

which follows a t-distribution with degrees of freedom = n – 2.

13. For a simple linear regression with regression equation: bXaY += ,

∑∑

=

=

−

−−= n

i i

n

i ii

XX

YYXXb

12

1

)(

))(( =

∑∑

=

=

−

−n

i i

n

i ii

XnX

YXnYX

122

1 , and XbYa −= .

14. Classification of correlation

0.90 – 1.00 0.70 – 0.89 0.50 – 0.69 0.30 – 0.49 0.00 – 0.29

15. Classification of Cramer’s V index

0.75 – 1.00 0.50 – 0.74 0.25 – 0.49 0.00 – 0.24

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Table 1. Standard Normal Table

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359 0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753 0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141 0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517 0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879

0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224 0.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549 0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852 0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133 0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621 1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830 1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015 1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177 1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319

1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441 1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545 1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633 1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706 1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767

2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817 2.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4857 2.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .4890 2.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4916 2.4 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .4936

2.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .4952 2.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4964 2.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4974 2.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4981 2.9 .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .4986

3.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990 3.1 .4990 .4991 .4991 .4991 .4992 .4992 .4992 .4992 .4993 .4993 3.2 .4993 .4993 .4994 .4994 .4994 .4994 .4994 .4995 .4995 .4995 3.3 .4995 .4995 .4995 .4996 .4996 .4996 .4996 .4996 .4996 .4997 3.4 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4998

Example: 𝑃 0 < 𝑍 < 1.96 = 0.475.

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Table 2. Standard Normal table

Area .000 .001 .002 .003 .004 .005 .006 .007 .008 .009

0.00 .0000 .0025 .0050 .0075 .0100 .0125 .0150 .0175 .0200 .0225 0.01 .0250 .0275 .0300 .0325 .0350 .0376 .0401 .0426 .0451 .0476 0.02 .0501 .0526 .0551 .0576 .0601 .0627 .0652 .0677 .0702 .0727 0.03 .0752 .0777 .0802 .0828 .0853 .0878 .0903 .0928 .0953 .0979 0.04 .1004 .1029 .1054 .1079 .1105 .1130 .1155 .1180 .1206 .1231

0.05 .1256 .1281 .1307 .1332 .1357 .1383 .1408 .1433 .1459 .1484 0.06 .1509 .1535 .1560 .1585 .1611 .1636 .1661 .1687 .1712 .1738 0.07 .1763 .1789 .1814 .1840 .1865 .1891 .1916 .1942 .1967 .1993 0.08 .2018 .2044 .2070 .2095 .2121 .2147 .2172 .2198 .2224 .2249 0.09 .2275 .2301 .2326 .2352 .2378 .2404 .2430 .2455 .2481 .2507

0.10 .2533 .2559 .2585 .2611 .2637 .2663 .2689 .2715 .2741 .2767 0.11 .2793 .2819 .2845 .2871 .2897 .2923 .2949 .2976 .3002 .3028 0.12 .3054 .3081 .3107 .3133 .3160 .3186 .3212 .3239 .3265 .3292 0.13 .3318 .3345 .3371 .3398 .3424 .3451 .3477 .3504 .3531 .3557 0.14 .3584 .3611 .3638 .3664 .3691 .3718 .3745 .3772 .3799 .3826

0.15 .3853 .3880 .3907 .3934 .3961 .3988 .4015 .4042 .4070 .4097 0.16 .4124 .4151 .4179 .4206 .4234 .4261 .4288 .4316 .4343 .4371 0.17 .4399 .4426 .4454 .4482 .4509 .4537 .4565 .4593 .4621 .4649 0.18 .4676 .4704 .4732 .4761 .4789 .4817 .4845 .4873 .4901 .4930 0.19 .4958 .4986 .5015 .5043 .5072 .5100 .5129 .5157 .5186 .5215

