In: Statistics and Probability
In a simple random sample of 70 automobile dealers registered in a certain state, 30 of them were found to have emission levels that exceeded a state standard. At the 95% confidence interval, the proportion of automobiles whose emissions exceeded the state standard is closest to?
Solution :
Given that,
n = 70
x = 30
Point estimate = sample proportion = = x / n = 30/70=0.429
1 - = 1- 0.429=0.571
At 95% confidence level the z is ,
= 1 - 95% = 1 - 0.95 = 0.05
/ 2 = 0.05 / 2 = 0.025
Z/2 = Z0.025 = 1.96 ( Using z table )
Margin of error = E = Z/2 * ((( * (1 - )) / n)
= 1.96 (((0.429*0.571) / 70)
E = 0.116
A 95% confidence interval is ,
- E < p < + E
0.429-0.116 < p < 0.429+0.116
(0.313 , 0.545)