Question

A report on consumer financial literacy summarized data from a representative sample of 1,663 adult Americans. Based on data from this sample, it was reported that over half of U.S. adults would give themselves a grade of A or B on their knowledge of personal finance. This statement was based on observing that 934 people in the sample would have given themselves a grade of A or B.
(a) Construct and interpret 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. (Use a table or technology. Round your answers to three decimal places.) , (b) Is the confidence interval from part (a) consistent with the statement that a majority of adult Americans would give themselves a grade of A or B? Explain why or why not. Because this confidence interval , the interval consistent with the statement that a majority of adult Americans would give themselves a grade of A or B.

Answer 1

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.

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