Z-test [BETTER]

A tag_hash_106_-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. Z-test tests the mean of a distribution. For each significance level in the confidence interval, the Z-test has a single critical value (for example, 1.96 for 5% two tailed) which makes it more convenient than the Student's t-test whose critical values are defined by the sample size (through the corresponding degrees of freedom). Both the Z-test and Student's t-test have similarities in that they both help determine the significance of a set of data. However, the z-test is rarely used in practice because the population deviation is difficult to determine.

Z-tests are closely related to t-tests, but t-tests are best performed when the data consists of a small sample size, i.e., less than 30. Also, t-tests assume the standard deviation is unknown, while z-tests assume it is known.

Z-test

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If the standard deviation of the population is unknown and the sample size is greater than or equal to 30, then the assumption of the sample variance equaling the population variance should be made using the z-test. Regardless of the sample size, if the population standard deviation for a variable remains unknown, a t-test should be used instead.

A z-score, or z-statistic, is a number representing how many standard deviations above or below the mean population the score derived from a z-test is. Essentially, it is a numerical measurement that describes a value's relationship to the mean of a group of values. If a z-score is 0, it indicates that the data point's score is identical to the mean score. A z-score of 1.0 would indicate a value that is one standard deviation from the mean. Z-scores may be positive or negative, with a positive value indicating the score is above the mean and a negative score indicating it is below the mean.

If we have a sample size of less than 30 and do not know the population variance, we must use a t-test. This is how we judge when to use the z-test vs the t-test. Further, it is assumed that the z-statistic follows a standard normal distribution. In contrast, the t-statistics follows the t-distribution with a degree of freedom equal to n-1, where n is the sample size.

If the sample size is large enough, then the z-Test and t-Test will conclude with the same results. For a large sample size, Sample Variance will be a better estimate of Population variance, so even if population variance is unknown we can use the z-test using sample variance.

Early Z is not something you can turn on or off through any particular state you set on the device. The hardware will perform z-testing as early as it can and in such a way that you didn't know it was anywhere other than after the pixel shader.

There are certain things you can do though that can restrict the hardware in its ability to do z-testing as early as it might otherwise have been able to. Alpha testing, use of 'discard' (killing the pixel) and alpha to coverage will all certainly disable early-z writes as the pixel shaders need to be run before the hardware can determine whether to write a depth value or not. If you're using alpha testing / discard and don't need z-writes, then turn them off and you stand the best possible chance of early-z being available.

Modifying/writing 'depth' in the pixel shader is a definite no-no if you want early Z. In this situation the hardware can't even perform an early test as it's not yet aware of what the depth of the pixel is until you've decided it in the pixel shader, it can neither perform an early z-test nor an early z write.

If you need to write depth from the pixel shader but can guarantee you'll only be writing a depth value greater than or equal to the one the rasterizer has produced you can use the rather undocumented SV_DepthGreater output semantic. Since you're promising not to write a depth value less than the interpolated depth the hardware can still try and perform an early-z test but then defer the z write until the end of the pixel shader. (There is an SV_DepthLessEqual equivalent if you happen to be using an inverted z-test/z-buffer).

Since it must appear that z-testing happens at the end of the pipeline, using UAVs in the pixel shader will also disable early-z. Since the render target is now not the only output and since the DirectX specification says it must appear that z-testing happens at the end, it follows that your UAV writes should happen even for pixels that will eventually fail the z-test. For this reason an attribute was added in Shader Model 5 called [earlydepthstencil] which tells DirectX that you're happy for early-z to occur (if possible) and not to run the pixel shader even though you've got UAV writes happening.

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In my dataset, I have data on 58 people, so the standard deviation, mean and distributions are all known. My data is being compared to the U.S. population norm data collected by a survey company. The only thing that is known from their data is the mean and the standard deviation. So I would like to run a one-sample z-test that compares my mean against theirs with the null hypothesis being: "There is no difference in the means between the two populations". A couple of questions:

2. Why is it so difficult to find a tutorial on this via Google? Any time I find anything regarding doing a z-test, I mostly get t-test instructions instead, or, if I do get a z-test instructions, I get tutorials on knowing the variance and not specifically how to do a z-test in SAS, but by-hand calculations instead. And if I do get an SAS tutorial, there is nothing said in any example about inputting anything about the variance for PROC FREQ (I'm assuming this is the z-test). So, I gather from your questions, that you have to know the variance for a z-test, then why isn't it talked about in tutorials or specified in PROC FREQ examples, like h0 is?

h = ztest(x,m,sigma) returnsa test decision for the null hypothesis that the data in the vector x comesfrom a normal distribution with mean m and astandard deviation sigma, using the z-test.The alternative hypothesis is that the mean is not m.The result h is 1 if the testrejects the null hypothesis at the 5% significance level, and 0 otherwise.

h= ztest(x,m,sigma,Name,Value) returnsa test decision for the z-test with additionaloptions specified by one or more name-value pair arguments. For example,you can change the significance level or conduct a one-sided test.

The prop.test( ) command performs one- and two-sample tests for proportions, and gives a confidence interval for a proportion as part of the output. For example, in the Age at Walking example, let's test the null hypothesis that 50% of infants start walking by 12 months of age. By default, R will perform a two-tailed test. The variable 'walkby12' that takes on the value of 1 for infants who walked by 1 year of age, and 0 for infants who did not start walking until after they were a year old. Using the table( ) command shows that, in this sample, 36/50=.72 of the infants walked by 1 year. The prop.test( ) procedure will perform the z-test comparing this proportion to the hypothesized value; input for the prop.test is the number of events (36), the total sample size (50), the hypothesized value of the proportion under the null (p=0.50 for a null value of 50%). Specifying 'correct=TRUE' tells R to use the small sample correction when calculating the confidence interval (a slightly different formula), and specifying 'correct=FALSE' tells R to use the usual large sample formula for the z-test for a proportion (since categorical data are not normally distributed, the usual z-statistic formula for the confidence interval for a proportion is only reliable with large samples - with at least 5 events and 5 non-events in the sample).

The prop.test( ) command performs a two-sample test for proportions, and gives a confidence interval for the difference in proportions as part of the output. The z-test comparing two proportions is equivalent to the chi-square test of independence, and the prop.test( ) procedure formally calculates the chi-square test. The p-value from the z-test for two proportions is equal to the p-value from the chi-square test, and the z-statistic is equal to the square root of the chi-square statistic in this situation.

For the one-sample z-test, the null hypothesis isthat the mean of the population from which x is drawn is mu.For the standard two-sample z-tests, the null hypothesis is that thepopulation mean for x less that for y is mu.

Our critical values are based on two things: the directionality of the test and the level of significance. We decided in step 1 that a two-tailed test is the appropriate directionality. We were given no information about the level of significance, so we assume that = 0.05 is what we will use. As stated earlier in the chapter, the critical values for a two-tailed z-test at = 0.05 are z* = 1.96. This will be the criteria we use to test our hypothesis. We can now draw out our distribution so we can visualize the rejection region and make sure it makes sense

We have seen the critical values for z-tests at = 0.05 levels of significance several times. To find the values for = 0.01, we will go to the standard normal table and find the z-score cutting of 0.005 (0.01 divided by 2 for a two-tailed test) of the area in the tail, which is zcrit* = 2.575. Notice that this cutoff is much higher than it was for = 0.05. This is because we need much less of the area in the tail, so we need to go very far out to find the cutoff. As a result, this will require a much larger effect or much larger sample size in order to reject the null hypothesis. 006ab0faaa