Find the one-sided confidence interval of the form Tplug—in = (U, B at asymptotic (confidence) level 95% for 0 given by the plug-in method Specify B below in terms of 11, U and V (0) (Enter hattheta for 0, v for V (e) If applicable, for any numerical a enter q(alpha) for the 1 — a quantile qa of the standard normal distribution, i.e. P (Z < qa)=1- a). For, example enter q(0.01) for qo.01 You ma}r also use numerical values for qa as long as they are accurate to at least 4 decimal places.) B = STANDARD NOTATION You have used 0 of 3 attempts Submit Save Confidence Interval Continued 1 point possible (graded, results hidden) We observe 0 =0.62 for 11 = 100. (Give an answer accurate to at least 3 decimal places.)
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To find the one-sided confidence interval using the plug-in method, we need to calculate the asymptotic confidence level and determine the values of U and B.
The plug-in method involves replacing the unknown parameters with their estimates. In this case, we have 0 = 0.62 and n = 100.
First, let's calculate the estimate of the standard deviation (V(0)):
V(0) = (0(1-0))/n
= (0.62(1-0))/100
≈ 0.0062 (rounded to 4 decimal places)
Next, we need to find the quantile qa such that P(Z < qa) = 1 - α, where α is the confidence level. Since the confidence level is 95%, α = 0.05. We can find qa by using a standard normal distribution table or calculator. For simplicity, let's assume that we have qa = q(0.05) = -1.645 (accurate to at least 4 decimal places).
Now, we can calculate U and B:
U = 0 + qa * sqrt(V(0))
= 0.62 + (-1.645) * sqrt(0.0062)
≈ 0.62 - 0.064 (rounded to 3 decimal places)
B = U + 2 * sqrt(V(0))
= 0.62 - 0.064 + 2 * sqrt(0.0062)
≈ 0.62 - 0.064 + 2 * 0.079 (rounded to 3 decimal places)
Therefore, the one-sided confidence interval in the form Tplug—in = (U, B) at the asymptotic (confidence) level 95% for the given values is approximately (0.556, 0.758).
To find the one-sided confidence interval of the form Tplug—in = (U, B at asymptotic (confidence) level 95% for 0 given by the plug-in method, we need to calculate the upper bound B.
Given that 0 = 0.62 and n = 100, we can compute the standard deviation s using the formula:
s = sqrt((U*(1-U))/n)
By substituting the given values into the formula, we get:
s = sqrt((0.62*(1-0.62))/100)
s ≈ 0.0489988 (rounded to at least 4 decimal places)
Next, we calculate the quantile qa using the formula:
qa = q(1 - a)
Since we have an asymptotic (confidence) level of 95%, a = 0.05, so qa = q(0.05).
Using a standard normal distribution table or calculator, we can find that q(0.05) ≈ -1.644854 (rounded to at least 4 decimal places).
Finally, we can calculate the upper bound B using the formula:
B = 0 + s * qa
Substituting the values we have calculated, we get:
B ≈ 0 + 0.0489988 * (-1.644854)
B ≈ -0.0805958 (rounded to at least 3 decimal places)
Therefore, the one-sided confidence interval, using the plug-in method, is given by:
(Tplug—in = (U, B) ≈ (0, -0.0806)