Answer:
a) 95% confidence interval for the proportion of all adult Americans who would give themselves a grade of A or B on their financial knowledge of personal finance = (0.498, 0.547)
This means we are 95% confident that the true proportion all adult Americans who would give themselves a grade of A or B on their financial knowledge of personal finance is within the range 49.8% and 54.7%.
b) The confidence interval from part (a) is consistent with the statement that a majority of adult Americans would give themselves a grade of A or B because the interval obtained contains proportions that are greater than 50% indicating that there is significant evidence that the true proportion of all adult Americans who would give themselves a grade of A or B on their financial knowledge of personal finance is more than half of the total population.
Step-by-step explanation:
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 = (820/1570) = 0.5223
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 1570 is obtained from the z-tables.
Critical value = 1.960
Standard error of the mean = σₓ = √[p(1-p)/n]
p = sample proportion = 0.5223
n = sample size = 1570
σₓ = √(0.5223×0.4777/1570) = 0.0126063049 = 0.01261
95% Confidence Interval = (Sample proportion) ± [(Critical value) × (standard Error)]
CI = 0.5223 ± (1.96 × 0.01261)
CI = 0.5223 ± 0.02471
95% CI = (0.4975916424, 0.5470083576)
95% Confidence interval = (0.4976, 0.5470)
We are 95% confident that the true proportion all adult Americans who would give themselves a grade of A or B on their financial knowledge of personal finance is within the range 49.8% and 54.7%.
b) The confidence interval from part (a) is consistent with the statement that a majority of adult Americans would give themselves a grade of A or B because the interval obtained contains proportions that are greater than 50% indicating that there is significant evidence that the true proportion of all adult Americans who would give themselves a grade of A or B on their financial knowledge of personal finance is more than half of the total population.
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