In: Statistics and Probability
In a survey of 1010 Canadian adults, 780 say that the energy situation in Canada is very or fairly serious.
1. Find the point estimate for the population proportion
2. Construct a 95% confidence interval for the population proportion.
a) The critical value
b) The margin of error
c) The lower limit of the interval
d) The upper limit of the interval
3. Find the minimum sample size needed to estimate the population proportion at the 99% confidence level in order to ensure that the estimate is accurate within 5 % of the population proportion.
a) The critical value
b) The margin of error
c) The sample size
Solution :
Given that
n =1010
x = 780
1 ) The point estimate
= = x / n = 780 /1010 = 0.772
1 - = 1 - 0.772 = 0.228
2 ) At 95% confidence level the z is ,
= 1 - 95% = 1 - 0.95 = 0.05
/ 2 = 0.05 / 2 = 0.025
Z/2 = Z0.025 = 1.960
a ) The critical value = 1.960
b ) Margin of error = E = Z / 2 * (( * (1 - )) / n)
=1.960 * (((0.772 * 0.228) / 1010)
= 0.026
Margin of error = 0.026
A 95 % confidence interval for population proportion p is ,
- E < P < + E
0.772 - 0.026 < p < 0.772 + 0.026
0.746 < p < 0.798
c) The lower limit = 0.746
d) The upper limit =0.798
3 ) Given that,
= 0.772
1 - = 1 - 0.772 = 0.228
margin of error = E = 5% = 0.05
At 99% confidence level the z is ,
= 1 - 99% = 1 - 0.99 = 0.01
/ 2 = 0.01 / 2 = 0.005
Z/2 = Z0.005 = 2.576
a) The critical value = 2.576
Sample size = n = ((Z / 2) / E)2 * * (1 - )
= (2.576 / 0.05)2 *0.772 * 0.228
=467
n = sample size = 467