To prevent erosion along the sides of many highways, the highway department has

Question

To prevent erosion along the sides of many highways, the highway department has spent a large sum of money planting grass. Because the cost of labor and equipment is a substantial part of the department's budget, the department wants to make sure that the grass seed being planted will help prevent erosion. Before purchasing a certain brand of seed, the department examined a random sample of 600 seeds. In the sample, 500 seeds germinated.

A. Construct a 95% confidence interval for p.

B. The seed compant has claimed that the probability of germination is at least .9. Do you agree? Explain

Explanation / Answer

PART A.
TRADITIONAL METHOD
given that,
possibile chances (x)=500
sample size(n)=600
success rate ( p )= x/n = 0.8333
I.
sample proportion = 0.8333
standard error = Sqrt ( (0.8333*0.1667) /600) )
= 0.0152
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.05
from standard normal table, two tailed z /2 =1.96
margin of error = 1.96 * 0.0152
= 0.0298
III.
CI = [ p ± margin of error ]
confidence interval = [0.8333 ± 0.0298]
= [ 0.8035 , 0.8632]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
possibile chances (x)=500
sample size(n)=600
success rate ( p )= x/n = 0.8333
CI = confidence interval
confidence interval = [ 0.8333 ± 1.96 * Sqrt ( (0.8333*0.1667) /600) ) ]
= [0.8333 - 1.96 * Sqrt ( (0.8333*0.1667) /600) , 0.8333 + 1.96 * Sqrt ( (0.8333*0.1667) /600) ]
= [0.8035 , 0.8632]
-----------------------------------------------------------------------------------------------
interpretations:
1. We are 95% sure that the interval [ 0.8035 , 0.8632] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the true population proportion

PART B.
no we don't agree since the confidence calculated explains the proportion value
is lies in between 80% to 80%

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