Determine whether the following $H_0$ can be accepted or not using hypothesis testing
$H_0 : p=0.5$
$H_1 : p > 0.5 $
where p is the probability of heads from a coin flip.
Let $W_1$ be the number of heads from 10 coin flips and and $W_2$ be the number of heads from 1000 coin flips.
Accept $H_0$ if $W_1 leq c_1 $ and $W_2 leq c_2$, otherwise accept $H_1$. Determine the value of $c_1$ and $c_2$, or in other words the number of heads, using a level 0.05 test so that we may accept $H_0$.
I know that coin flips follow a binomial distribution, however I am not sure how to determine the two values using that. So far I've only solved problems with normal distributed random events. I am aware of CLT but is n=10 large enough to use it? For n=1000 it seems more reasonable. So do I have to use the fact that they are binomial distributed or is there some formula or trick that I am unaware of? Any help is appreciated!
statistics binomial-distribution hypothesis-testing
asked Nov 29 at 8:14
CruZ
406
add a comment |
$H_0 : p=0.5$
$H_1 : p > 0.5 $
where p is the probability of heads from a coin flip.
Let $W_1$ be the number of heads from 10 coin flips and and $W_2$ be the number of heads from 1000 coin flips.
Accept $H_0$ if $W_1 leq c_1 $ and $W_2 leq c_2$, otherwise accept $H_1$. Determine the value of $c_1$ and $c_2$, or in other words the number of heads, using a level 0.05 test so that we may accept $H_0$.
I know that coin flips follow a binomial distribution, however I am not sure how to determine the two values using that. So far I've only solved problems with normal distributed random events. I am aware of CLT but is n=10 large enough to use it? For n=1000 it seems more reasonable. So do I have to use the fact that they are binomial distributed or is there some formula or trick that I am unaware of? Any help is appreciated!
statistics binomial-distribution hypothesis-testing
asked Nov 29 at 8:14
CruZ
406
add a comment |
$H_0 : p=0.5$
$H_1 : p > 0.5 $
where p is the probability of heads from a coin flip.
Let $W_1$ be the number of heads from 10 coin flips and and $W_2$ be the number of heads from 1000 coin flips.
Accept $H_0$ if $W_1 leq c_1 $ and $W_2 leq c_2$, otherwise accept $H_1$. Determine the value of $c_1$ and $c_2$, or in other words the number of heads, using a level 0.05 test so that we may accept $H_0$.
I know that coin flips follow a binomial distribution, however I am not sure how to determine the two values using that. So far I've only solved problems with normal distributed random events. I am aware of CLT but is n=10 large enough to use it? For n=1000 it seems more reasonable. So do I have to use the fact that they are binomial distributed or is there some formula or trick that I am unaware of? Any help is appreciated!
statistics binomial-distribution hypothesis-testing
asked Nov 29 at 8:14
CruZ
406
$H_0 : p=0.5$
$H_1 : p > 0.5 $
where p is the probability of heads from a coin flip.
Let $W_1$ be the number of heads from 10 coin flips and and $W_2$ be the number of heads from 1000 coin flips.
Accept $H_0$ if $W_1 leq c_1 $ and $W_2 leq c_2$, otherwise accept $H_1$. Determine the value of $c_1$ and $c_2$, or in other words the number of heads, using a level 0.05 test so that we may accept $H_0$.
I know that coin flips follow a binomial distribution, however I am not sure how to determine the two values using that. So far I've only solved problems with normal distributed random events. I am aware of CLT but is n=10 large enough to use it? For n=1000 it seems more reasonable. So do I have to use the fact that they are binomial distributed or is there some formula or trick that I am unaware of? Any help is appreciated!
statistics binomial-distribution hypothesis-testing
statistics binomial-distribution hypothesis-testing
asked Nov 29 at 8:14
CruZ
406
asked Nov 29 at 8:14
CruZ
406
edited Nov 29 at 8:31
asked Nov 29 at 8:14
CruZ
406
asked Nov 29 at 8:14
CruZ
406
asked Nov 29 at 8:14
CruZ
406
406
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
For $n=1000$ it is certainly reasonable to just use a normal approximation. Be careful to use a one-tailed test.
For $n$ too small to use CLT (10 is on the small side), you can go back to the basic definition of a binomial distribution to write
$$alpha=Pr(W_1>c_1,|,H_0)=sum_{i=c_1+1}^{10}binom{10}{i}left(frac{1}{2}right)^{10}text{,}$$
and solve for the smallest value of $c_1$ that leaves this probability below 0.05.
