Chapter 11 Chi Square Distribution and goodness of fit

What is Goodness of Fit? This application is a way to evaluate whether population fits a distribution. How closely does observed data match an idealized situation?

χ 2 Goodness of Fit (cont) Why right tailed? For both χ 2 tests, the idea is to see if the measure between the observed and expected frequencies is too large (i.e. to the right) to be due to chance alone.

Do we still have hypotheses? Why yes, we do. H O : Population fits the given distribution H A : Population does not fit the given distribution It will look VERY similar… with just one small change

χ 2 Goodness of Fit (cont) d.f. = k – 1 where k = number of categories in the distribution

The table to the right is the zodiac signs of 256 heads of the largest 400 companies. The natural null would be that the birth dates are divided equally among all the signs of the zodiac. Compute the test statistic to evaluate how closely the observed data matches this idealized situation. BirthsSign 23Aries 20Taurus 18Gemini 23Cancer 20Leo 19Virgo 18Libra 21Scorpio 19Sagittarius 22Capricorn 24Aquarius 29Pisces

BirthsSign 23Aries 20Taurus 18Gemini 23Cancer 20Leo 19Virgo 18Libra 21Scorpio 19Sagittarius 22Capricorn 24Aquarius 29Pisces 1. Compute the null and alternate hypotheses. 2. Check the conditions 3. Compute χ Find the degrees of freedom. 5. Conclude

The difference? The previous lesson was using χ 2 to evaluate independence. This lesson uses χ 2 to see how well data fits expected models.