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## Statistics and probability

### Course: Statistics and probability>Unit 12

Lesson 5: More significance testing videos

# Small sample hypothesis test

Sal walks through an example of a hypothesis test where he determines if there is sufficient evidence to conclude that a new type of engine meets emission requirements. Created by Sal Khan.

## Want to join the conversation?

• I've always been confused about what a degree of freedom is. My textbook is very unclear, and wikipedia isn't much help either. Wikipedia describes degrees of freedom as "the number of values in the final calculation of a statistic that are free to vary", which is very vague. Is anyone really clear on what this is? I've seen it used all the time in hypothesis tests, but it's always baffled me •   Correct me if I'm wrong, but the way I see it can be illustrated by the following example. Lets say you have de letters A, B, C and D and you have four boxes under which those letters are hidden, called 1, 2, 3 and 4. The letters are randomly hidden under the boxes so you have to guess them. You open box 1 and see the letter C, so that one's out. Box 2 reveiles A, so that one's out aswell and the letters B and D are left. However, if you open the third box and you see the letter, you automatically know what's below box number 4 aswell. So if box 3 reveiles the letter D, you automatically know that B is below box 4. Hence, you have one degree of freedom less, since the last letter is known when the previous three boxes are lifted and there's no need to lift up the last box.
• Correction @; it should use "<=" instead of "=", that is: P( xbar <= 17.7 | H0 )<0.01. p-values are the probability of the statistic coming out "more extreme" than what was observed. This makes sense since we are working with a one-sided test that rejects only if the mean is low. (For the two-sided test, you double the probability to represent both tails.) • I agree that the probability phrasing in the video is incorrect. It should be <=. Since this is a continuous distribution, the probability of getting any single value is actually zero. So, P(xbar = 17.7|H0 is true) = 0). We are truly looking for the probability of getting a value of xbar more extreme than the observed value of 17.7. Later in the video, Sal shifts gears to examining for a value that is more extreme (than the t-statistic), but that "more extreme than" bit should have been present from the beginning of the analysis.
• Sal, I am struggling a bit with the reasoning on this question. If the null states we failed to meet new emissions standards, then if we have a low probability of getting 17.17 (a lower emission average), then wouldn't that mean the null is true? • Or phrased without statistical jargon: Assuming that this type of engine does NOT meet the emission standard, there is a fairly low chance (<1%) that, when we randomly test 10 engines, we'll get an average result like 17.17ppm (which suggests the contrary -- that the engine DOES meet the standard). Therefore it is safe to conclude that, in reality, the engine DOES in fact meet the standard.
• Why do we actually use s / sqrt(N) and not s / sqrt(N-1) ? I thought that we used the latter if the sample size is small, or am I wrong? When do you use the one or the other? • Dividing by n-1 is used when we calculate the standard deviation, s. Once we've done that, we've already adjusted for the bias. The calculation of s / sqrt(n) is calculating the standard error of the sample mean (well, an estimate of it). This calculation uses just sqrt(n) in the denominator.
• I have a basic question on the null hypothesis (H0). Why wasn't the null hypothesis stated as x<20? Is it because the question mentioned Type 1 error or is there some other reasoning for assessing the problems in general? • Hypothesis tests are designed to prove the alternative hypothesis, so we try to put what we want to show into H1, and use the opposite of it as the null, Ho.

And yes, this is related to Type I Error - which is the probability of incorrectly deciding that H1 is true. So in this case, if we rejected Ho (that is, conclude the new engine design meets the emission requirements), then there is only a 1% chance that we made a mistake.

• What does Sal mean by a normalised t-distribution? • Very basic question, but been a long time since i've done any statistics. So may I please ask you how you found the standarddiviation? Left my calculator at school, so cant try, but is it. )(15,6 - 17,17)^2)* 1/10 + .......((13,9- 17,17)^2)*1/10 ? • When referencing the t-table, why did Sal decide to use the one tailed test rather than the two-tailed test? • Why is the null hypothesis u=20 ppm and not u is greater than or equal to 20 ppm?  