# When is a test statistic considered unusual?

I am taking an introductory stats course this semester and have two questions regarding the following problem:

Suppose that 8% of college students are vegetarians. Determine if the following statements are true or false, and explain your reasoning:

A random sample of 125 college students where 12% are vegetarians would be considered unusual.

After testing if the sample size meets the conditions for a normal distribution i calculated the standard error using $SE=\sqrt{\frac{{\hat{p}(1-\hat{p})}}{n}}$ using $\hat{p}$ instead of $p$ the population parameter (which is unknown). The standard error was:

$$SE = \sqrt{\frac{0.08(1-0.08)}{125}} = 0.0243$$

next i calculated the test statistic for $\hat{p}$ using $Z = \frac{\hat{p} -p}{SE}$ :

$$Z=\frac{0.12-0.08}{0.0243} = 1.65$$

This means that the point estimate $\hat{p}$ lies 1.65 standard errors away from the mean ($p$). I have two questions:

1.) How many standard deviations do we consider to be 'abnormal' when it comes to a normal distribution? The answer to this question is false based on the calculated Z-score - after how many standard deviations would it become true?

2.) What is the difference between standard deviation and standard error? Am i correct in saying that standard deviation is a measure of spread of values against the mean of a sample itself whilst standard error is a measure of spread of samples against a population mean?

RA Fisher's initial rule of thumb was that any excess of two standard errors was worth investigating further. For a Gaussian distribution, looking at both tails, this had a probability of about $4.55\%$ of occurring, which he rounded to $5\%$: this $5\%$ is consequentially in common use as a measure of statistical significance.

It is arbitrary, and does not correspond to $2$ or $1.96$ standard errors in non-Gaussian cases. Examples include $t$-tests with small samples or $\chi^2$-tests.

The standard error of a statistic (often the mean) is the standard deviation of its sampling distribution: it gets its name as being a measure of the scale of the possible error in estimating a parameter.

This is usually not the standard deviation of the population: for example, if the population has an expected value of $\mu$ and standard deviation of $\sigma$ and you take $n$ independent samples with replacement then the sample mean has expected value $\mu$ and a standard deviation of $\frac\sigma{\sqrt{n}}$, which is rather smaller than $\sigma$; intuitively, a larger sample size is likely to lead to smaller errors.