Chapter 11: Q 15. (page 718)

Mendel and the peas Gregor Mendel $(1822 - 1884)$ , an Austrian monk, is considered the father of genetics. Mendel studied the inheritance of various traits in pea plants. One such trait is whether the pea is smooth or wrinkled. Mendel predicted a ratio of $3$ smooth peas for every 1 wrinkled pea. In one experiment, he observed $423$ smooth and $133$ wrinkled peas. Assume that the conditions for inference are met.
a. Carry out a chi-square test for goodness of fit for the genetic model that Mendel predicted.
b. In Chapter $9$ Exercise $49$ you tested Mendel’s prediction using a one-sample z test for a proportion. The hypotheses were $H_{0} : p = 0.75$ and $H_{a} : p \neq 0.75$ where $p =$ true proportion of smooth peas. Calculate the $z$ statistic and $P -$ value for this test. How do these values compare to the values from part (a)?

Expert verified

Part (a) There is insufficient evidence to disprove Mendel's assertion.

Part (b) There is insufficient evidence to disprove Mendel's assertion.

Step by step solution

The null and alternative hypotheses:

$H 0 : p_{1} = \frac{3}{3 + 1} = \frac{3}{4} = 0.75, p_{2} = 1 - 0.75 = 0.25 H_{a} : A t l e a s t o n e o f t h e p_{i} ’ s i s i n c o r r e c t .$

Using excel,

$P - v a l u e > 0.05, f a i l t o r e j e c t H_{0}$

There is insufficient evidence to disprove Mendel's assertion.

Using chi-square test,

Using $z$ test,

$Z = 0.588$

$P -$ value $= 0.2784$

Decision: $P - v a l u e > 0.05, f a i l t o r e j e c t H_{0}$

We can conclude from both outcomes that there is insufficient evidence to dismiss Mendel's belief.

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