The formula to set out the lower and the upper margin of a confidence interval when given the proportion (no standard deviation) is
Lower margin = p - Z* √[(pq) ÷ n]
Upper margin = p + Z* √[(pq) ÷ n]
Where:
p is the sample proportion
q is 1 - p
Z* is the z-score for the confidence level
n is the number size
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p = (68% of 737) ÷ 737 = 501.16 ÷ 737 = 0.68
q = 1 - p = 1 - 0.68 = 0.32
Z* = 2.58 (refer to the table attached below)
n = 737
substituting these values into the formula, we have
lower margin = 0.68 - (2.58) √[(0.68×0.32) ÷ 737]
lower margin = 0.68 - (2.58) √(0.0002952510176...)
lower margin = 0.68 - 0.04433180431
lower margin = 0.6357 (rounded to four decimal places)
lower margin = 63.57%
upper margin = 0.68 + 0.04433180431
upper margin = 0.7243 (rounded to four decimal places)
upper margin = 72.43%
The confidence interval is between 63.57% and 72.43%. In other words, we can say that between 63.57% and 72.43% of jury members are immigrant.
The claim of 69.46% is within the confidence interval, hence we can conclude that the selection of grand jury duty is not biased against the immigrant.
Lower margin = p - Z* √[(pq) ÷ n]
Upper margin = p + Z* √[(pq) ÷ n]
Where:
p is the sample proportion
q is 1 - p
Z* is the z-score for the confidence level
n is the number size
--------------------------------------------------------------------------------------------------------------
p = (68% of 737) ÷ 737 = 501.16 ÷ 737 = 0.68
q = 1 - p = 1 - 0.68 = 0.32
Z* = 2.58 (refer to the table attached below)
n = 737
substituting these values into the formula, we have
lower margin = 0.68 - (2.58) √[(0.68×0.32) ÷ 737]
lower margin = 0.68 - (2.58) √(0.0002952510176...)
lower margin = 0.68 - 0.04433180431
lower margin = 0.6357 (rounded to four decimal places)
lower margin = 63.57%
upper margin = 0.68 + 0.04433180431
upper margin = 0.7243 (rounded to four decimal places)
upper margin = 72.43%
The confidence interval is between 63.57% and 72.43%. In other words, we can say that between 63.57% and 72.43% of jury members are immigrant.
The claim of 69.46% is within the confidence interval, hence we can conclude that the selection of grand jury duty is not biased against the immigrant.