You might ask, "Why double the probabilities if your Ha contains a non-equal-to alternative?" Think of the not-equal-to alternative as the combination of the greater-than alternative and the less-than alternative. If you’ve got a positive test statistic, its p-value only accounts for the greater-than portion of the not-equal-to alternative; double it to account for the less-than portion. (The doubling of one p-value is possible because the Z-distribution is symmetric.) If the p-value is really close to 0.05 (like 0.051 or 0.049), the results should be considered marginally ­significant — the decision could go either way. is the critical t*-value from the t-distribution with n – 1 degrees of freedom (where n is the sample size). For example, suppose you want to find p(Z < 2.13). Using the z-table below, find the row for 2.1 and the column for 0.03. Intersect that row and column to find the probability: 0.9834. Therefore p(Z < 2.13) = 0.9834.

If the results are likely to have occurred under the claim, then you fail to reject Ho (like a jury decides not guilty). If the results are unlikely to have occurred under the claim, then you reject Ho (like a jury decides guilty). The cutoff point between rejecting Ho and failing to reject Ho is another whole can of worms that I dissect in the next section (no pun intended). Placing observations (or points) on a scatterplot is similar to playing the game Battleship. Each observation has two coordinates; the first corresponds to the first piece of data in the pair (that’s the X coordinate; the amount that you go left or right). The second coordinate corresponds to the second piece of data in the pair (that’s the Y-coordinate; the amount that you go up or down). You place the point representing that observation at the intersection of the two coordinates. For example, suppose a pizza place claims their delivery times are 30 minutes or less on average but you think it’s more than that. You conduct a hypothesis test because you believe the null hypothesis, H o, that the mean delivery time is 30 minutes max, is incorrect. Your alternative hypothesis (H a) is that the mean time is greater than 30 minutes. The above figure shows examples of what various correlations look like, in terms of the strength and direction of the relationship. Figure (a) shows a correlation of nearly +1, Figure (b) shows a correlation of –0.50, Figure (c) shows a correlation of +0.85, and Figure (d) shows a correlation of +0.15.

wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, 28 people, some anonymous, worked to edit and improve it over time. A large p-value (> 0.05) indicates weak evidence against the null hypothesis, so you fail to reject the null hypothesis. Your 95% confidence interval for the difference between the average lengths for these two varieties of sweet corn is 1 inch, plus or minus 0.9273 inches. (The lower end of the interval is 1 – 0.9273 = 0. 0727 inches; the upper end is 1 + 0. 9273 = 1. 9273 inches.) Notice all the values in this interval are positive. That means Corn-e-stats is estimated to be longer than Stats-o-sweet, based on your data.

If you are a confused consumer when it comes to links and correlations, take heart; this article can help. You’ll gain the skills to dissect and evaluate research claims and make your own decisions about those headlines and sound bites that you hear each day alerting you to the latest correlation. You’ll discover what it truly means for two variables to be correlated, when a cause-and-effect relationship can be concluded, and when and how to predict one variable based on another. If the p-value is between 0.05 and 0.01 (but not super close to 0.05), the results are considered statistically significant — reject Ho. Suppose you want to estimate with 95% confidence the difference between the mean (average) lengths of the cobs of two varieties of sweet corn (allowing them to grow the same number of days under the same conditions). Call the two varieties Corn-e-stats (group 1) and Stats-o-sweet (group 2). Assume that you don’t know the population standard deviations, so you use the sample standard deviations instead — suppose they turn out to be s1 = 0.40 and s2 = 0.50 inches, respectively. Suppose the sample sizes, n1 and n2, are each only 15. If the p-value is greater than (but not super-close to) 0.05, the results are considered non-significant — you fail to reject Ho.Critical values ( z *-values) are an important component of confidence intervals (the statistical technique for estimating population parameters).

For example, when testing Ho: p = 0.25 versus Ha: p < 0.25, the p-value turns out to be 0.1056. This is because the test statistic was –1.25, and when you look this number up on the Z-table (in the appendix) you find a probability of 0.1056 of being less than this value. If you had been testing the two-sided alternative, Ha: p ≠ 0.25, the p-value would be 2 * 0.1056, or 0.2112.When you test a hypothesis about a population, you find a p-value and use your test statistic to decide whether to reject the null hypothesis. Mastering the process of how to find a p-value from a test statistic is vital for identifying a statistical error in our hypothesis testing. You can use the z-table to find a full set of "less-than" probabilities for a wide range of z-values. To use the z-table to find probabilities for a statistical sample with a standard normal (Z-) distribution, follow the steps below. Using the t-table, locate the row with 14 degrees of freedom and look for 2.35. However, this exact value doesn’t lie in this row, so look for the values on either side of it: 2.14479 and 2.62449. The upper-tail probabilities appear in the column headings; the column heading for 2.14479 is 0.025, and the column heading for 2.62449 is 0.01.