This is known as the (binomial) exact test since no approximations are used. Some authors take issue with saying the test is "at significance level 0.05" when the actual level is much lower. Of course, with 10 flips it is impossible to get that close to 0.05.
Note that we usually do not talk about "accepting" $H_0$ so much as failing to reject it, because not (enough) evidence that $H_0$ is false is not the same as evidence that $H_0$ is in fact true, in this frequentist setting.
answered Nov 29 at 10:40
obscurans
748110
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%2f3018346%2fdetermine-whether-the-following-h-0-can-be-accepted-or-not-using-hypothesis-te%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
For $n=1000$ it is certainly reasonable to just use a normal approximation. Be careful to use a one-tailed test.
For $n$ too small to use CLT (10 is on the small side), you can go back to the basic definition of a binomial distribution to write
$$alpha=Pr(W_1>c_1,|,H_0)=sum_{i=c_1+1}^{10}binom{10}{i}left(frac{1}{2}right)^{10}text{,}$$
and solve for the smallest value of $c_1$ that leaves this probability below 0.05.
This is known as the (binomial) exact test since no approximations are used. Some authors take issue with saying the test is "at significance level 0.05" when the actual level is much lower. Of course, with 10 flips it is impossible to get that close to 0.05.
Note that we usually do not talk about "accepting" $H_0$ so much as failing to reject it, because not (enough) evidence that $H_0$ is false is not the same as evidence that $H_0$ is in fact true, in this frequentist setting.
answered Nov 29 at 10:40
obscurans
748110
add a comment |
For $n=1000$ it is certainly reasonable to just use a normal approximation. Be careful to use a one-tailed test.
For $n$ too small to use CLT (10 is on the small side), you can go back to the basic definition of a binomial distribution to write
$$alpha=Pr(W_1>c_1,|,H_0)=sum_{i=c_1+1}^{10}binom{10}{i}left(frac{1}{2}right)^{10}text{,}$$
and solve for the smallest value of $c_1$ that leaves this probability below 0.05.
This is known as the (binomial) exact test since no approximations are used. Some authors take issue with saying the test is "at significance level 0.05" when the actual level is much lower. Of course, with 10 flips it is impossible to get that close to 0.05.
Note that we usually do not talk about "accepting" $H_0$ so much as failing to reject it, because not (enough) evidence that $H_0$ is false is not the same as evidence that $H_0$ is in fact true, in this frequentist setting.
answered Nov 29 at 10:40
obscurans
748110
add a comment |
For $n=1000$ it is certainly reasonable to just use a normal approximation. Be careful to use a one-tailed test.
For $n$ too small to use CLT (10 is on the small side), you can go back to the basic definition of a binomial distribution to write
$$alpha=Pr(W_1>c_1,|,H_0)=sum_{i=c_1+1}^{10}binom{10}{i}left(frac{1}{2}right)^{10}text{,}$$
and solve for the smallest value of $c_1$ that leaves this probability below 0.05.
This is known as the (binomial) exact test since no approximations are used. Some authors take issue with saying the test is "at significance level 0.05" when the actual level is much lower. Of course, with 10 flips it is impossible to get that close to 0.05.
Note that we usually do not talk about "accepting" $H_0$ so much as failing to reject it, because not (enough) evidence that $H_0$ is false is not the same as evidence that $H_0$ is in fact true, in this frequentist setting.
answered Nov 29 at 10:40
obscurans
748110
For $n=1000$ it is certainly reasonable to just use a normal approximation. Be careful to use a one-tailed test.
For $n$ too small to use CLT (10 is on the small side), you can go back to the basic definition of a binomial distribution to write
$$alpha=Pr(W_1>c_1,|,H_0)=sum_{i=c_1+1}^{10}binom{10}{i}left(frac{1}{2}right)^{10}text{,}$$
and solve for the smallest value of $c_1$ that leaves this probability below 0.05.
This is known as the (binomial) exact test since no approximations are used. Some authors take issue with saying the test is "at significance level 0.05" when the actual level is much lower. Of course, with 10 flips it is impossible to get that close to 0.05.
Note that we usually do not talk about "accepting" $H_0$ so much as failing to reject it, because not (enough) evidence that $H_0$ is false is not the same as evidence that $H_0$ is in fact true, in this frequentist setting.
answered Nov 29 at 10:40
obscurans
748110
edited Nov 29 at 10:45
answered Nov 29 at 10:40
obscurans
748110
answered Nov 29 at 10:40
obscurans
748110
answered Nov 29 at 10:40
obscurans
748110
748110
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3018346%2fdetermine-whether-the-following-h-0-can-be-accepted-or-not-using-hypothesis-te%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
