Respuesta :
Answer:
Step-by-step explanation:
Confidence interval is written as
Sample proportion ± margin of error
Margin of error = z × √pq/n
Where
z represents the z score corresponding to the confidence level
p = sample proportion. It also means probability of success
q = probability of failure
q = 1 - p
p = x/n
Where
n represents the number of samples
x represents the number of success
a) From the information given,
Margin of error = 0.02
p = 0.3
q = 1 - 0.3 = 0.7
To determine the z score, we subtract the confidence level from 100% to get α
α = 1 - 0.5 = 0.05
α/2 = 0.05/2 = 0.025
This is the area in each tail. Since we want the area in the middle, it becomes
1 - 0.05 = 0.975
The z score corresponding to the area on the z table is 1.96. Thus, confidence level of 95% is 1.96
Therefore,
0.02 = 1.96 × √(0.3 × 0.7)/n
0.02/1.96 = √0.21/n
0.0102 = √0.21/n
Taking square of both sides, it becomes
0.00010404 = 0.21/n
n = 0.21/0.00010404
n = 2018
Sample size = 2018
b) if n = 2018
x = 520
Then
p = 520/2018 = 0.26
Point estimate of the proportion of smokers in the population is 0.26
c) q = 1 - 0.26 = 0.74
the 95% confidence interval for the proportion of smokers in the population is
0.26 ± 1.96 × √(0.26)(0.74)/2018
= 0.26 ± 0.019
Answer:
a) n = 2017
b) Point estimate is 0.2578
c) 95% Confidence Interval is Minimum = 0.2387, Maximum 0.2769
Step-by-step explanation:
Here we have
At 95%, we have
[tex]z_{\alpha /2}[/tex] = 1.96
To determine sample size, we have
[tex]n = \frac{(z_{\alpha /2})^2 \hat p \hat q}{E^2}[/tex]
Where:
[tex]\hat p[/tex] = 0.3
[tex]\hat q[/tex] = [tex]1-\hat p[/tex] = 0.7
E = 0.02
Therefore, n = 2016.84 ≈2017
b) The point estimate is given by
[tex]\hat p =\frac{x}{n} = \frac{520}{2017}[/tex] = 0.2578
c) The confidence interval is given by;
[tex]CI=\hat{p}\pm z\times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}[/tex]
Which gives
[tex]CI=0.2578\pm 1.96\times \sqrt{\frac{0.2578(1-0.2578)}{2017}}[/tex]
Hence CI = Min = 0.2387 to Max = 0.2769
