What is the first additional assumption needed to turn a descriptive summary into a hypothesis test?

The data must be a random sample from the population of interest. Non-random samples invalidate the test because they break the link between the sample and the population.

What sample size is required for the Central Limit Theorem to apply in a test about the mean?

Typically n ≥ 30. If the population is normally distributed, you can use a smaller sample size.

What sample facts are needed for a one-sample t-test about the mean?

You need the sample mean (x̄), sample standard deviation (s), and sample size (n).

What sample facts are needed for a one-sample z-test about a proportion?

You need the sample proportion (p̂) and the sample size (n). The sample must also satisfy np₀ ≥ 10 and n(1-p₀) ≥ 10.

What is a critical value?

The threshold that the test statistic must exceed to reject the null hypothesis at a given significance level (α). It depends on the sampling distribution of the test statistic.

Can descriptive statistics alone determine statistical significance?

No. Descriptive statistics summarise the sample but do not account for sampling variability or the chosen significance level.

What is the most common mistake when moving from descriptive statistics to a hypothesis test?

Using the wrong sample size in the standard error calculation, which changes the test statistic and the conclusion.

From Descriptive Statistics to a Critical Value Explained

You have a sample. Five numbers: 5, 7, 9, 11, 13. You calculate the mean (9) and the sample standard deviation (3.16). That is descriptive statistics: you summarised what you have. The step from descriptive statistics to a critical value demands additional assumptions and sample facts that descriptive summaries do not supply.

A Worked Example

Take the sample mean of 9 and the sample standard deviation of 3.16 from the worked example. Without more information, those numbers describe only those five data points. To test whether the population mean could be 8, you need the sample size (n = 5) and the hypothesised population mean (μ₀ = 8). Those are sample facts, not summary statistics.

You also need assumptions. The data must be a random sample from the population of interest. If you grabbed the nearest five people in a coffee shop, you cannot test a population mean. The observations must be independent. Repeated measurements on the same person violate that.

Calculation Steps

With n = 5, x̄ = 9, s = 3.16, and μ₀ = 8, compute the test statistic. The formula for a one-sample t-test is t = (x̄ - μ₀) / (s / √n). Plug the numbers: t = (9 - 8) / (3.16 / √5) = 1 / (3.16 / 2.236) = 1 / 1.414 = 0.707.

That t-value of 0.707 is a test statistic, not a critical value. To decide whether 0.707 is large enough to reject the null hypothesis, you need a critical value. That is the threshold the test statistic must exceed. At this point, you need a critical value calculator that accounts for the significance level (α) and degrees of freedom (n - 1 = 4). For α = 0.05 in a two-tailed test with 4 degrees of freedom, the critical t-value is approximately 2.776.

Your test statistic (0.707) does not exceed 2.776. You fail to reject the null hypothesis. The sample does not provide enough evidence that the population mean differs from 8.

Why the Method Works

The explanatory gap between a data summary and a hypothesis test is not magic. It is a set of conditions and numbers you must bring. Here is what you need, in order.

Additional Assumptions

Random sampling is the first requirement. A convenience sample or a self-selected group breaks the link between the sample and the population. The sample size must be at least 30 for the Central Limit Theorem to apply to tests about the mean, unless the population is normally distributed. For your sample of 5, you must assume the population is normal. For tests about proportions, the sample must satisfy np₀ ≥ 10 and n(1-p₀) ≥ 10, where p₀ is the hypothesised proportion.

Additional Sample Facts

Beyond the descriptive summaries, you need the sample mean (x̄) and sample standard deviation (s) for a one-sample t-test about the mean. For a one-sample z-test about a proportion, you need the sample proportion (p̂). In both cases, you need the sample size (n) to compute the standard error and degrees of freedom.

The Critical Threshold

The critical value belongs only after the descriptive stage and test choice. It depends on the sampling distribution of the test statistic. For a t-test, it comes from the t-distribution with n-1 degrees of freedom. For a z-test on proportions, it comes from the standard normal distribution. Choose your significance level (α) before you look up the critical value. The test statistic must exceed that threshold to reject the null hypothesis.

Failure Case and Limit

The most common failure in this step is forgetting that descriptive statistics alone cannot determine statistical significance. They do not account for sampling variability or the chosen significance level. Your sample mean of 9 might look obviously different from the hypothesised mean of 8, but with n = 5 and a sample standard deviation of 3.16, the variability is too high to conclude anything. The test statistic (0.707) is far below the critical value (2.776).

Another failure mode is using the wrong sample size. If you used n = 5 in the standard error calculation but your sample actually had 50 data points, the standard error would be much smaller, and the test statistic much larger. A small error in n changes the conclusion from 'fail to reject' to 'reject'.

A final limit: the assumptions may not hold. If the sample is not random, or if the population is not normal and n is small, the test is invalid. Your careful calculation of the t-value and critical value is wasted on bad input.