Question

Watching grass grow The germination rate of seeds is defined as the proportion of seeds that sprout and grow when properly planted and watered. A certain variety of grass seed usually has a germination rate of $0.80$. A company wants to see if spraying the seeds with a chemical that is known to increase germination rates in other species will increase the germination rate of this variety of grass. The company researchers spray a random sample of $400$grass seeds with the chemical, and $339$of the seeds germinate. Do these data provide convincing evidence at the $\alpha =0.05$ significance level that the chemical is

effective for this variety of grass?

Short answer

The chemical are effective of grass of germination process.

Step-by-step solution

Given Information

It is given that $n=400$

$x=339$

$p=0.05$

Simplification

Null hypothesis is: $H_0:p=0.80$

Alternate hypothesis: $H_a:p\ne 0.80$

Sample proportion is $\hat{p}=\frac{x}{n}$

$\hat{p}=\frac{339}{400}=0.8475$

Standard Deviation is: ${\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}$

${\sigma }_{\hat{p}}=\sqrt{\frac{0.8(1-0.8)}{400}}=0.02$

$Z$score is $Z=\frac{\hat{p}-p}{{\sigma }_{\hat{p}}}$

$Z=\frac{0.8475-0.80}{0.02}=2.375$

The probability value is $P(x<Z)=0.99123$

$P(x>Z)=1-0.99123=0.0087745$

$P=0.0087745<0.05$, null hypothesis is rejected.$H_0:p=0.80$

Chemical is effective for variety of grass.