Answer:
a) 95% Confidence interval = (0.536, 0.584)
This means that we are 95% confident that between 53.6% and 58.4% of Americans would give themselves grade of A or B on their financial knowledge of personal finance.
b) The result of the 95% confidence interval agrees with the claim that majority of Americans would give themselves a grade of A or B on their financial knowledge of personal finance because the interval obtained for possible values that the population proportion of Americans that would give themselves a grade of A or B on their financial knowledge of personal finance lies completely on the side that is greater than a proportion of 0.50 which indicates that truly, majority of Americans would give themselves a grade of A or B on their financial knowledge of personal finance.
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 = proportion of Americans in the sample that would give themselves grade of A or B on their financial knowledge of personal finance = p = (934/1663) = 0.56
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 1663 is obtained from the z-tables as the sample size is large enough for the sample properties to approximate the population properties.
Critical value = 1.960 (from the z-tables)
Standard error of the mean = σₓ = √[p(1-p)/n]
p = 0.56
n = sample size = 1663
σₓ = √[0.56×0.44/1663] = 0.0121723457 = 0.01217
95% Confidence Interval = (Sample proportion) ± [(Critical value) × (standard Error)]
CI = 0.56 ± (1.96 × 0.01217)
CI = 0.56 ± 0.0238532
95% CI = (0.5361468, 0.5838532)
95% Confidence interval = (0.536, 0.584)
b) The result of the 95% confidence interval agrees with the claim that majority of Americans would give themselves a grade of A or B on their financial knowledge of personal finance because the interval obtained for possible values that the population proportion of Americans that would give themselves a grade of A or B on their financial knowledge of personal finance lies completely on the side that is greater than a proportion of 0.50 which indicates that truly, majority of Americans would give themselves a grade of A or B on their financial knowledge of personal finance.
Hope this Helps!!!
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