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                      neville
               Welcome to DISCUS 8 !

        Workbook for Hypothesis Testing




          copyright Neville Hunt & Sidney Tyrrell 1995




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DISCUS 8 is a workbook about hypothesis testing.

It begins by looking at the traditional problem of tossing a
coin to see if it is fair. This is used to introduce the important
ideas of statistical significance, Type I and Type II errors,
and the power of a test.

Later we look more formally at the relationship between
power and significance in the context of tests of means.

Finally we look at how to conduct a chi-squared test of
association in a contingency table.

Self-teaching notes and questions are provided on separate
workcards which are designed to be USED with the workbook.

There is a spare spreadsheet at the end of the workbook which
can be used for recording results or doing calculations.




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             USING the fair coins spreadsheet

Aim: To introduce the concept of statistical significance.
A coin is tossed 8 times and the number of heads recorded.
Press F9 to toss the coins again.

Based on the number of heads a decision must be made on
whether the coin is fair (i.e. accepting the null hypothesis) or
unfair (i.e. rejecting the null hypothesis).
Rejecting a fair coin is a mistake, called a Type I error.
The question is, how many heads do we need to get to decide
that the coin is unfair? 8 heads out of 8? What about 7 out of 8?
Don't forget 0 out of 8. You must decide where to draw the line(s).
Actually, all the coins used are fair so ideally you should never
reject the null hypothesis! How well can you do?

Use workcard 8.1 to learn more                    DISCUS 8.1




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                                 8.1       Testing fair coins

 Hº: p=0.5




                                                                                                    Coins
 H¹: p#0.5
                 Heads     6       J J J J J   J
 p=Pr(Caras)      Distribuição do número de Caras assumindo uma moeda honesta
 Amostras
                    35.0
     58                                                              Accept Hº
                    30.0
 Hº rejeitar
                    25.0
                    20.0
% Hº rejeição
                    15.0
     0.0
                    10.0
   Nível de
 Significancia       5.0
      (%)            0.0
    14.8                   0           1         2      3        4       5         6     7     8

                 Prob      0.4         3.1     10.9   21.9      27.3   21.9      10.9   3.1   0.4    %




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uma moeda honesta




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            USING the unfair coins spreadsheet

Aim: To understand the concept of the power of a test.
This is an extension of DISCUS 8.1 to consider unfair coins.
How good (or powerful) is our test at detecting when a coin is unfair?

If the actual value of p is NOT 0.5, we would like the test to reject
the null hypothesis (that the coin is fair) 100% of the time.
In reality the test sometimes gets it wrong and accepts the coin as
fair when in fact it is not. This is called a Type II error.
A powerful test minimises the risk of making this error.

There are three things on the spreadsheet which affect the power
of the test. Two you can change, one is fixed.
Can you see what they are and how they affect power?

Use workcard 8.2 to learn more.                       DISCUS 8.2




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                               8.2   Testing unfair coins

Hº: p=0.5




                                                                                             Coins
H¹: p#0.5
                 Heads     7         J J J J J J  J
p=Pr(Heads)       Distribution of number of heads assuming a fair coin
  Actual p
     0.6            35.0
                                                       Accept Hº
 Samples            30.0
    101             25.0
 Hº rejects         20.0
     22             15.0
% Hº rejects
                    10.0
    21.8
                     5.0
Sig. level (%)
                     0.0
     7.0                   0         1      2      3        4       5       6     7     8

                 Prob      0.4       3.1   10.9   21.9   27.3      21.9   10.9   3.1   0.4    %




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   USING the spreadsheet for testing the significance of means

Aim: To gain a better understanding of the concepts involved
     in hypothesis tests for a population mean. In particular,
     what is meant by significance and the power of a test.
The given population has a mean of 0 and s.d. of 5.

The screen shows, in black, the distribution of the sampling
means for your chosen sample size.
Superimposed in red is the distribution of sampling means one
would expect if the null hypothesis were true.

You can change any of the numbers in blue. See how different
values affect the rejection regions and the power of the test.

Use workcard 8.3 to learn more.                       DISCUS 8.3




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                         8.3 Teste de significancia para média

                                                                 Verdadeira µ

           Rejeitar Hº                           Rejeitar Hº




   160.0   165.0    170.0           175.0    180.0            185.0    190.0       195.0    200.0


