Question

A garden center wants to store leftover packets of vegetable seeds for sale the following spring, but the center is concerned that the seeds may not germinate at the same rate a year later. The manager finds a packet of last year’s green bean seeds and plants them as a test. Although the packet claims a germination rate of 92%, only 176 of 200 test seeds sprout. Perform a significance test atα= 0.05 to determine if the seeds have lost viability during a year in storage (i.e., is the proportion of seeds that will sprout less than 92%).

Hypotheses:

Test statistic :

P-value:

Conclusion:

Answer 1

Solution :

This is the left tailed test .

The null and alternative hypothesis is

H₀ : p = 0.92

H_a : p < 0.92

$\hat p$ = x / n = 176/200 = 0.88

P₀ = 0.92

1 - P₀ = 1-0.92 = 0.08

Test statistic = z

= $\hat p$ - P₀ / [$\sqrt$P_{0 *} (1 - P₀ ) / n]

= 0.88-0.92/ [$\sqrt$0.92*(0.08) /200 ]

= -2.085

P(z < -2.085 ) = 0.0185

P-value = 0.0185

$\alpha$ = 0.05    

p=0.0185<0.05

Reject the null hypothesis .

There is sufficient evidence to suggest that the population proportion pp is less than p₀​, at the α=0.05 significance level .