= Solution
Use the <Pearson chi-squared goodness-of-fit test>. Under the <null hypothesis>, each count has the specified <binomial distribution>, and no parameter is estimated from the data. The seven mutually exclusive classes give $7-1=6$ <degrees of freedom>. Their expected counts are all at least five, so the usual <chi-squared asymptotic approximation> does not require pooling. The computed statistic is
$$
X^2=\frac{(-2)^2}{5}+\frac{(-9)^2}{30}+\frac{10^2}{75}+\frac{10^2}{100}
+\frac{(-13)^2}{75}+\frac{2^2}{30}+\frac{2^2}{5}
=\boxed{9.02}.
$$
Since $9.02<12.59$, \b[do not reject fairness at the 5% level]. The approximate $p$-value is $0.172$. This conclusion concerns agreement with the specified independent fair-coin count model; a nonrejection is not proof of fairness.
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