Question 984195
 
Given:
Claim: proportion of pancake-lovers less than 0.25
Sample of 700 people netted 160 pancake-eaters.
Test at the 0.95 level the validity of the claim.
  
Solution:
  
Define hypotheses:
  
H0 (null hypothesis) : proportion of pancake-eaters is greater than or equal to 0.25
  
Ha (alternate hypothesis) : proportion of pancake-eaters is less than 0.25.
Critical Z-score for 95% confidence (left-tail), Zc = -1.645
  
population proportion, p = 0.25
  
sample size, n=700,
  
sample proportion, ps 
  = 160/700
  = 0.2286
  
sample standard error, sigma_p 
  = sqrt(p(1-p)/n)
  = sqrt(0.25(0.75)/700)
  = 0.01637
  
sample z-score, Zp 
  = (ps-p)/sigma_p
  = ((160/700-0.25)/0.01637)
  = -1.309
  
Since Zp > Zc at 95% confidence level, there is insufficient evidence to reject the null hypothesis H0, hence we conclude that the proportion of pancake eaters is greater than or equal to 25%, 
and reject the claim (Ha) that there are less than 25% of pancake eaters.