Answer
Question 3
Option A is correct.
Margin of Error = 0.071
Question 4
Option A is correct.
95% Confidence interval = 0.062 < p < 0.121
Step-by-step explanation
Question 3
Confidence Interval for the population proportion is basically an interval of range of values where the true population proportion can be found with a certain level of confidence.
Mathematically,
Confidence Interval = (Sample proportion) ± (Margin of error)
Margin of Error is the width of the confidence interval about the mean.
Now, here, we are given the confidence interval and asked to find the margin of error.
Sample Proportion = Population proportion = p
Let Margin of Error be E
Lower limit of confidence interval = p - E = 0.179
Upper limit of confidence interval p + E = 0.321
Now, we have a simultaneous equation
p - E = 0.179
p + E = 0.321
Add the two equations
2p = 0.179 + 0.321
2p = 0.5
Divide both sides by 2
p = 0.25
We can then solve for E
p - E = 0.179
0.25 - E = 0.179
E = 0.25 - 0.179
E = 0.071
Question 4
Confidence Interval for the population proportion is basically an interval of range of values where the true population proportion can be found with a certain level of confidence.
Mathematically,
Confidence Interval = (Sample proportion) ± (Margin of error)
Sample proportion = p = (33/362) = 0.0912
Margin of Error is the width of the confidence interval about the mean.
It is given mathematically as,
Margin of Error = (Critical value) × (standard Error)
Critical value at 95% confidence interval for sample size of 362 is obtained from the z-tables.
Critical value = 1.960
Standard error or deviation of the sample proportion = σₓ = √[p(p – 1)/n]
p = 0.0912
1 - p = 0.9088
n = sample size = 362
σₓ = √[(0.0912 × 0.9088)/362] = 0.0151
95% Confidence Interval = (Sample proportion) ± [(Critical value) × (standard Error)]
CI = 0.0912 ± (1.96 × 0.0151)
CI = 0.0912 ± 0.0297
95% CI = (0.062, 0.121)
95% Confidence interval = 0.062 < p < 0.121
Hope this Helps!!!