Using [Hint_1] show that the sampling distribution of MLE is given by OMLE 0 N Fn-1m-1 i.e., the sampling distribution of MLE is an F distribution with n - 1 degrees of freedom in the numerator, and m - 1 degrees of freedom in the denominator.
Using [Hint_1] show that the sampling distribution of MLE is given by OMLE 0 N Fn-1m-1 i.e., the sampling distribution of MLE is an F distribution with n - 1 degrees of freedom in the numerator, and m - 1 degrees of freedom in the denominator.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![In this question, we will set up and test the appropriate hypothesis for verifying a claim from
astronomy.
• The Bubble space telescope is an optical telescope that was launched into space in
1990 to collect data from distant stars and glaxies, and has proven to be an
invaluable source of information to many astronomers and cosmologists.
• Recently, a team of researchers from NASA led the design and development of the
Webb space telescope, which was claimed to be capable of higher resolution images
and less error-prone in comparison to its predecesor, the Bubble space telescope.
• In order to lend credence to the claim that the newly designed Webb space telescope was
capable of more accurate measurements compared to the Bubble space telescope, the
scientists from NASA were required to show sufficient evidence for this claim in a
formal statistical setting.
In order to test the hypothesis that the measurement error from the Webb telescope was indeed smaller
than the measurement error from the Bubble telescope, the researchers designed the following
experiment:
A collection of n and m observations were taken of a fixed object in the observable universe by, the
Bubble telescope and the Webb telescope, respectively.
Let
X1, X2,..., X,
N(μ, ox)
denote the collection of n observations taken by the Bubble Telescope, and let
Y₁, Y2,..., Ymd N(μ, o)
denote the collection of n observations taken by the Webb Telescope. The mean μ is the true
(unknown) brightness of the object, and the (unknown) variances o and o represent the
measurement error from the Bubble and Webb telescopes, respectively.
Using this information, answer the following questions.
[Hint 1]
In class, we saw that if we are given two random variables X~X and Y~ x the distribution
of (X/n)/(Y/m) is given by
Also, in the midterm we saw that
X/n
Y/m
Fnm
T (k, 2) = X₂k¹
i.e., A Gamma distribution with parameters (k, 2) is a Chi-squared distribution with 2k degrees of
freedom. Similarly, if c> 0 is some constant then
cX~ I (a₁, c),
i.e., multiplying a Gamma distribution by c results in a new Gamma distribution with the same
shape a₁ but the scale becomes cß.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7c2748e-5ba1-4f2f-bc7d-c6e1ab8bba12%2F1bf509e7-9d5a-4f37-b9c8-4a73eae1b21b%2Fw9rsty_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In this question, we will set up and test the appropriate hypothesis for verifying a claim from
astronomy.
• The Bubble space telescope is an optical telescope that was launched into space in
1990 to collect data from distant stars and glaxies, and has proven to be an
invaluable source of information to many astronomers and cosmologists.
• Recently, a team of researchers from NASA led the design and development of the
Webb space telescope, which was claimed to be capable of higher resolution images
and less error-prone in comparison to its predecesor, the Bubble space telescope.
• In order to lend credence to the claim that the newly designed Webb space telescope was
capable of more accurate measurements compared to the Bubble space telescope, the
scientists from NASA were required to show sufficient evidence for this claim in a
formal statistical setting.
In order to test the hypothesis that the measurement error from the Webb telescope was indeed smaller
than the measurement error from the Bubble telescope, the researchers designed the following
experiment:
A collection of n and m observations were taken of a fixed object in the observable universe by, the
Bubble telescope and the Webb telescope, respectively.
Let
X1, X2,..., X,
N(μ, ox)
denote the collection of n observations taken by the Bubble Telescope, and let
Y₁, Y2,..., Ymd N(μ, o)
denote the collection of n observations taken by the Webb Telescope. The mean μ is the true
(unknown) brightness of the object, and the (unknown) variances o and o represent the
measurement error from the Bubble and Webb telescopes, respectively.
Using this information, answer the following questions.
[Hint 1]
In class, we saw that if we are given two random variables X~X and Y~ x the distribution
of (X/n)/(Y/m) is given by
Also, in the midterm we saw that
X/n
Y/m
Fnm
T (k, 2) = X₂k¹
i.e., A Gamma distribution with parameters (k, 2) is a Chi-squared distribution with 2k degrees of
freedom. Similarly, if c> 0 is some constant then
cX~ I (a₁, c),
i.e., multiplying a Gamma distribution by c results in a new Gamma distribution with the same
shape a₁ but the scale becomes cß.
![5.
Using [Hint_1] show that the sampling distribution of MLE is given by
OMLE
0
Fn-1,m-1
i.e., the sampling distribution of MLE is an F distribution with n 1 degrees of freedom in the
numerator, and m - 1 degrees of freedom in the denominator.
By fixing the Type-l error probability to be equal to cx, i.e.,
a = P(Type-I Error)
derive an expression for the rejection region R(0; a) in terms of Fn-1,m-1;a, the a quantile of the
Fn-1,m-1 distribution from part#5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7c2748e-5ba1-4f2f-bc7d-c6e1ab8bba12%2F1bf509e7-9d5a-4f37-b9c8-4a73eae1b21b%2Flrilx2b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5.
Using [Hint_1] show that the sampling distribution of MLE is given by
OMLE
0
Fn-1,m-1
i.e., the sampling distribution of MLE is an F distribution with n 1 degrees of freedom in the
numerator, and m - 1 degrees of freedom in the denominator.
By fixing the Type-l error probability to be equal to cx, i.e.,
a = P(Type-I Error)
derive an expression for the rejection region R(0; a) in terms of Fn-1,m-1;a, the a quantile of the
Fn-1,m-1 distribution from part#5
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