Question

1. The numbers 811, 84.7, 77.8, 76.4 represent the temperature measured at a certain location and at a certain time. This is regarded as a random sample from a distribution N(m σ2), where m is the true temperature, and σ2 measures the precision of the met hod. a) Find an unbiased estimate for m. b) Find an unbiased estimate for ơ2 if it is known that m-: 79.0. c) Find an unbiased estimate for σ2 if m is unknown.

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Answer 1

SolutionA:

unbiased estimated of m=sample mean

=m=sample mean=81.1+84.7+77.8+76.4/4=320/4=80

m=80

SolutionB:

unbiased estimated of sigma^2=sample variance

given m=79

X	XBAR	X-XBAR	(X-XBAR)^2
81.1	79	2.1	4.41
84.7	79	5.7	32.49
77.8	79	-1.2	1.44
76.4	79	-2.6	6.76
		TOTAL	45.1

s^2=45.1/4-1=45.1/3=15.033

ANSWER:

s=15.0333

unbiased estimator for sigma^2 when m is known=15.0333

Solutionc:

here m is not given

so sample mean=m=80

X	XBAR	X-XBAR	(X-XBAR)^2
81.1	80	1.1	1.21
84.7	80	4.7	22.09
77.8	80	-2.2	4.84
76.4	80	-3.6	12.96
		TOTAL	41.1

s^2=45.1/4-1=41.1/3=13.7

ANSWER:

unbiased estimator for sigma^2 when m is not known=13.7