Noted television personality Stuart Smalley claims Im good e

Noted television personality Stuart Smalley claims “I’m good enough, I’m smart enough, and gosh darn it, people like me.” To ascertain whether the last part of the statement is true, a national polling organization asked 900 individuals randomly sampled from the American public, “Gosh darn it, do you like Stuart?” 423 of the 900 said they liked Stuart and the other 477 said they didn’t.

a.To what population can the pollsters make inference?

b. With 95% condence estimate the proportion of the population that likes Stuart.

c. Tell what is meant by “95% condence” here.

d. When asked to clarify his statement, Stuart said he meant that a majority of the American public likes him. Does the result in b. contradict this claim? Why?

Solution

a)
No.of liked Stuart (x)=423
Sample Size(n)=900
Sample proportion = x/n =0.47
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 = [ 0.47 ±Z a/2 ( Sqrt ( 0.47*0.53) /900)]
= [ 0.47 - 1.96* Sqrt(0.0003) , 0.47 + 1.96* Sqrt(0.0003) ]
= [ 0.4374,0.5026]~ [43.74% to 50.26%]
c)
Interpretations:
1) We are 95% sure that the interval [43.74% to 50.26%] 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 contain the true proportion

d)
It conradict the statement, as the reason Lower interval is below 50%. It can only be true if lower is above 50%