 Hipótese Nula Hº:µ =                  174            Tamanho da Amostra (n)                25
 Média População (µ)                   184              Nível de Significancia              0.1
 Desv. Pad. População                  14                     Poder do Teste               0.973




z                                     1.644854       Upcrit                                178.6056
step                                     0.592       upper                                    193.8
sterror                                    2.8       lower                                    164.2

step0                                   0.392    step1                               0.392 step2       0
black                               normal       rednormal                                 blue       shading
                            174.2    0.000312    ##                            0.000311672 178.6056    0
                            174.6    0.000504    ##                            0.000503787 184.5256    0
                            175.0    0.000799    ##                            0.000798514 178.6056    0
                            175.4    0.001241    ##                            0.001241099 178.6056    0
                            175.8    0.001892    ##                            0.001891551 178.6056    0
                            176.2    0.002827    ##                            0.002826947 178.8653    0
                            176.6    0.004143    ##                            0.004142905 178.8653    0
                            176.9    0.005954    ##                            0.005953607 179.125     0
                            177.3     0.00839    ##                            0.008389638 179.125     0
                            177.7    0.011593    ##                            0.011592952 179.3848    0
                            178.1    0.015708    ##                            0.015708427 179.3848    0
                            178.5    0.020872    ##                            0.020871766 179.6445    0
                            178.9    0.027194    ##                            0.027194026 179.6445    0
                            179.3    0.034744    ##                            0.034743668 179.9042    0
                            179.7    0.043528    ##                            0.043527692 179.9042    0




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 180.1   0.053474   ##                     0.053474095   180.1639   0
 180.5   0.064418   ##                     0.064418273   180.1639   0
 180.9   0.076096   ##                     0.076096124   180.4236   0
 181.3   0.088146   ##                     0.088146247   180.4236   0
 181.6   0.100123   ##                      0.10012279   180.6834   0
 182.0   0.111519   ##                     0.111519262   180.6834   0
 182.4   0.121802   ##                     0.121802067   180.9431   0
 182.8   0.130451   ##                     0.130450955   180.9431   0
 183.2   0.137002   ##                     0.137002247   181.2028   0
 183.6    0.14109   ##                     0.141089907   181.2028   0
 184.0   0.142479   ##                     0.142479386   181.4625   0
 184.4    0.14109   ##                     0.141089907   181.4625   0
 184.8   0.137002   ##                     0.137002247   181.7222   0
 185.2   0.130451   ##                     0.130450955   181.7222   0
 185.6   0.121802   ##                     0.121802067    181.982   0
 186.0   0.111519   ##                     0.111519262    181.982   0
 186.4   0.100123   ##                      0.10012279   182.2417   0
 186.7   0.088146   ##                     0.088146247   182.2417   0
 187.1   0.076096   ##                     0.076096124   182.5014   0
 187.5   0.064418   ##                     0.064418273   182.5014   0
 187.9   0.053474   ##                     0.053474095   182.7611   0
 188.3   0.043528   ##                     0.043527692   182.7611   0
 188.7   0.034744   ##                     0.034743668   183.0208   0
 189.1   0.027194   ##                     0.027194026   183.0208   0
 189.5   0.020872   ##                     0.020871766   183.2806   0
 189.9   0.015708   ##                     0.015708427   183.2806   0
 190.3   0.011593   ##                     0.011592952   183.5403   0
 190.7    0.00839   ##                     0.008389638   183.5403   0
 191.1   0.005954   ##                     0.005953607      183.8   0
 191.4   0.004143   ##                     0.004142905      183.8   0
 191.8   0.002827   ##                     0.002826947   184.0597   0
 192.2   0.001892   ##                     0.001891551   169.3944   0
 192.6   0.001241   ##                     0.001241099   169.3944   0
 193.0   0.000799   ##                     0.000798514   163.4744   0
 193.4   0.000504   ##                     0.000503787   163.4744   0
 193.8   0.000312   ##                     0.000311672   169.3944   0
                    ##                               0   169.3944   0
                    ##                     0.142464286   169.1347   0
                                                         169.1347   0
                                                          168.875   0
                                                          168.875   0
                                                         168.6152   0
                                                         168.6152   0
                                                         168.3555   0
                                                         168.3555   0
                                                         168.0958   0
                                                         168.0958   0
                                                         167.8361   0
                                                         167.8361   0
                                                         167.5764   0
                                                         167.5764   0
                                                         167.3166   0
                                                         167.3166   0
                                                         167.0569   0
                                                         167.0569   0
                                                         166.7972   0




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                                           166.7972   0
                                           166.5375   0
                                           166.5375   0
                                           166.2778   0
                                           166.2778   0
                                            166.018   0
                                            166.018   0
                                           165.7583   0
                                           165.7583   0
                                           165.4986   0
                                           165.4986   0
                                           165.2389   0
                                           165.2389   0
                                           164.9792   0
                                           164.9792   0
                                           164.7194   0
                                           164.7194   0
                                           164.4597   0
                                           164.4597   0
                                              164.2   0
                                              164.2   0




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Locrit     169.3944
label     Rejeitar Hº
label2    Verdadeira µ

step3       -0.12014   step4       0.3798602
green-lowershading     gr-upper   shading    box       mu
    174.2          0          184          0     164.2       0
    174.2 0.000312      178.60559          0     193.8       0
 174.0799 0.000268      178.60559 0.0222725       184        0
 174.0799          0    178.98545 0.0286604       184 0.232143
 173.9597          0    178.98545          0 185.184 0.232143
 173.9597 0.00023       179.36531          0 184.592 0.232143
 173.8396 0.000197      179.36531 0.0362079
 173.8396          0    179.74517 0.0449088
 173.7194          0    179.74517          0
 173.7194 0.000168      180.12503          0
 173.5993 0.000144      180.12503 0.0546847
 173.5993          0    180.50489 0.0653743
 173.4792          0    180.50489          0
 173.4792 0.000122      180.88475          0
  173.359 0.000104      180.88475 0.0767283




