Null hypothesis : [tex]H_0:p=0.25[/tex]
Alternative hypothesis : [tex]H_a:p\neq0.25[/tex]
Given : A genetic experiment with peas resulted in one sample of offspring that consisted of 429 green peas and 159 yellow peas.
i.e. [tex]\hat{p}=\dfrac{159}{429}=0.370629370629\approx0.37[/tex]
For 95% level of confidence, significance level :[tex]\alpha:1-0.95=0.05[/tex]
Critical value of z =[tex]z_{\alpha/2}=1.96[/tex]
Confidence interval for population proportion :
[tex]\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]
[tex]=0.37\pm(1.96)\sqrt{\dfrac{0.37(1-0.37)}{429}}\\\\=0.37\pm0.0456876228438\\\\\approx0.37\pm0.05\\\\=(0.37-0.05,0.37+0.05)=(0.32,0.42)[/tex]
Since 0.25 is not contained in the confidence interval , it means is not reasonable that the true proportion is 0.25 (25%).
Thus, the results contradicts the expectations.
