In Exercise 9.17, suppose that the populations are normally distributed with σ 1 2 = σ 2 2 = σ 2 . Show that ∑ i = 1 n ( X i − X ¯ ) 2 + ∑ i = 1 n ( Y i − Y ¯ ) 2 2 n − 2 is a consistent estimator of σ 2 . 9.17 Suppose that X 1 , X 2 ,…, X n and Y 1 , Y 2 ,…, Y n are independent random samples from populations with means µ 1 and µ 2 and variances σ 1 2 and σ 2 2 , respectively. Show that X ¯ − Y ¯ is a consistent estimator of µ 1 – µ 2.
In Exercise 9.17, suppose that the populations are normally distributed with σ 1 2 = σ 2 2 = σ 2 . Show that ∑ i = 1 n ( X i − X ¯ ) 2 + ∑ i = 1 n ( Y i − Y ¯ ) 2 2 n − 2 is a consistent estimator of σ 2 . 9.17 Suppose that X 1 , X 2 ,…, X n and Y 1 , Y 2 ,…, Y n are independent random samples from populations with means µ 1 and µ 2 and variances σ 1 2 and σ 2 2 , respectively. Show that X ¯ − Y ¯ is a consistent estimator of µ 1 – µ 2.
In Exercise 9.17, suppose that the populations are normally distributed with
σ
1
2
=
σ
2
2
=
σ
2
. Show that
∑
i
=
1
n
(
X
i
−
X
¯
)
2
+
∑
i
=
1
n
(
Y
i
−
Y
¯
)
2
2
n
−
2
is a consistent estimator of σ2.
9.17 Suppose that X1, X2,…, Xn and Y1, Y2,…,Yn are independent random samples from populations with means µ1 and µ2 and variances
σ
1
2
and
σ
2
2
, respectively. Show that
X
¯
−
Y
¯
is a consistent estimator of µ1 –µ2.
Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.
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