Talk:Z-test

Z score
Hey, shouldn't the z score be $$\frac{X - \mu}{\sigma}$$? as in Standard_score?

I think there is confusion with standardization. The above formula is used to standardize each data point (X) so that the population has a mean of zero and a range of unity. 194.176.105.153 (talk) 11:24, 15 November 2013 (UTC)

SD / SE unit?
Hi all,

in the line "a mean score of 96 is -2.47 standard deviation units from the population mean", shouldn't it say "standard error units"? I'm not sure about this, so if you know for sure, please change or don't.

Regards WikiBasti 12:16, 25 November 2005 (UTC)

Example misleading
This sentence in the article: "What this tells us is that 49.32% plus 50% or 99.32% of the students who took the test scored better than did the fifth grade students in our sample" is misleading. It's not true that 99.32% of all students do better: what's true is that 99.32% of the time, a randomly selected group of 55 students have a higher average score than these 55 students had. MvH Jan 16, 2006.
 * I think there's actually a factual error in the example. "Next we look the z score up in a Z table and we find that a z score of −2.47 is 49.32%". Typically, Z tables provide cumulative probabilities, not percentages difference from the mean, so -2.47 would produce 0.0068. Furthermore, the example currently is very wordy and would greatly benefit from editing, and possibly the inclusion of an image of a bell curve with alpha shaded and -2.47 indicated. Any volunteers? Evlshout (talk) 02:51, 15 May 2009 (UTC)
 * I decided to restructure the example, hopefully eliminating this and other issues that have been raised. I agree that a figure would be nice, I'll try to get back to it. Skbkekas (talk) 04:09, 15 May 2009 (UTC)

Two-sided test
I think it should be clarified at the bottom of this article that this is a two-sided test as opposed to a one-sided test.

Simple random sample
The article states that the sample must be a simple random sample from the population, but I can't make sense of this. Isn't this the null hypothesis, and consequently, isn't the test only meaningful if the sample is NOT random? The test is used to see if the sample that you drew from the population behaves like a simple random sample...it's only interesting if it doesn't -anon


 * I think you are referring to Z-tests for the population mean. These are tests of the null hypothesis that the population mean is a given value, say zero, against the alternative that the population mean is something else (say, any non-zero value).  Regardless of whether the mean is zero or non-zero, the data can still be a simple random sample.  This doesn't mean that the data necessarily are a simple random sample, but the simplest form of Z-test for expected values is derived under this assumption.  Whether or not the data can be modeled as a simple random sample is usually a product of the design of a study, so is known to the researcher.  See the sampling (statistics) article for a discussion of different types of sampling.  It is rare that one would want to test whether the data formed a simple random sample.  The test is more commonly about the expected value. Skbkekas (talk) 03:24, 15 May 2009 (UTC)

Error in article?
Reject null hypothesis? Shouldn't the null hypothesis be rejected in the example since the Z score of -2.47 is outside the 5% confidence interval?

"In the case of our sample mean, the z score of −2.47 which provides us a value of 49.32% means that 49.32% plus 50% or 99.32% of the population scored closer to the population mean than did our sample of students. Since our sample is outside of this area by 1.82%, we can't reject the null hypothesis because the value of 1.82% is less than 5%, our confidence level.

Therefore we can conclude with a 95% confidence level that the test performance of the students in our sample were within the normal variation."

It looks like whoever wrote the above is confusing tail probability and the z-test.

Error Regarding Population Standard Deviation and Sample Standard Deviation?
In the formula under "First calculate the standard error (SE) of the mean:" shouldn't the sigma instead be 's'? I believe the question here is whether your value falls into your estimated confidence based on your sample. I am not sure what the population standard deviation is used here. I am not certain about this, but it seems intuitively wrong to me The Sleepwalker (talk) 07:11, 5 June 2008 (UTC)

203.143.164.204 (talk) 03:12, 2 April 2008 (UTC)

Edits
I edited the top parts of this page to make it more parallel to the t-test page, which defines a t-test as being any test for which the test statistic follows a t-distribution (rather than just covering the one-sample and two-sample t-tests). Nevertheless, I agree that the one-sample/two-sample tests need to be covered in detail. Also, I generally disagree with the way these tests are presented now. Granted, the test statistic will only be exactly normal if the variance is known, however if the sample size is not too small, many t-statistics are very well approximated by a normal distribution. All statistical testing requires approximations. It simply is not an accurate reflection of how modern professional and academic statisticians think to say that the Z-test can only be applied if the population is known. Skbkekas (talk) 15:24, 23 April 2009 (UTC)

Equation
I think the equation is $$ P(z) = \frac{1}{1 + e^{-\Phi z}} \! $$, but I can't find a source. Dude1818 (talk) 03:07, 16 April 2010 (UTC)

Definition way too general
The majority of test statistics are distributed asymptotically normally, thanks to the CLT, delta-method, and continuous-mapping theorem. Yet I never heard anybody calling them "Z-tests". Either this terminology is outdated and completely fell out of use, or it is exclusive to some small subfield (like sociology, or I don't know), in which case this should be specified explicitly. //  st pasha  »  07:24, 22 February 2011 (UTC)
 * Isn't 'z-test' more or less a synonym for 'Wald test'? I think mathematical-statistical texts would prefer the latter, but less technical sources may prefer the former. I guess 'Wald test' only applies when the estimate is obtained by maximum likelihood though. Should we consider merging the two articles? Qwfp (talk) 08:48, 22 February 2011 (UTC)
 * Wald test, poor, poor article :( It is definitely not the same, since Wald test has asymptotically &chi;$2 k$ distribution; it is applicable for any estimation procedure, as long as the resulting estimator is asymptotically normal.  //  st pasha  »  09:22, 22 February 2011 (UTC)
 * I'm not in favor of specializing this into an article about the comparison of two population means, if that is what is being proposed. That topic is well covered in the Student's t-test article. Granted, people don't generally call something a "Z-test" and leave it at that, since so many tests would fall into that category. But it is not uncommon when discussing a testing situation to say that the test can be carried out "as a Z-test" (after suitable standardization of the test statistic). Since some readers may come to this page looking for information about the two-sample test, it might be useful to highlight this distinction more in the article's introduction. Skbkekas (talk) 15:14, 23 February 2011 (UTC)
 * Two dictionaries that I have give definitions of "z-test" limited to the case where the variance of the population is known and the distribution is normal. One specifically included the two-sample case. If it is to be extended beyond that, citations are needed for such usage.

Example
The following paragraph in the example seems to have an error in the first sentence. Shouldn't it state that 'with probability 0.014, a simple random sample ....'?

"Another way of stating things is that with probability 1 − 0.014 = 0.986, a simple random sample of 55 students would have a mean test score within 4 units of the population mean. We could also say that with 98.6% confidence we reject the null hypothesis that the 55 test takers are comparable to a simple random sample from the population of test-takers."

MarkThoreau (talk) 18:19, 6 March 2021 (UTC)