### About this deal

Find the standard deviation of all the x-values (call it) and the standard deviation of all the y-values (call it).

The temptation is to say, “Well, I knew Corn-e-stats corn was longer because its sample mean was 8.5 inches and Stat-o-sweet was only 7.5 inches on average. Why do I even need a confidence interval?” All those two numbers tell you is something about those 30 ears of corn sampled. You also need to factor in variation using the margin of error to be able to say something about the entire populations of corn. After delving into the process of how to find a p-value from a test statistic and understanding its significance in hypothesis testing, we now transition to a critical stage: making conclusions. To find the answer using the z-table, find where the row for 1.5 intersects with the column for 0.00; this value is 0.9332. The z-table shows only "less than" probabilities so it gives you exactly what you need for this question. Note: No probability is exactly at one single point, so:

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. where t* is the critical value from the t-distribution with n1 + n2 – 2 degrees of freedom; n1 and n2 are the two sample sizes, respectively; and s1 and s2 are the two sample standard deviations. This t*-value is found on the following t-table by intersecting the row for df = n1 + n2 – 2 with the column for the confidence level you need, as indicated by looking at the last row of the table.

However, a reader whose significance level is 0.01 wouldn’t have enough evidence (based on your sample) to reject Ho because the p-value of 0.026 is greater than 0.01. These results wouldn’t be statistically significant. Understanding how to get a p-value from a test statistic is essential for assessing whether the results of your test are likely to have occurred by chance, assuming the null hypothesis is true. However, this may lead you to wonder whether it’s okay to say “Accept Ho” instead of “Fail to reject Ho.” The answer is a big no.To draw conclusions about Ho (reject or fail to reject) based on a p-value, you need to set a predetermined cutoff point where only those p-values less than or equal to the cutoff will result in rejecting Ho. This cutoff point is called the alpha level (α), or significance level for the test. Critical values ( z *-values) are an important component of confidence intervals (the statistical technique for estimating population parameters). However, if you plan to make decisions about Ho by comparing the p-value to your significance level, you must decide on your significance level ahead of time. It wouldn’t be fair to change your cutoff point after you’ve got a sneak peek at what’s happening in the data.

If the p-value is less than or equal to your significance level, then it meets your requirements for having enough evidence against Ho; you reject Ho. While 0.05 is a very popular cutoff value for rejecting Ho, cutoff points and resulting decisions can vary — some people use stricter cutoffs, such as 0.01, requiring more evidence before rejecting Ho, and others may have less strict cutoffs, such as 0.10, requiring less evidence. If the p-value is less than 0.01 (very small), the results are considered highly statistically significant — reject Ho.

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, Ho, that the mean delivery time is 30 minutes max, is incorrect. Your alternative hypothesis (Ha) is that the mean time is greater than 30 minutes. If you know the standard deviation for a population, then you can calculate a confidence interval (CI) for the mean, or average, of that population. When a statistical characteristic that’s being measured (such as income, IQ, price, height, quantity, or weight) is numerical, most people want to estimate the mean (average) value for the population. You estimate the population mean, μ, by using a sample mean, x̄, plus or minus a margin of error. The result is called a confidence interval for the population mean, μ.