0.20 .5244 .5272 .5301 .5330 .5359 .5388 .5417 .5446 .5475 .5504 0.21 .5533 .5563 .5592 .5621 .5651 .5680 .5709 .5739 .5769 .5798 0.22 .5828 .5858 .5887 .5917 .5947 .5977 .6007 .6037 .6067 .6097 0.23 .6128 .6158 .6188 .6219 .6249 .6280 .6310 .6341 .6371 .6402 0.24 .6433 .6464 .6495 .6526 .6557 .6588 .6619 .6650 .6682 .6713

0.25 .6744 .6776 .6807 .6839 .6871 .6903 .6934 .6966 .6998 .7030 0.26 .7063 .7095 .7127 .7159 .7192 .7224 .7257 .7290 .7322 .7355 0.27 .7388 .7421 .7454 .7487 .7520 .7554 .7587 .7621 .7654 .7688 0.28 .7721 .7755 .7789 .7823 .7857 .7891 .7926 .7960 .7995 .8029 0.29 .8064 .8098 .8133 .8168 .8203 .8238 .8274 .8309 .8344 .8380

0.30 .8416 .8451 .8487 .8523 .8559 .8596 .8632 .8668 .8705 .8742 0.31 .8778 .8815 .8852 .8890 .8927 .8964 .9002 .9039 .9077 .9115 0.32 .9153 .9191 .9230 .9268 .9307 .9345 .9384 .9423 .9462 .9502 0.33 .9541 .9581 .9620 .9660 .9700 .9741 .9781 .9822 .9862 .9903 0.34 .9944 .9985 1.003 1.007 1.011 1.015 1.019 1.024 1.028 1.032

0.35 1.036 1.041 1.045 1.049 1.054 1.058 1.063 1.067 1.071 1.076 0.36 1.080 1.085 1.089 1.094 1.098 1.103 1.108 1.112 1.117 1.122 0.37 1.126 1.131 1.136 1.141 1.146 1.150 1.155 1.160 1.165 1.170 0.38 1.175 1.180 1.185 1.190 1.195 1.200 1.206 1.211 1.216 1.221 0.39 1.227 1.232 1.237 1.243 1.248 1.254 1.259 1.265 1.270 1.276

0.40 1.282 1.287 1.293 1.299 1.305 1.311 1.317 1.323 1.329 1.335 0.41 1.341 1.347 1.353 1.359 1.366 1.372 1.379 1.385 1.392 1.398 0.42 1.405 1.412 1.419 1.426 1.433 1.440 1.447 1.454 1.461 1.468 0.43 1.476 1.483 1.491 1.499 1.506 1.514 1.522 1.530 1.538 1.546 0.44 1.555 1.563 1.572 1.580 1.589 1.598 1.607 1.616 1.626 1.635

0.45 1.645 1.655 1.665 1.675 1.685 1.695 1.706 1.717 1.728 1.739 0.46 1.751 1.762 1.774 1.787 1.799 1.812 1.825 1.838 1.852 1.866 0.47 1.881 1.896 1.911 1.927 1.943 1.960 1.977 1.995 2.014 2.034 0.48 2.054 2.075 2.097 2.120 2.144 2.170 2.197 2.226 2.257 2.290 0.49 2.326 2.366 2.409 2.457 2.512 2.576 2.652 2.748 2.878 3.090

Example: For 𝑃 0 < 𝑍 < 𝑧 = 0.475, 𝑧 = 1.96.

HKU/SON/2018 5

Table 3. t Table

Level of significance for one-sided tests 0.10 0.05 0.025 0.01 0.005

df Level of significance for two-sided tests 0.20 0.10 0.05 0.02 0.01

1 3.078 6.314 12.706 31.821 63.657 2 1.886 2.920 4.303 6.965 9.925 3 1.638 2.353 3.182 4.541 5.841 4 1.533 2.132 2.776 3.747 4.604 5 1.476 2.015 2.571 3.365 4.032

6 1.440 1.943 2.447 3.143 3.707 7 1.415 1.895 2.365 2.998 3.499 8 1.397 1.860 2.306 2.896 3.355 9 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.169

11 1.363 1.796 2.201 2.718 3.106 12 1.356 1.782 2.179 2.681 3.055 13 1.350 1.771 2.160 2.650 3.012 14 1.345 1.761 2.145 2.624 2.977 15 1.341 1.753 2.131 2.602 2.947

