Pr(Z ≥ x | H 0) = 1-Φ(x) = the area to the right of x Pr(Z ≤ x | H 0) = Φ(x) = the area to the left of x To compute these probabilities, we can use the cumulative distribution function, (cdf) of N(0,1), which for a real number, x, is defined as:Īlso, p-values can be nicely depicted as the area under the probability density function (pdf) of N(0,1), due to: Right-tailed Z-test: extreme values are those greater than z, so Left-tailed Z-test: extreme values are those smaller than z, so The symmetry of the normal distribution gives: Two-tailed Z-test: extreme values are those whose absolute value exceeds |z|, so those smaller than -|z| or greater than |z|. (The probability of an event calculated under the assumption that H 0 is true will be denoted as Pr(event | H 0).) It is the alternative hypothesis which determines what more extreme means: To find the p-value, you have to calculate the probability that the test statistic, Z, is at least as extreme as the value we've actually observed, z, provided that the null hypothesis is true. Provided that I live in a world where the null hypothesis holds, how probable is it that the value of the test statistic will be at least as extreme as the z- value I've got for my sample? Hence, a small p-value means that your result is very improbable under the null hypothesis, and so there is strong evidence against the null hypothesis - the smaller the p-value, the stronger the evidence. More intuitively, p-value answers the questions: In general, you are strongly advised to report the p-value of your tests!įormally, the p-value is the smallest level of significance at which the null hypothesis could be rejected. However, with help of modern computers, we can do it fairly easily, and with decent precision. Two approaches can be used in order to arrive at that decision: the p-value approach, and critical value approach - and we cover both of them! Which one should you use? In the past, the critical value approach was more popular because it was difficult to calculate p-value from Z-test.
#Graphing calculator with statcalc feature how to
In sections below, we will explain to you how to use the value of the test statistic, z, to make a decision, whether or not you should reject the null hypothesis. However, if the sample is sufficiently large, then the central limit theorem guarantees that Z is approximately N(0,1). If our data does not follow a normal distribution, or if the population standard deviation is unknown (and thus in the formula for Z we substitute the population standard deviation σ with sample standard deviation), then the test statistics Z is not necessarily normal. As Z is the standardization ( z-score) of S n/n, we can conclude that the test statistic Z follows the standard normal distribution N(0,1), provided that H₀ is true.
+ x n follows the normal distribution, with mean n * μ 0 and variance n² * σ. In what follows, the uppercase Z stands for the test statistic (treated as a random variable), while the lowercase z will denote an actual value of Z, computed for a given sample drawn from N(μ,σ²). X̄ is the sample mean, i.e., x̄ = (x 1 +. , x n be an independent sample following the normal distribution N(μ, σ²), i.e., with a mean equal to μ, and variance equal to σ².
0 kommentar(er)
