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Heterozygous plants with red flowers (descendants of plants with red and white flowers) should have, according to Mendel's law, descendants in the proportion $3:1$ (plants with red flowers: plant with white flowers). In a crossing experiment we get 20 descendants with red flowers and 3 descendants with white flowers. Does this result confirm or contradict the theory?

My attempt

This is binomial, because we have either red or white, in

$H_0 : \pi = \dfrac{3}{4}$

$H_1 : \pi \neq \dfrac{3}{4}$ (two-sided)

We can use a graphing calculator with the option binomcdf:

$P (x \geq 20 | n = 23 , p = \dfrac{3}{4}) \approx 0.137$

$0.137 > \dfrac{1}{2} \alpha $, so $H_0$ is not rejected, Mendel's laws hold.

Have I done this correct? Keep in mind, this is just very basic statistics, I know that if you go deeper into this you can find more accurate ways, but is the main idea behind this correct?

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up vote 0 down vote accepted

Here is a sample hypothesis test at the $10\%$ significance level ($\alpha=0.10$) for your data. Firstly, as you have done you state your null and alternative hypotheses: $H_{0}: p = \frac{3}{4}$, $H_{1}: p\not=\frac{3}{4}$, where $p$ is the proportion of red flowers to white flowers.

Under our null hypothesis, we have the distribution $X\sim B(23,\frac{3}{4})$. A reasonable approximation to this distribution would be $X_{\sim}\stackrel{\text{approx}}{\sim}\mathcal{N}(\frac{69}{4},\frac{69}{16})$.

We now find the probability of $X\geq20$:

$$\begin{align*}P(X\geq20)&\approx P(X_{\sim}\geq19.5) \\ &\approx 1-P(X_{\sim}\leq19.5) \\ &\approx 1-P(Z\leq\frac{19.5-17.25}{\sqrt{4.3125}}) \\ &\approx 1-\Phi(1.083) \\ &\approx 1-0.8606=0.1394>0.05\end{align*}$$

As you can see the approximation yielded a result very similar to the actual distribution (with an absolute error of $+0.0024$).

Therefore we can accept $H_{0}$, i.e. there is insufficient evidence at the $10\%$ significance level to reject Mendel's theory.

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