hypothesis testing and confidence interval for a population proportion
I'm trying to understand why a basic undergraduate intro to statistics still uses the normal approximation to do hypothesis tests for population proportions. It used to be that the computations were much easier using a normal approximation, but this is no longer the case.
Technically speaking, one "should" use the hypergeometric distribution, but since the population size is usually extremely large and rarely known, it seems best in these cases to use a binomial distribution instead.
Example. Let's say we want to test $H_0:p=.46$ versus $H_a:p<.46$. We get a sample proportion of $559/1267$. Using the normal approximation (in particular, using 1-PropZTest in the TI-84) we find that the P-value is
$$Pleft(left.hat{p}leqfrac{559}{1267}right|p=.46right)approx.0897$$
This is not hard to compute with the binomcdf function on the TI-84. This is because
$$Pleft(left.hat{p}leqfrac{559}{1267}right|p=.46right)approx P(xleq 559)$$
where $x$ is the number of "successes" in 1267 "trials" of a binomial distribution with $p=.46$. Using binomcdf on the TI-84 we obtain a P-value of approximately $.0942$. This is not an insignificant difference. The relative error here is about $-4.8%$.
If we use continuity correction with the normal distribution then we obtain $.0943$ which is a lot better of an approximation, but this is just as difficult if not more so to compute than the even more accurate binomial approximation.
Question 1. Are there any "real-world" circumstances under which it is appropriate to use a normal approximation without continuity correction?
It seems to me that the answer must be "no," but I am to instruct students with a textbook (Sullivan) that uses normal approximation without continuity correction for hypothesis testing. In fact, it gets worse because Sullivan rounds midway through his calculations which increases the error. His answer to the above was $.0869$, inflating the relative error to a whopping $-7.8%$!
Question 2. Is this a serious flaw in the Sullivan textbook, or should I just live with such approximations?
Now, we can certainly use the binomial distribution for a 1-tailed test, as the P-value in that case has a clear interpretation as above. P-values for symmetric distributions in the two-tailed case are also natural. But the binomial distribution is nonsymmetric.
Question 3. How do we interpret the P-value for a two-tailed proportion test using the binomial distribution?
And similarly:
Question 4. How do we interpret a confidence interval for a binomial distribution?
Thanks in advance!
statistics education hypothesis-testing
asked 2 days ago
Ben WBen W
1,995615
add a comment |
I'm trying to understand why a basic undergraduate intro to statistics still uses the normal approximation to do hypothesis tests for population proportions. It used to be that the computations were much easier using a normal approximation, but this is no longer the case.
Technically speaking, one "should" use the hypergeometric distribution, but since the population size is usually extremely large and rarely known, it seems best in these cases to use a binomial distribution instead.
Example. Let's say we want to test $H_0:p=.46$ versus $H_a:p<.46$. We get a sample proportion of $559/1267$. Using the normal approximation (in particular, using 1-PropZTest in the TI-84) we find that the P-value is
$$Pleft(left.hat{p}leqfrac{559}{1267}right|p=.46right)approx.0897$$
This is not hard to compute with the binomcdf function on the TI-84. This is because
$$Pleft(left.hat{p}leqfrac{559}{1267}right|p=.46right)approx P(xleq 559)$$
where $x$ is the number of "successes" in 1267 "trials" of a binomial distribution with $p=.46$. Using binomcdf on the TI-84 we obtain a P-value of approximately $.0942$. This is not an insignificant difference. The relative error here is about $-4.8%$.
If we use continuity correction with the normal distribution then we obtain $.0943$ which is a lot better of an approximation, but this is just as difficult if not more so to compute than the even more accurate binomial approximation.
Question 1. Are there any "real-world" circumstances under which it is appropriate to use a normal approximation without continuity correction?
It seems to me that the answer must be "no," but I am to instruct students with a textbook (Sullivan) that uses normal approximation without continuity correction for hypothesis testing. In fact, it gets worse because Sullivan rounds midway through his calculations which increases the error. His answer to the above was $.0869$, inflating the relative error to a whopping $-7.8%$!
Question 2. Is this a serious flaw in the Sullivan textbook, or should I just live with such approximations?
Now, we can certainly use the binomial distribution for a 1-tailed test, as the P-value in that case has a clear interpretation as above. P-values for symmetric distributions in the two-tailed case are also natural. But the binomial distribution is nonsymmetric.
