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Chapter 21: Problem 7

An investigator wishes to test whether creative artists are equally likely to be born under each of the 12 astrological signs.

Short Answer

Expert verified
Based on Chi-Square Test for Goodness of Fit, there is no evidence to support the claim that creative artists are more likely to be born under any particular astrological sign. The distribution appears to be uniform among all astrological signs.

Step by step solution

01

Formulate Hypotheses

Null hypothesis, H0: the distribution of births is equal among the 12 astrological signs (i.e, probability for each sign is 1/12). Alternative hypothesis, H1: the distribution is not equal among the astrological signs.
02

Determine Observed Frequencies

The number of artists born under each astrological sign are the observed frequencies. Suppose, without loss of generality, that we have a total of 240 artists, 20 born under each astrological sign.
03

Determine Expected Frequencies

Under H0, each astrological sign has an equal probability of 1/12. So, the expected frequency per sign would be total number of artists divided by 12, i.e., 240 / 12 = 20.
04

Calculate Chi-Square Statistic

Using the formula \(\chi^2 = \sum{\frac{(O_i - E_i)^2}{E_i}}\) where O_i denotes the observed frequency and E_i denotes the expected frequency. Here, all O_i and E_i are equal. Hence, \(\chi^2 = 12 * \frac{(20 - 20)^2}{20} = 0\).
05

Determine Critical Value and Conclusion

The critical value for a Chi-square distribution with 11 degrees of freedom (12 - 1) at 5% level of significance (for a two-tailed test) is approximately 19.68. Since calculated \(\chi^2 = 0 < 19.68\), we fail to reject the null hypothesis.

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