Central limit theorem … need help
$begingroup$
So far I have used Chebychev's inequality to calculate n = 157, and my initial thinking for applying the central limit theorem is -1.2815 = (5-0n)/(n* sqrt(391/n)) because P(Z > -1.2815) = 90%, however this isn't giving me the correct value for n so I'm not sure what to try now.
"You want to determine the melting point
c
of a new material. You have
n
specimens on each of which you make a measurement of the melting point in degrees Kelvin, giving you a dataset
m
1
,…,
m
n
. We model this with random variables
M
i
=c+
U
i
, where
U
i
is the random measurement error. It is known that
E[
U
i
]=0
and Var
(
U
i
)=391
for each
i
, and that we may consider the random variables
M
1
,
M
2
,…
as independent.
how many measurements do you need to perform to be
90%
sure that the average of the measurements is within
5
degrees of
c?
Use the normal approximation (central limit theorem rule of thumb) to find a value for
n
. You can use that the
0.95
quantile of the standard normal distribution is
q(0.95)≈1.645
."
probability probability-theory central-limit-theorem
asked Dec 28 '18 at 14:05
KiryneKiryne
135
$endgroup$
add a comment |
$begingroup$
So far I have used Chebychev's inequality to calculate n = 157, and my initial thinking for applying the central limit theorem is -1.2815 = (5-0n)/(n* sqrt(391/n)) because P(Z > -1.2815) = 90%, however this isn't giving me the correct value for n so I'm not sure what to try now.
"You want to determine the melting point
c
of a new material. You have
n
specimens on each of which you make a measurement of the melting point in degrees Kelvin, giving you a dataset
m
1
,…,
m
n
. We model this with random variables
M
i
=c+
U
i
, where
U
i
is the random measurement error. It is known that
E[
U
i
]=0
and Var
(
U
i
)=391
for each
i
, and that we may consider the random variables
M
1
,
M
2
,…
as independent.
how many measurements do you need to perform to be
90%
sure that the average of the measurements is within
5
degrees of
c?
Use the normal approximation (central limit theorem rule of thumb) to find a value for
n
. You can use that the
0.95
quantile of the standard normal distribution is
q(0.95)≈1.645
."
probability probability-theory central-limit-theorem
asked Dec 28 '18 at 14:05
KiryneKiryne
135
$endgroup$
add a comment |
$begingroup$
So far I have used Chebychev's inequality to calculate n = 157, and my initial thinking for applying the central limit theorem is -1.2815 = (5-0n)/(n* sqrt(391/n)) because P(Z > -1.2815) = 90%, however this isn't giving me the correct value for n so I'm not sure what to try now.
"You want to determine the melting point
c
of a new material. You have
n
specimens on each of which you make a measurement of the melting point in degrees Kelvin, giving you a dataset
m
1
,…,
m
n
. We model this with random variables
M
i
=c+
U
i
, where
U
i
is the random measurement error. It is known that
E[
U
i
]=0
and Var
(
U
i
)=391
for each
i
, and that we may consider the random variables
M
1
,
M
2
,…
as independent.
how many measurements do you need to perform to be
90%
sure that the average of the measurements is within
5
degrees of
c?
Use the normal approximation (central limit theorem rule of thumb) to find a value for
n
. You can use that the
0.95
quantile of the standard normal distribution is
q(0.95)≈1.645
."
probability probability-theory central-limit-theorem
asked Dec 28 '18 at 14:05
KiryneKiryne
135
$endgroup$
So far I have used Chebychev's inequality to calculate n = 157, and my initial thinking for applying the central limit theorem is -1.2815 = (5-0n)/(n* sqrt(391/n)) because P(Z > -1.2815) = 90%, however this isn't giving me the correct value for n so I'm not sure what to try now.
"You want to determine the melting point
c
of a new material. You have
n
specimens on each of which you make a measurement of the melting point in degrees Kelvin, giving you a dataset
m
1
,…,
m
n
. We model this with random variables
M
i
=c+
U
i
, where
U
i
is the random measurement error. It is known that
E[
U
i
]=0
and Var
(
U
i
)=391
for each
i
, and that we may consider the random variables
M
1
,
M
2
,…
as independent.
how many measurements do you need to perform to be
90%
sure that the average of the measurements is within
5
degrees of
c?
