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The calculations involved in the test transform the deviations into a single chi square value by using the following formula.
chi square = (observed - expected)2 + (observed - expected)2
expected expected
This first group represents This second group represents
the number of individuals the number individuals we
we observed in our proposed observed that did not fit into
4cm/kg category, we will our proposed 4cm/kg category
call this category class 1. we will call this category class 2.
This class will include all This class will include all
individuals from 3.5cm/kg individuals not in the 3.5cm/kg
to 4.5cm/kg. to 4.5 cm/kg class.
*In this class we are *In this class we are expecting
expecting to have all 16 to have zero observations,
subject observations, so remember our hypothesis, so
the expected value is 16 the expected value is zero
Procedure.
Let's use data from a group of people like those in exercise one of the previous laboratory exercise regarding the scientific method and hypothesis testing. Let's assume that we have the following results. Of the 16 people, the sample person plus 15 others, we have the following values relating height in centimeters to weight in kilograms. Two of the people sampled gave results between 3.5 and 4.5cm/kg and the remaining fourteen people gave results other than 3.5 to 4.5cm/kg. So we use the formula to calculate the chi square value.
chi square = (2 - 16)2 + (14-0)2 first take 2 minus 16, a negative 14 is ok
16 16 you may not always have a negative answer
then take 14 minus 0, answer 14
chi square = (-14)2 + 142 then square the negative 14 from the above
16 16 step and get 196, (-14 times -14)
then square the 14 from the above
step and get 196, (14 times 14)
chi square = 196 + 196 now divide 196 by 16
16 16 and divide 196 by 16
chi square = 12.25 +12.25 add the two values together
chi square = 24.50
The next step is to determine the degrees of freedom, which is equal to one less than the number of categories or classes possible. The number of classes we have in the above sample set is 2, so the degrees of freedom, (df) is equal to 2 - 1, or 1 df. The classes are as follows.
Class one, values from 3.5 to 4.5cm/kg.
Class two, values other than 3.5 to 4.5cm/kg.
We now need to use the table below to determine whether or not our chi square value is significant or insignificant.
Probability values
0.90 0.80 0.70 0.50 0.30 0.20 0.10 0.05 0.01
df
1 0.016 0.064 0.15 0.46 1.07 1.64 2.71 3.84 6.64
2 0.21 0.45 0.71 1.39 2.41 3.22 4.61 5.99 9.21
3 0.58 1.00 1.42 2.37 3.67 4.64 6.25 7.82 11.35
4 1.06 1.65 2.20 3.36 4.88 5.99 7.78 9.49 13.28
5 1.61 2.34 3.00 4.35 6.06 7.29 9.24 11.07 15.09
Using 1 degree of freedom, our calculated chi square value of 24.50 is greater than 3.84. The chi square value 3.84 corresponds to a probability value of 0.05. This means that random sampling error cannot be the cause of our measured results; something else has produced the 24.50 chi square value.
We must make one final decision as to whether the observed results can be explained by the hypothesis. We must decide on the level of significance to use. The most widely used level of significance is 0.05. Using this level helps to minimize the chance of accepting a wrong hypothesis, while at the same time not rejecting a correct hypothesis.
A chi square value of 3.84 has a probability value of 0.05.
A chi square value of 24.50 has a probability value of much less than 0.01
The probability value associated with our calculated chi square value of 24.50 is much less than 0.05, farther to the right of the probability values and much less than 0.01. This tells us that sampling error is not the cause of this calculated chi square value the cause must be our hypothesis.
Since our calculated chi square probability value is much less than 0.05, the deviation between the observed and expected values is significant, therefore the hypothesis proposed to account for the data collected must be rejected.
Therefore we reject our hypothesis that height in centimeters when divided by weight in kilograms is equal to 4cm/kg.
**If the calculated chi square value were less than 3.84 then the hypothesis would be accepted because the deviation between the observed and expected values is insignificant.
Sample size and chi square test validity.
The chi square test is valid only when whole numbers are used; it will not work with values that are expressed in fractional or percentage form. The test is also very sensitive to small sample sizes. As a rule of thumb, the expected number in each class should be greater than or equal to five for the test to be accurate. Finally, the decision to accept or reject the hypothesis must not be taken as proof that the data collected are or are not correct and reflect all people that could be sampled. A statistical test can only support or fail to support a proposed hypothesis. The test can never actually prove the hypothesis is true or false. By failing to reject a proposed hypothesis, we do not necessarily mean that there is nothing wrong with it, only that we have failed to detect anything wrong with it.
Student data and determination of hypothesis correctness
Enter all of the following required work on the following blank sheet or sheets of paper. At this point you are required to use your data collected from the first laboratory exercise and determine if the proposed hypothesis is accepted or must be rejected based on the data you collected. You may use either the height to weight data or the arm length to body height data. Be sure to state your hypothesis on the chi square calculation page or pages. You need only perform one chi square test with one set of data. Use the same probability value of 0.05 for your chi square calculation. Keep in mind that your data will most likely be different than the data used in the previous example. The pattern for calculating chi square will be the same only the numbers will change because you have different data. Be sure to show all steps of the test as in the example. Once you have finished with the chi square calculation explain your reason for accepting or rejecting your stated hypothesis. You will need to keep all of this information for later – both the lab quiz and a subsequent lab will require you to refer to this lab exercise.
You are strongly encouraged to work through the sample chi square test and make sure that you thoroughly understand the sample chi square test before proceeding with your own data and chi square test. Keep in mind that the sample chi square test may or may not be indicative of your data and your final chi square test outcome.
When you have finished this lab proceed with the quiz. Laboratory quiz for this lab is worth 10 points.
All of your chi square work must be placed below.
You may use either the height to weight data or the arm length to body height data; state which data set you are using. You may need this during the quiz.
State your hypothesis. You may need this during the quiz.
Be sure to show all steps of the test as in the example. You may need this during the quiz.
Once you have finished with the chi square calculation explain your reason for accepting or rejecting your stated hypothesis. You may need this during the quiz.

The calculations involved in the test transform the deviations into a single chi square value by using the following formula.

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