# WELCOME SCREEN

Document Sample

```					6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

neville
Welcome to DISCUS 8 !

Workbook for Hypothesis Testing

copyright Neville Hunt & Sidney Tyrrell 1995

Page 1
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

Page 2
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

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.

Page 3
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

Page 4
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

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?

Page 5
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

Page 6
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

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    %

Page 7
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

uma moeda honesta

Page 8
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

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?

Page 9
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

Page 10
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

8.2   Testing unfair coins

Hº: p=0.5

Coins
H¹: p#0.5
Heads     7         J J J J J J  J
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    %

Page 11
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

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.

Page 12
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

Page 13
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

8.3 Teste de significancia para média

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
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

Page 14
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

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

Page 15
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

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

Page 16
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

Locrit     169.3944
label     Rejeitar Hº

step3       -0.12014   step4       0.3798602
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

Page 17
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

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

Page 18
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

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

Page 19
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

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.

Page 20
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

Page 21
6ed9e22d-f53f-4b80-b839-e7be00ceb82e.xls

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

Page 22

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 2 posted: 7/9/2011 language: English pages: 22