# DRAWING INFERENCES FROM DATA

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DRAWING INFERENCES FROM DATA
THE CHI SQUARE TEST

Video on chi square test

As you complete lab exercises you will see that actual data do not always fit the expected pattern exactly. The question then arises “ Is the observed value significantly different from the value that would be expected?” Another way of asking this question is, “Is the difference between the observed and expected values likely due to chance?”

You can answer this question by submitting the data to a mathematical test. One such test is the Chi Square Test.

Let’s say you and your partner decide to measure the number of times one of you could accurately predict the number on a card from a choice of 5 cards.

By chance alone, your partner could guess the number 1 out of 5 times or 20%. This means that for every 100 tries, your partner should be able to guess the number on the card 20 times by chance alone.

However, when you actually conduct the experiment, your partner was able to guess correctly 22 times out of 100 tries. Does this mean that your partner is psychic? How do we determine whether your partner’s actual performance would be outside the realm of chance?

This is where we use the Chi Square analysis.
In the chart below, we will call the correct guesses “HITS” and the incorrect guesses “MISSES”. Set up the chart as follows: EXPECTED OBSERVED HITS 20 22 MISSES 80 78

STEP 2: FORMULATE THE NULL HYPOTHESIS
Make a statement that assumes that there is no difference between the observed and expected values. Such a statement is called the null hypothesis. In this case the null hypothesis might be stated “There is no significant difference between the experimental results and those which would be expected.”

STEP 3: DETERMINE THE CHI SQUARE VALUE
The values from the table are then inserted into a formula that will find a number called the Chi Square (X2) value:

The Greek symbol Σ means that the chi square value is the sum of all terms represented in the formula. In this case, there are two observed values and two expected values. Thus, there will be two parts, or terms, in the equation. + (O +E)2 ________________ E

STEP 4: DETERMINE THE NUMBER OF DEGREES OF FREEDOM
The number of degrees of freedom (d.f.) is always one less than the number of terms in the formula used above. In this case, there is one degree of freedom (2 terms – 1 equals 1).

STEP 5: DETERMINE WHERE YOUR CHI SQUARE VALUE IS IN THE TABLE BELOW:
Now that you know that the chi square value and the number of degrees of freedom, you can then turn to the chi square table below. Compare your chi square value with those in the row that corresponds to one degree of freedom.

Find your chi square value on the chart.

0.25 is located in the body of the chart between 0.15 and 0.46.

STEP 6: DETERMINE THE P VALUE
After determining the position of the number most nearly matching your value, look at the head of the column it is in. The number at the head of the column is the probability (P) that the results obtained in the experiment differ from the expected results by chance. BIOLOGISTS GENERALLY REJECT THE NULL HYPOTHESIS IF THE VALUE OF P IS LESS THAN 0.05.

0.25 falls between the columns headed by P values of 0.50 and 0.70.

In this illustration, the value of P falls between 0.50 and 0.70.
STEP 7: DETERMINE WHETHER YOU WILL ACCEPT OR REJECT THE NULL HYPOTHESIS: In this illustration, the value of P falls between 0.50 and 0.70. Clearly, the experimenter must ACCEPT THE NULL HYPOTHESIS.

What does this mean? Accepting the null hypothesis means that the results of the experiment differ from the expected only by chance. Thus the experimenter can conclude that the subject did not exhibit psychic powers in this particular experiment.

The smaller the p value the more significant the results are said to be!

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