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 173.359          0   181.26461   0.0884119
173.2389          0   181.26461           0
173.2389   8.84E-05   181.64447           0
173.1187   7.49E-05   181.64447   0.1000168
173.1187          0   182.02433   0.1110816
172.9986          0   182.02433           0
172.9986   6.33E-05   182.40419           0
172.8785   5.34E-05   182.40419   0.1211206
172.8785          0   182.78405   0.1296584
172.7583          0   182.78405           0
172.7583    4.5E-05   183.16391           0
172.6382   3.79E-05   183.16391   0.1362669
172.6382          0   183.54377   0.1406006
 172.518          0   183.54377           0
 172.518   3.18E-05   183.92363           0
172.3979   2.66E-05   183.92363   0.1424264
172.3979          0   184.30349   0.1416449
172.2778          0   184.30349           0
172.2778   2.23E-05   184.68335           0
172.1576   1.86E-05   184.68335   0.1382987
172.1576          0   185.06321   0.1325691
172.0375          0   185.06321           0
172.0375   1.55E-05   185.44307           0
171.9173   1.29E-05   185.44307   0.1247594
171.9173          0   185.82293   0.1152686
171.7972          0   185.82293           0
171.7972   1.07E-05    186.2028           0
171.6771   8.87E-06    186.2028   0.1045577
171.6771          0   186.58266   0.0931124
171.5569          0   186.58266           0
171.5569   7.33E-06   186.96252           0
171.4368   6.06E-06   186.96252   0.0814078
171.4368          0   187.34238   0.0698766
171.3166          0   187.34238           0
171.3166   4.99E-06   187.72224           0
171.1965   4.11E-06   187.72224   0.0588849
171.1965          0    188.1021   0.0487173
171.0764          0    188.1021           0
171.0764   3.37E-06   188.48196           0
170.9562   2.76E-06   188.48196   0.0395703
170.9562          0   188.86182   0.0315546
170.8361          0   188.86182           0
170.8361   2.26E-06   189.24168           0
170.7159   1.85E-06   189.24168   0.0247037
170.7159          0   189.62154   0.0189876
170.5958          0   189.62154           0
170.5958    1.5E-06    190.0014           0
170.4757   1.22E-06    190.0014   0.0143279
170.4757          0   190.38126   0.0106146
170.3555          0   190.38126           0
170.3555   9.94E-07   190.76112           0
170.2354   8.06E-07   190.76112   0.0077202
170.2354          0   191.14098   0.0055127
170.1152          0   191.14098           0
170.1152   6.52E-07   191.52084           0
169.9951   5.26E-07   191.52084   0.0038646




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169.9951          0    191.9007    0.0026598
 169.875          0    191.9007            0
 169.875   4.24E-07   192.28056            0
169.7548   3.41E-07   192.28056    0.0017973
169.7548          0   192.66042    0.0011923
169.6347          0   192.66042            0
169.6347   2.74E-07   193.04028            0
169.5145    2.2E-07   193.04028    0.0007765
169.5145          0   193.42014    0.0004965
169.3944          0   193.42014            0
169.3944   1.76E-07        193.8           0
                           193.8   0.0003117




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            USING the chi-squared test spreadsheet

Aim: To demonstrate the calculations involved in a chi-squared
     test of association and understand the idea of
     independence in a two-way table.
A 3x2 contingency table is given, highlighted in pale blue which
shows the sex and eye-colour of a random sample of students.
The lower table shows the numbers expected in each category
IF sex and eye-colour are independent.

The chi-squared statistic measures how well the Observed and
Expected frequencies match. The larger the differences between
them, the larger the chi-squared statistic. Whether or not the
differences can be regarded as statistically significant depends
on the significance level used. See if you can discover how the test works.

Use workcard 8.4 to learn more.                      DISCUS 8.4




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                          8.4 Chi-squared test of association

Hº: Eye colour independent of sex            H¹: Eye colour is related to sex

 Observed    Male   Female      Totals              O-E     Male    Female   Totals
   Blue       48      12         60                 Blue     6.1     -6.1     0.0
  Brown       12      10         22                Brown    -3.4      3.4     0.0
  Other       14      10         24                Other    -2.8      2.8     0.0
  Totals      74      32         106               Totals    0.0      0.0     0.0

 Expected    Male   Female      Totals          (O-E)²/E    Male    Female   Totals
   Blue      41.9    18.1        60               Blue      0.89     2.06     2.96
  Brown      15.4     6.6        22              Brown      0.73     1.70     2.43
  Other      16.8     7.2        24              Other      0.45     1.05     1.50
  Totals      74      32         106             Totals     2.08     4.81     6.89

Significance level               5.0%        Observed Chi² statistic       6.89
Critical value of Chi²           5.99        Sufficient evidence to reject Hº




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