1 Calculate the test statistic z used in testing the followi

1. Calculate the test statistic z used in testing the following:

a. Ho: p 0.70   vs. Ha: p > 0.70 with the sample n = 300 and x = 224.

b. Using the classical approach or the p-value approach state your decision and explain how you arrived at it if = 0.05.

c. Construct the 99% confidence interval for the true population proportion and write a confidence statement.

Solution

a)
No. Of Success chances Observed (x)=224
Number of objects in a sample provided(n)=300
No. Of Success Rate ( P )= x/n = 0.75
b)
Set Up Hypothesis
Under The Null Hypothesis H0:P<0.7
Under The Alternate Hypothesis H1: P>0.7
Test Statistic
Success Probability ( Po )=0.7
Failure Probability ( Qo) = 0.3
we use Test Statistic (Z) for Single Proportion = P-Po/Sqrt(PoQo/n)
Zo=0.74667-0.7/(Sqrt(0.21)/300)
Zo =1.76
| Zo | =1.76
Critical Value
The Value of |Z | at LOS 0.05% is 1.64
We got |Zo| =1.764 & | Z | =1.64
Make Decision
Hence Value of | Zo | > | Z | and Here we Reject Ho
P-Value: Right Tail - Ha : ( P > 1.76383 ) = 0.03888
Hence Value of P0.05 > 0.03888,Here we Reject Ho

c)
Confidence Interval For Proportion
CI = p ± Z a/2 Sqrt(p*(1-p)/n)))
x = Mean
n = Sample Size
a = 1 - (Confidence Level/100)
Za/2 = Z-table value
CI = Confidence Interval
Mean(x)=224
Sample Size(n)=300
Sample proportion = x/n =0.7467
Confidence Interval = [ 0.7467 ±Z a/2 ( Sqrt ( 0.7467*0.2533) /300)]
= [ 0.7467 - 2.58* Sqrt(0.0006) , 0.7467 + 2.58* Sqrt(0.0006) ]
= [ 0.6819,0.8115]