Independence: If the occurrence of one of the factors in a ...



Independence: If the occurrence of one of the factors in a given sample affects the occurrence of the other, then the two factors are not independent of each other.

The χ2 test examines the difference between the ______________________ and ______________________ values.

Part I. Calculating Expected Values:

The following contingency table shows the results of a sample of 400 randomly selected adults classified according to gender and regular exercise.

| |Pets |No Pets |Sum |

|Male | | | |

|Female | | | |

|Sum | | | |

| |Pets |No Pets |Sum |

|Male | | | |

|Female | | | |

|Sum | | | |

Practice – Find the expected values for the following contingency tables.

1. Students who use pencils or pens vs. right-handed or left-handed

Observed Values Expected Values

| |Pencils |Pens |Sum |

|Right- |31 |22 | |

|handed | | | |

|Left- |20 |27 | |

|handed | | | |

| Sum | | | |

The Formal χ2 Test for independence.

Step 1: State the Null Hypothesis (for our class survey)

H0: _________________________ and _____________________ are INDEPENDENT.

The alternative Hypotheis:

H1: _________________________ and _____________________ are NOT INDEPENDENT.

Step 2: Degrees of Freedom

To find the χ2 distribution you must determine the degrees of freedom (df) for each contingency table.

df = (r-1)(c-1) for a contingency table which is r x c in size.

Calculate the df - ___________________________________________________

Step 3: The problem states the significance level

For this problem, let’s say that it is a 5% significance level

Step 4: State the rejection inequality χ2 > k, where k is obtained from the table of critical values.

We reject Ho if ____________________

Step 5: Find χ2using matrices

1) Put Observed values in a 2x2 Matrix A(Note: your calculator automatically finds the

expected values and inserts them into Matrix B)

2) Press STAT , ►, ►, TESTS, go down 3) Press ▼, ▼, to Calculate, Press

to C: χ2 – Test, Press Enter Enter

[pic] [pic]

χ2 = ________________________________

Step 6: If the rejection inequality is true, then we accept H0. If it is not true, then we reject H0.

______________ _______ _______________

χ2 Critical Value

Step 7: Look at p – value. If p > significance level we accept H0. If p < .05, we reject H0.

P = _________________ ___________ .05, which is further evidence to _______________ H0

Step 8: Conclusion

Example : A survey was given to randomly chosen high school students from years 9 to 12 on possible changes to the school’s canteen. The contingency table shows the results. At a 5% level, test whether there is a significant difference between the proportion of students wanting a change in the canteen across the four year groups.

| |9 |10 |11 |12 |

|Change |7 |9 |13 |14 |

|No Change |14 |12 |9 |7 |

Step 1: State H0 – the null hypothesis. This is a statement that the two variables are considered independent.

H0 – ___________________________________________________

Step 2: Calculate the df

df - ___________________________________________________

Step 3: The problem states the significance level ______________________

Step 4: State the rejection inequality χ2 > k, where k is obtained from the table of critical values.

We reject Ho if ____________________

Step 5: Find χ2 using the contingency table and your calculator

Expected values:

χ2 = ________________________________

Step 6: If the rejection inequality is true, then we accept H0. If it is not true, then we reject H0.

______________ __________ _______________

χ2 Critical Value

Step 7: Look at p – value. If p > significance level we accept H0. If p < .05, we reject H0.

P = __________

Step 8: Conclusion:

1. Students who use pencils or pens vs. right-handed or left-handed

Observed Values Expected Values

| |Pencils |Pens |Sum |

|Right- |31 |22 | |

|handed | | | |

|Left- |20 |27 | |

|handed | | | |

| Sum | | | |

Step 1: State H0 – the null hypothesis. This is a statement that the two variables are considered independent.

H0 – ___________________________________________________

Step 2: Calculate the df

df - ___________________________________________________

Step 3: The problem states the significance level ______________________

Step 4: State the rejection inequality χ2 > k, where k is obtained from the table of critical values.

We reject Ho if ____________________

Step 5: Find χ2 using the contingency table and your calculator

Expected values:

χ2 = ________________________________

Step 6: If the rejection inequality is true, then we accept H0. If it is not true, then we reject H0.

______________ __________ _______________

χ2 Critical Value

Step 7: Look at p – value. If p > significance level we accept H0. If p < .05, we reject H0.

P = __________

Step 8: Conclusion:

-----------------------

χ2 = Σ [pic] where[pic]is an observed frequency and [pic]is an expected frequency.

To find the table of expected values:

Step 1: Sum all rows and columns

Step 2: Multiply the sums for each row and column and divide by the lower right corner box.

| |Pencils |Pens |

|Right- | | |

|handed | | |

|Right- | | |

|handed | | |

| |Pencils |Pens |

|Right- | | |

|handed | | |

|Right- | | |

|handed | | |

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