16 1.337 1.746 2.120 2.583 2.921 17 1.333 1.740 2.110 2.567 2.898 18 1.330 1.734 2.101 2.552 2.878 19 1.328 1.729 2.093 2.539 2.861 20 1.325 1.725 2.086 2.528 2.845

21 1.323 1.721 2.080 2.518 2.831 22 1.321 1.717 2.074 2.508 2.819 23 1.319 1.714 2.069 2.500 2.807 24 1.318 1.711 2.064 2.492 2.797 25 1.316 1.708 2.060 2.485 2.787

26 1.315 1.706 2.056 2.479 2.779 27 1.314 1.703 2.052 2.473 2.771 28 1.313 1.701 2.048 2.467 2.763 29 1.311 1.699 2.045 2.462 2.756 30 1.310 1.697 2.042 2.457 2.750

40 1.303 1.684 2.021 2.423 2.704 50 1.299 1.676 2.009 2.403 2.678 60 1.296 1.671 2.000 2.390 2.660 70 1.294 1.667 1.994 2.381 2.648 80 1.292 1.664 1.990 2.374 2.639

90 1.291 1.662 1.987 2.368 2.632

100 1.290 1.660 1.984 2.364 2.626 110 1.289 1.659 1.982 2.361 2.621 120 1.289 1.658 1.980 2.358 2.617 ∞ 1.282 1.645 1.960 2.326 2.576

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Table 4. z Table

Level of significance for one-sided tests 0.10 0.05 0.025 0.01 0.005 0.001 0.0005

Level of significance for two-sided tests 0.20 0.10 0.05 0.02 0.01 0.002 0.001

1.2816 1.6449 1.9600 2.3263 2.5758 3.0903 3.2906

HKU/SON/2018 7

Table 5. 2χ table

Level of significance df 0.05 0.025 0.01 0.005 0.001

1 3.84 5.02 6.63 7.88 10.83 2 5.99 7.38 9.21 10.60 13.82 3 7.81 9.35 11.34 12.84 16.27 4 9.49 11.14 13.28 14.86 18.47 5 11.07 12.83 15.09 16.75 20.52

6 12.59 14.45 16.81 18.55 22.46 7 14.07 16.01 18.48 20.28 24.32 8 15.51 17.53 20.09 21.95 26.12 9 16.92 19.02 21.67 23.59 27.88

10 18.31 20.48 23.21 25.19 29.59

11 19.68 21.92 24.72 26.76 31.26 12 21.03 23.34 26.22 28.30 32.91 13 22.36 24.74 27.69 29.82 34.53 14 23.68 26.12 29.14 31.32 36.12 15 25.00 27.49 30.58 32.80 37.70

16 26.30 28.85 32.00 34.27 39.25 17 27.59 30.19 33.41 35.72 40.79 18 28.87 31.53 34.81 37.16 42.31 19 30.14 32.85 36.19 38.58 43.82 20 31.41 34.17 37.57 40.00 45.31

21 32.67 35.48 38.93 41.40 46.80 22 33.92 36.78 40.29 42.80 48.27 23 35.17 38.08 41.64 44.18 49.73 24 36.42 39.36 42.98 45.56 51.18 25 37.65 40.65 44.31 46.93 52.62

26 38.89 41.92 45.64 48.29 54.05 27 40.11 43.19 46.96 49.64 55.48 28 41.34 44.46 48.28 50.99 56.89 29 42.56 45.72 49.59 52.34 58.30 30 43.77 46.98 50.89 53.67 59.70

40 55.76 59.34 63.69 66.77 73.40 50 67.50 71.42 76.15 79.49 86.66 60 79.08 83.30 88.38 91.95 99.61 70 90.53 95.02 100.43 104.21 112.32 80 101.88 106.63 112.33 116.32 124.84

90 113.15 118.14 124.12 128.30 137.21

100 124.34 129.56 135.81 140.17 149.45 110 135.48 140.92 147.41 151.95 161.58 120 146.57 152.21 158.95 163.65 173.62 130 157.61 163.45 170.42 175.28 185.57