Question 3. How do we interpret the P-value for a two-tailed proportion test using the binomial distribution?
And similarly:
Question 4. How do we interpret a confidence interval for a binomial distribution?
Thanks in advance!
statistics education hypothesis-testing
asked 2 days ago
Ben WBen W
1,995615
Not familiar with the textbook. So I will leave these as comments. Q1: No reason I can think of not to use continuity correction. But de Moivre–Laplace should be mentioned to students because it was a historical achievement. If one is going to use technology, one might as well skip to using something likeprop.testin R in one shot. One can choose the pedagogical approach based on what one wants the students to take away. Q3) p-value by its philosophical definition doesn't depend on whether the sampling distribution is symmetric or not. Q4) Similar comment holds for conf intervals.
– Just_to_Answer
23 hours ago
add a comment |
I'm trying to understand why a basic undergraduate intro to statistics still uses the normal approximation to do hypothesis tests for population proportions. It used to be that the computations were much easier using a normal approximation, but this is no longer the case.
Technically speaking, one "should" use the hypergeometric distribution, but since the population size is usually extremely large and rarely known, it seems best in these cases to use a binomial distribution instead.
Example. Let's say we want to test $H_0:p=.46$ versus $H_a:p<.46$. We get a sample proportion of $559/1267$. Using the normal approximation (in particular, using 1-PropZTest in the TI-84) we find that the P-value is
$$Pleft(left.hat{p}leqfrac{559}{1267}right|p=.46right)approx.0897$$
This is not hard to compute with the binomcdf function on the TI-84. This is because
$$Pleft(left.hat{p}leqfrac{559}{1267}right|p=.46right)approx P(xleq 559)$$
where $x$ is the number of "successes" in 1267 "trials" of a binomial distribution with $p=.46$. Using binomcdf on the TI-84 we obtain a P-value of approximately $.0942$. This is not an insignificant difference. The relative error here is about $-4.8%$.
If we use continuity correction with the normal distribution then we obtain $.0943$ which is a lot better of an approximation, but this is just as difficult if not more so to compute than the even more accurate binomial approximation.
Question 1. Are there any "real-world" circumstances under which it is appropriate to use a normal approximation without continuity correction?
It seems to me that the answer must be "no," but I am to instruct students with a textbook (Sullivan) that uses normal approximation without continuity correction for hypothesis testing. In fact, it gets worse because Sullivan rounds midway through his calculations which increases the error. His answer to the above was $.0869$, inflating the relative error to a whopping $-7.8%$!
Question 2. Is this a serious flaw in the Sullivan textbook, or should I just live with such approximations?
Now, we can certainly use the binomial distribution for a 1-tailed test, as the P-value in that case has a clear interpretation as above. P-values for symmetric distributions in the two-tailed case are also natural. But the binomial distribution is nonsymmetric.
Question 3. How do we interpret the P-value for a two-tailed proportion test using the binomial distribution?
And similarly:
Question 4. How do we interpret a confidence interval for a binomial distribution?
Thanks in advance!
statistics education hypothesis-testing
asked 2 days ago
Ben WBen W
1,995615
I'm trying to understand why a basic undergraduate intro to statistics still uses the normal approximation to do hypothesis tests for population proportions. It used to be that the computations were much easier using a normal approximation, but this is no longer the case.
Technically speaking, one "should" use the hypergeometric distribution, but since the population size is usually extremely large and rarely known, it seems best in these cases to use a binomial distribution instead.
Example. Let's say we want to test $H_0:p=.46$ versus $H_a:p<.46$. We get a sample proportion of $559/1267$. Using the normal approximation (in particular, using 1-PropZTest in the TI-84) we find that the P-value is
$$Pleft(left.hat{p}leqfrac{559}{1267}right|p=.46right)approx.0897$$
This is not hard to compute with the binomcdf function on the TI-84. This is because
$$Pleft(left.hat{p}leqfrac{559}{1267}right|p=.46right)approx P(xleq 559)$$
where $x$ is the number of "successes" in 1267 "trials" of a binomial distribution with $p=.46$. Using binomcdf on the TI-84 we obtain a P-value of approximately $.0942$. This is not an insignificant difference. The relative error here is about $-4.8%$.
