In a Gallup poll adult survey subjects were asked Do you ha

In a Gallup poll, adult survey subjects were asked “ Do you have a gun in your home?” the respondents consist of 413 people who answered “yes,” and 646 other respondents who either answered “no” or had no opinion.

a) What percentage of respondents answered “yes”? Percentage: = _________

b) Construct a 95% confidence interval estimate of the percentage of all adults who answer “yes” when asked if they have a gun in their home. Confidence interval: (____ , _____)

c) Can we safely conclude that less than 50% of adults answer “yes” when asked if they have a gun in their home? Why or why not?

Solution

a)
Mean(x)=413
Sample Size(n)=646
Sample proportion = x/n =0.6393
b)
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

Confidence Interval = [ 0.6393 ±Z a/2 ( Sqrt ( 0.6393*0.3607) /646)]
= [ 0.6393 - 1.96* Sqrt(0.0004) , 0.6393 + 1.96* Sqrt(0.0004) ]
= [ 0.6023,0.6763]

c)
Set Up Hypothesis
Null, H0:P<0.5
Alternate,safely conclude that less than 50% of adults answer “yes” H1: P<0.5
Test Statistic
No. Of Success chances Observed (x)=413
Number of objects in a sample provided(n)=646
No. Of Success Rate ( P )= x/n = 0.6393
Success Probability ( Po )=0.5
Failure Probability ( Qo) = 0.5
we use Test Statistic (Z) for Single Proportion = P-Po/Sqrt(PoQo/n)
Zo=0.63932-0.5/(Sqrt(0.25)/646)
Zo =7.082
| Zo | =7.082
Critical Value
The Value of |Z | at LOS 0.05% is 1.64
We got |Zo| =7.082 & | Z | =1.64
Make Decision
Hence Value of | Zo | > | Z | and Here we Reject Ho
P-Value: Right Tail - Ha : ( P > 7.08201 ) = 0
Hence Value of P0.05 > 0,Here we Reject Ho