The election of Argentine cardinal Jorge Mario Bergoglio as

The election of Argentine cardinal Jorge Mario Bergoglio as the new Pope in March 2013 had a major impact on the Vatican\'s perception in global news. In the period March 2012 - March 2013, under Francis\' predeccessor Pope Benedict, 30% of twitter conversations had a positive tone and 70% had a negative tone. Between March 2013 and January 2014, 84% of the 13 million twitter conversations contained a positive tone (PewResearch). Is there evidence that the difference is real? State and test the appropriate hypotheses. Check conditions.

Solution

Z-Test For Proportion
Set Up Hypothesis
Under The Null Hypothesis H0:P=0.3
Under The Alternate Hypothesis H1: P!=0.3
Test Statistic
No. Of Success chances Observed (x)=10.92
Number of objects in a sample provided(n)=13
No. Of Success Rate ( P )= x/n = 0.84
Success Probability   ( Po )=0.3
Failure Probability ( Qo) = 0.7
we use Test Statistic (Z) for Single Proportion = P-Po/Sqrt(PoQo/n)
Zo=0.84-0.3/(Sqrt(0.21)/13)
Zo =4.25
| Zo | =4.25
Critical Value
The Value of |Z ?| at LOS 0.01% is 2.58
We got |Zo| =4.249 & | Z ? | =2.58
Make Decision
Hence Value of | Zo | > | Z ?| and Here we Reject Ho
P-Value: Two Tailed ( double the one tail ) - Ha : ( P != 4.2487 ) = 0.00002
Hence Value of P0.01 > 0,Here we Reject Ho
YEs, we have evidence that there is diffrence in real