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CHISQDIST

Calculates values for a $χ^{2} - d i s t r ib u t i o n$ .

Syntax

CHISQDIST(x, k, Cumulative)

x is the number, at which you will evaluate the $χ^{2} - d i s t r ib u t i o n$ .

k sets the degrees of freedom for the $χ^{2} - d i s t r ib u t i o n$

Constraint: k must be a positive integer

Cumulative is a logical value.

In the case Cumulative=TRUE() the cumulative distribution function is used, in the case Cumulative=FALSE() the probability density function. This parameter is optional. It is set to TRUE() if missing.

Semantic

CHISQDIST(x,k,FALSE()) returns values of the probability density function for the $χ^{2} - d i s t r ib u t i o n$ :

$= ⎩ ⎨ ⎧ 0, \frac{x ^{\frac{k}{2} - 1} e ^{- \frac{x}{2}}}{2 ^{\frac{k}{2}} Γ ( \frac{k}{2} )} d t, if x \leq 0 if x > 0$

CHISQDIST(x,k,TRUE()) returns the left tail probability for the $χ^{2} - d i s t r ib u t i o n$ :

$= ⎩ ⎨ ⎧ 0, \int_{0}^{x} \frac{t ^{\frac{x}{2} - 1} e ^{- \frac{t}{2}}}{2 ^{\frac{k}{2}} Γ ( \frac{k}{2} )} d t, if x \leq 0 if x > 0$

Example

CHISQDIST(2.3,15,FALSE())returns approximately 0,000209862

CHISQDIST(1.5,2,TRUE())returns approximately 0,5276334

CHISQDIST(18,15,TRUE())returns approximately 0,73733444

Remarks

If you need CHISQDIST(x,k,TRUE()) with a non integer parameter k, then use GAMMADIST(x,k/2,2) instead.

Application:

Testing for Independence of Gender and Movie Genre Preference

A movie streaming company wants to determine if there is a relationship between a person's gender and their preferred movie genre. They conduct a survey of 300 people and ask them to choose their favorite genre from a list of three: Action, Comedy, and Sci-Fi.

1. The Data

First, they collect the observed frequencies.

Observed Frequencies

		Action	Comedy	Sci-Fi	Total
	A	B	C	D	E
1	Male	50	70	30	150
2	Female	40	80	30	150
3	Total	90	150	60	300

2. The Null Hypothesis

The null hypothesis (H0) is that there is no relationship between gender and movie genre preference; that is, they are independent. The alternative hypothesis (Ha) is that there is a relationship.

3. Calculating Expected Frequencies

If gender and genre preference are independent, we would expect the distribution of preferences to be the same for both males and females. The expected frequency for each cell is calculated as:

Expected Frequency = (Row Total * Column Total) / Grand Total

For example, the expected frequency for "Male - Action" would be:

Expected Frequency (Male - Action) = (150 * 90) / 300 = 45

Here is the full table of expected frequencies:

Expected Frequencies

		Action	Comedy	Sci-Fi	Total
	A	B	C	D	E
1	Male	45	75	30	150
2	Female	45	75	30	150
3	Total	90	150	60	300

4. Calculating the Chi-Squared Statistic

Next, we calculate the chi-squared statistic (χ2) using the formula:

$χ^{2} = \sum \frac{( O - E ) ^{2}}{E}$

where O is the observed frequency and E is the expected frequency.

$χ^{2} = \frac{( 50 - 45 ) ^{2}}{45} + \frac{( 70 - 75 ) ^{2}}{75} + \frac{( 30 - 30 ) ^{2}}{30} + \frac{( 40 - 45 ) ^{2}}{45} + \frac{( 80 - 75 ) ^{2}}{75} + \frac{( 30 - 30 ) ^{2}}{30}$

$χ^{2} = \frac{25}{45} + \frac{25}{75} + 0 + \frac{25}{45} + \frac{25}{75} + 0$

$χ^{2} = 0.5556 + 0.3333 + 0 + 0.5556 + 0.3333 + 0$

$χ^{2} \approx 1.7778$

5. Using the CHISQDIST Function

Now we use the CHISQDIST function to find the probability of getting a chi-squared statistic of 1.7778 or higher, assuming the null hypothesis is true. The function requires two arguments: the chi-squared value and the degrees of freedom (df).

Chi-Squared value: 1.7778
Degrees of Freedom (df): The degrees of freedom for a chi-squared test of independence is calculated as (rows−1)×(columns−1). In our example, we have 2 rows (Male, Female) and 3 columns (Action, Comedy, Sci-Fi). df=(2−1)×(3−1)=1×2=2

The function call would be:

CHISQDIST(1.7778, 2)

This function will return the p-value. Assuming we set a significance level (α) of 0.05, we will compare the p-value to this alpha.

Let's assume the result of the CHISQDIST function is approximately 0.589.

6. Conclusion

P-value: 0.589
Significance level (α): 0.05

Since the p-value (0.589) is greater than the significance level (0.05), we do not have enough evidence to reject the null hypothesis.

Conclusion: There is no statistically significant relationship between a person's gender and their preferred movie genre based on this sample data.

Result for CHISQDIST(1.7778, 2):

0.589

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