881 Hypothesizing without Content Given that the data follow

8.81 Hypothesizing without Content: Given that the data follows a binomial distribution, in a random sample of size 170 trails it was found that 72 were successes. Test the claim that the actual number of successes is 50%, what is the statistical decision at the 5% level of significance in testing the following hypothesis:

(b) One- tail hypothesis, testing the claim that the true proportion is 0.50.

(c) Two- tailed hypothesis, testing the claim that the true proportion is .50. Do you feel you can start with a stronger statement (one-tail) or the weaker statement (two- tail)?

(d) repeat part (c) at the 1% level of significance. Compare and interpret these results as they would relate to the population parameter.

Solution

a. ONE TAILED
Set Up Hypothesis
Null, H0:P=0.5
Alternate, H1: P<0.5
Test Statistic
No. Of Success chances Observed (x)=72
Number of objects in a sample provided(n)=170
No. Of Success Rate ( P )= x/n = 0.4235
Success Probability ( Po )=0.5
Failure Probability ( Qo) = 0.5
we use Test Statistic (Z) for Single Proportion = P-Po/Sqrt(PoQo/n)
Zo=0.42353-0.5/(Sqrt(0.25)/170)
Zo =-1.9941
P-Value: Left Tail -Ha : ( P < -1.99411 ) = 0.02307
Critical Value
The Value of |Z | at LOS 0.05% is 1.64
We got |Zo| =1.994 & | Z | =1.64
Make Decision
Hence Value of | Zo | > | Z | and Here we Reject Ho

b. TWO TAILED

Set Up Hypothesis
Null, H0:P=0.5
Alternate, H1: P!=0.5
Zo =-1.9941
| Zo | =1.9941
P-Value: Two Tailed ( double the one tail ) - Ha : ( P != -1.99411 ) = 0.04614
Critical Value
The Value of |Z | at LOS 0.05% is 1.96
We got |Zo| =1.994 & | Z | =1.96
Make Decision
Hence Value of | Zo | > | Z | and Here we Reject Ho

c.
No, both reflects the simiar strength in result

d.
ONE TAIL AT 0.01 LOS
Critical Value
The Value of |Z | at LOS 0.01% is 2.33
We got |Zo| =1.994 & | Z | =2.33
Make Decision
Hence Value of |Zo | < | Z | and Here we Do not Reject Ho

TWO TAIL AT 0.01 LOS
Critical Value
The Value of |Z | at LOS 0.01% is 2.58
We got |Zo| =1.994 & | Z | =2.58
Make Decision
Hence Value of |Zo | < | Z | and Here we Do not Reject Ho