The null hypothesis says that at least 20% of college students are left-handed. If we took a sample of 20 college students and let $X$ be the number of lefties in the sample. Calculate the p values if
$X=1$
$X=2$
Hint: If the null hypothesis is true, $X$ follows binomial distribution.
My attempt-
I tried applying the binomial formula for $X=1$
$${20 \choose 1} 0.5^{20}$$ and this is approx. 0.0001 and for $X=2$ $${20 \choose 2} 0.5^{20} + {20 \choose 1} 0.5^{20}$$ but getting my answer wrong.
With the null hypothesis being true, the binomial distribution will be B(0.2,20) since p=20%. The problem is a one-sided alternative problem, hence when X<= X_Critical_Value : We Accept the alternative X> X_Critical_Value : We retain the Null Hypothesis
for $ X=1 $ the $P=P(x<=1)=P(X=0)+P(X=1)=0.0115+0.0576=0.06912$
for $ X=2 $ the $P=P(x<=2)=P(X=0)+P(X=1)+P(X=2)=0.0115+0.0576+0.1369=0.206$