What scenarios make a t-test more appropriate than a z-test?

1) Small sample size, 2) unknown/estimated variance, 3) paired data. The small sample size IMO is the most important as the df grows, the "fatness" of the t-distribution grows closer and closer to the normal distribution.

In clinical trials, for large sample sizes, the t-test and z-test results are often similar due to the Central Limit Theorem, which states that the sampling distribution of the sample mean approaches a normal distribution as the sample size becomes large.

Since most has already been said. I will just add these conditions to consider in order for t-test to be effective especially for inference purposes Conditions 1. Independence: The samples must be independent (except for paired t-test). 2. Normality: The data should be approximately normally distributed, especially for small sample sizes. 3. Homogeneity of variance: For the independent two-sample t-test, the variances of the two groups should be approximately equal. If not, a variant called Welch's t-test can be used. When these conditions are met, a t-test is an appropriate and powerful tool for hypothesis testing in statistics.

A t-test and a z-test are both statistical tests used to determine if there is significant difference between the means of two groups, or to determine if a sample mean significantly differs from a (known) population mean but these tests are used under different conditions and assumptions. A t-test is used when the sample size is small and/or the standard deviation of the population is unknown, and accounts for variable distribution. On the other hand, z-test with a large sample size, standard deviation is known, and normal standard deviation is used.

when sample size is small. With large sample size you will get almost the same result.LArge means more than 120. Also there are other considerations, like known/unknown variances, but experience shows that the key is sample size.

Z-test has too many limitations: - must use large population size - or must have normal distribution - or have known population distribution deviation ..... Therefore, when in doubt, don't use z-tests, use t-test ;)

A z-test is used to determine whether two population means are different (when population means are known) and the sample size is large. An important assumption is that the data must fit approximately a normal distribution for the test to work. For performing the Z test, calculating mean and standard deviation is required.

The z test is the idealized condition. It is a powerful parametric test and can if we have a large sample size (one sample or two sample) and we know sigma (standard deviation of the population). Rarely, if ever, do we have the population standard deviation. Therefore, a more robust (less powerful) parametric test is normally used called a t test which takes into account sample size and adjusts the critical value accordingly (larger critical values for low samples sizes and smaller critical values for larger sample sizes). The t-test also allows for the use of the sample standard deviation to approximate the population standard deviation. Both test can be one sample or two sample tests.

Following are some conditions to be met to conduct a z test 1. Needs population standard deviation to be known. 2. Relies heavily on normality (of samples or population) 3. Needs a large sampling size (which is very difficult)

Z-test can be used in case of large samples to check the parameters of population using statistics of sample. This test is very effective if population variance and sample size greater than 30, as the sample increases, the efficiency of the test increases.

A t-test is more appropriate than a z-test when the population variance is unknown and needs to be estimated from the sample data. This is often the case in research studies involving small sample sizes, where the true population parameters are not readily available. The t-test accommodates the added uncertainty that comes with estimating the population standard deviation by using the sample standard deviation, adjusting the distribution used for hypothesis testing to be more conservative than the normal distribution used in a z-test. This adjustment is crucial for maintaining the accuracy of confidence intervals and significance tests, especially when sample sizes are smaller than 30.

A t-test is typically more appropriate than a z-test when the sample size is small (less than 30) and the population variance is unknown. In such cases, the t-test uses the sample standard deviation to estimate the population standard deviation, which provides a more accurate test statistic. On the other hand, a z-test is suitable when the population variance is known or when the sample size is larger than 30, ensuring that the sample mean is normally distributed due to the Central Limit Theorem. This makes the z-test a robust choice for larger sample sizes and known population variances.

For unknown variance, the T-test is more useful, as for the Z-test, the assumption is that the population mean is known. This is because the test is reasonably reliable even if the sample size is small.

The t-test is based on the student-t distribution, which is a modified version of the normal distribution adjusted to be used with small samples that do not have enough data to estimate the population variance.

To initiate a Z-test, consider a particular job scenario in which we need to compare variables such as averages or test hypotheses regarding a large group. Think back to when you have dealt with similar situations in the past; this might help you determine what works and what doesn't. It's critical to be explicit about the objects you're comparing and assessing. After that we can take care of it.

The central limit theorem states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. For Central limit theorem to work sample size should be large, given T test doesn't require Central limit theorem it works well with small sample size (<30 points)

A T-test is preferable if sample is relatively small to work with. Another reason a t-test is chosen when the population variance is unknown.

Given that the T-test doesn't require the Central limit theorem, it is more flexible and robust. It doesn't require distribution to be normal and thus works better with even small data samples, unknown variance, and samples with outliers.

The t-test is more appropriate than the z-test when handling small sample sizes or unknown population variances. It adjusts for data variability and uncertainty in estimating population standard deviation from the sample, making it highly robust and flexible. This adaptability makes the t-test suitable for various experimental designs, including those with unequal variances between groups or non-normally distributed data. Its ability to provide reliable confidence intervals and p-values in these scenarios makes it a preferred choice for many statistical analyses, especially in research with limited samples and imperfect data conditions.

This can be useful for search systems. If you compare 2 search algorithms, make a bunch of queries to both of them and measure a search quality metric (e.g. NDCG@k) for each response. The metric values for the same query and different algorithms are paired.

In a matched pairs design, a t-test is more appropriate than a z-test due to the specific structure of the data, where each subject is paired with another based on certain criteria, and differences within each pair are analyzed. The t-test is ideal for this setup because it effectively handles the differences between paired observations, assuming these differences are normally distributed. Since the sample variance of the differences must be estimated from the data itself—often with a relatively small number of pairs—a t-test, which adapts to the sample size and variability, provides a more reliable and robust analysis compared to a z-test that assumes a known population variance.

This test can be viewed as a one sample test on the differences. The assumption of the paired t-test is the normality of the differences between pairs. Actually, the sampling distribution of mean differences is what needs to be normal. This can be assumed if the differences themselves are normally distributed.

A t-test should always be paired with a nonparametric test, then compare the results. If they match, then there's an extra amount of what I call "management feel-good" evidence-- justification for management to make decisions and erroneously declare, "See statistic proves it." If they don't match, then a choice of which to believe presents itself. The important issues are was the experiment set up in ways reducing bias/confounding factors and do results corroborate or deny the claim for which the experiment was run, assuming the claim was developed using solid rationale. Statistics are simply tools helping researchers understand phenomena, not the arbiter of truth. For a simple decision, often graphical analysis provides enough evidence.

A t-test is more appropriate when the sample size is small or the population standard deviation is unknown, whereas a z-test is suitable for large sample sizes with a known population standard deviation. The t-test accounts for the uncertainty introduced by small sample sizes, providing more accurate results.

If your sample is small enough that you have to worry about the differences between t test and z test, it might be worthwhile to consider one of 'exact tests' that calculate valid p values regardless of underlying distributional assumptions. Permutation test is one such example.

As all standard problems using a z-test have already been discussed before, I only want to stress one important aspect: Although a t-Test may be an appropriate method in case the requirements for a z-Test are violated, it does not imply that a t-Test is an appropriate approach at all. Data are likely to follow a t-distribution in case the underlying data are (approximately) normal distributed. So it is highly recommended to also take into account non-parametric tests. Finally, boostrapping is a sophisticated approach to "test" independently of the underlying distribution of the data.