If we use continuity correction with the normal distribution then we obtain $.0943$ which is a lot better of an approximation, but this is just as difficult if not more so to compute than the even more accurate binomial approximation.
Question 1. Are there any "real-world" circumstances under which it is appropriate to use a normal approximation without continuity correction?
It seems to me that the answer must be "no," but I am to instruct students with a textbook (Sullivan) that uses normal approximation without continuity correction for hypothesis testing. In fact, it gets worse because Sullivan rounds midway through his calculations which increases the error. His answer to the above was $.0869$, inflating the relative error to a whopping $-7.8%$!
Question 2. Is this a serious flaw in the Sullivan textbook, or should I just live with such approximations?
Now, we can certainly use the binomial distribution for a 1-tailed test, as the P-value in that case has a clear interpretation as above. P-values for symmetric distributions in the two-tailed case are also natural. But the binomial distribution is nonsymmetric.
Question 3. How do we interpret the P-value for a two-tailed proportion test using the binomial distribution?
And similarly:
Question 4. How do we interpret a confidence interval for a binomial distribution?
Thanks in advance!
statistics education hypothesis-testing
statistics education hypothesis-testing
asked 2 days ago
Ben WBen W
1,995615
asked 2 days ago
Ben WBen W
1,995615
asked 2 days ago
Ben WBen W
1,995615
asked 2 days ago
Ben WBen W
1,995615
asked 2 days ago
Ben WBen W
1,995615
1,995615
Not familiar with the textbook. So I will leave these as comments. Q1: No reason I can think of not to use continuity correction. But de Moivre–Laplace should be mentioned to students because it was a historical achievement. If one is going to use technology, one might as well skip to using something likeprop.testin R in one shot. One can choose the pedagogical approach based on what one wants the students to take away. Q3) p-value by its philosophical definition doesn't depend on whether the sampling distribution is symmetric or not. Q4) Similar comment holds for conf intervals.
– Just_to_Answer
23 hours ago
add a comment |
Not familiar with the textbook. So I will leave these as comments. Q1: No reason I can think of not to use continuity correction. But de Moivre–Laplace should be mentioned to students because it was a historical achievement. If one is going to use technology, one might as well skip to using something likeprop.testin R in one shot. One can choose the pedagogical approach based on what one wants the students to take away. Q3) p-value by its philosophical definition doesn't depend on whether the sampling distribution is symmetric or not. Q4) Similar comment holds for conf intervals.
– Just_to_Answer
23 hours ago
Not familiar with the textbook. So I will leave these as comments. Q1: No reason I can think of not to use continuity correction. But de Moivre–Laplace should be mentioned to students because it was a historical achievement. If one is going to use technology, one might as well skip to using something like
prop.test in R in one shot. One can choose the pedagogical approach based on what one wants the students to take away. Q3) p-value by its philosophical definition doesn't depend on whether the sampling distribution is symmetric or not. Q4) Similar comment holds for conf intervals.– Just_to_Answer
23 hours ago
Not familiar with the textbook. So I will leave these as comments. Q1: No reason I can think of not to use continuity correction. But de Moivre–Laplace should be mentioned to students because it was a historical achievement. If one is going to use technology, one might as well skip to using something like
prop.test in R in one shot. One can choose the pedagogical approach based on what one wants the students to take away. Q3) p-value by its philosophical definition doesn't depend on whether the sampling distribution is symmetric or not. Q4) Similar comment holds for conf intervals.– Just_to_Answer
23 hours ago
add a comment |
0
active
oldest
votes
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%2f3062915%2fhypothesis-testing-and-confidence-interval-for-a-population-proportion%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3062915%2fhypothesis-testing-and-confidence-interval-for-a-population-proportion%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
Not familiar with the textbook. So I will leave these as comments. Q1: No reason I can think of not to use continuity correction. But de Moivre–Laplace should be mentioned to students because it was a historical achievement. If one is going to use technology, one might as well skip to using something like
prop.testin R in one shot. One can choose the pedagogical approach based on what one wants the students to take away. Q3) p-value by its philosophical definition doesn't depend on whether the sampling distribution is symmetric or not. Q4) Similar comment holds for conf intervals.– Just_to_Answer
23 hours ago