Use the normal approximation (central limit theorem rule of thumb) to find a value for
n
. You can use that the
0.95
quantile of the standard normal distribution is
q(0.95)≈1.645
."
probability probability-theory central-limit-theorem
probability probability-theory central-limit-theorem
asked Dec 28 '18 at 14:05
KiryneKiryne
135
asked Dec 28 '18 at 14:05
KiryneKiryne
135
asked Dec 28 '18 at 14:05
KiryneKiryne
135
asked Dec 28 '18 at 14:05
KiryneKiryne
135
asked Dec 28 '18 at 14:05
KiryneKiryne
135
135
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
begin{align}
P left(left| frac{sum_{i=1}^nM_i}{n}-cright|<5 right)&=P left(sqrt{n}left| frac{sum_{i=1}^nM_i}{n}-cright|<5sqrt{n} right)\
&approx Pleft(|Z|<frac{5sqrt{n}}{sqrt{Var(U_i)}}right)
end{align}
That is we want $$frac{5sqrt{n}}{sqrt{Var(U_i)}}ge 1.645$$
answered Dec 28 '18 at 14:25
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3054917%2fcentral-limit-theorem-need-help%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
begin{align}
P left(left| frac{sum_{i=1}^nM_i}{n}-cright|<5 right)&=P left(sqrt{n}left| frac{sum_{i=1}^nM_i}{n}-cright|<5sqrt{n} right)\
&approx Pleft(|Z|<frac{5sqrt{n}}{sqrt{Var(U_i)}}right)
end{align}
That is we want $$frac{5sqrt{n}}{sqrt{Var(U_i)}}ge 1.645$$
answered Dec 28 '18 at 14:25
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
add a comment |
$begingroup$
begin{align}
P left(left| frac{sum_{i=1}^nM_i}{n}-cright|<5 right)&=P left(sqrt{n}left| frac{sum_{i=1}^nM_i}{n}-cright|<5sqrt{n} right)\
&approx Pleft(|Z|<frac{5sqrt{n}}{sqrt{Var(U_i)}}right)
end{align}
That is we want $$frac{5sqrt{n}}{sqrt{Var(U_i)}}ge 1.645$$
answered Dec 28 '18 at 14:25
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
add a comment |
$begingroup$
begin{align}
P left(left| frac{sum_{i=1}^nM_i}{n}-cright|<5 right)&=P left(sqrt{n}left| frac{sum_{i=1}^nM_i}{n}-cright|<5sqrt{n} right)\
&approx Pleft(|Z|<frac{5sqrt{n}}{sqrt{Var(U_i)}}right)
end{align}
That is we want $$frac{5sqrt{n}}{sqrt{Var(U_i)}}ge 1.645$$
answered Dec 28 '18 at 14:25
Siong Thye GohSiong Thye Goh
101k1466118
$endgroup$
begin{align}
P left(left| frac{sum_{i=1}^nM_i}{n}-cright|<5 right)&=P left(sqrt{n}left| frac{sum_{i=1}^nM_i}{n}-cright|<5sqrt{n} right)\
&approx Pleft(|Z|<frac{5sqrt{n}}{sqrt{Var(U_i)}}right)
end{align}
That is we want $$frac{5sqrt{n}}{sqrt{Var(U_i)}}ge 1.645$$
answered Dec 28 '18 at 14:25
Siong Thye GohSiong Thye Goh
101k1466118
answered Dec 28 '18 at 14:25
Siong Thye GohSiong Thye Goh
101k1466118
answered Dec 28 '18 at 14:25
Siong Thye GohSiong Thye Goh
101k1466118
answered Dec 28 '18 at 14:25
Siong Thye GohSiong Thye Goh
101k1466118
101k1466118
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3054917%2fcentral-limit-theorem-need-help%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
