Log in Clay Branch 13 years agoPosted 13 years ago. Direct link to Clay Branch's post “At 4:43, you draw an arro...” At • (85 votes) Sal Khan 11 years agoPosted 11 years ago. Direct link to Sal Khan's post “Yes, it should point to n...” From the author:Yes, it should point to n, not s. (33 votes) Greg Mogged 11 years agoPosted 11 years ago. Direct link to Greg Mogged's post “In a problem, how do you ...” In a problem, how do you know when you need to use the Z chart vs the T table • (4 votes) Matthew Daly 11 years agoPosted 11 years ago. Direct link to Matthew Daly's post “If you know the standard ...” If you know the standard deviation of the population, use the z-table. If you don't but you have a large sample size (traditionally over 30, but some teachers might go up to 100 these days), then assume that the population standard deviation is the same as the sample standard deviation and use the z-table. But if you don't know the population standard deviation and have a relatively small sample size, then you use the t-table for greatest accuracy. (35 votes) Largo Terranova 12 years agoPosted 12 years ago. Direct link to Largo Terranova's post “Hello, I dont get it... W...” Hello, I dont get it... What is the difference between Z and t statistic? It's the same formula, for both, and the graph is not different either. A clue anyone? Thanks! • (8 votes) Jonathan Karan 12 years agoPosted 12 years ago. Direct link to Jonathan Karan's post “The Z-score and t-score t...” The Z-score and t-score tables themselves have different numbers in response to the fact that you can't have as much confidence in the data with a smaller sample size. You'll get a different value from Z=1.382 than t=1.382. (12 votes) symmetry 12 years agoPosted 12 years ago. Direct link to symmetry's post “Why is the mean of the t ...” Why is the mean of the t distribution zero and the mean of the z distribution equal to the population mean? • (19 votes) Rae Rohus 11 years agoPosted 11 years ago. Direct link to Rae Rohus's post “Below is what I need to f...” Below is what I need to figure out: Student Opie has a T-score of 60. Obtain the z-score and T-score for EACH student. (0 votes) adar2007 9 years agoPosted 9 years ago. Direct link to adar2007's post “What is the difference be...” What is the difference between a "normal" distribution and a normalized distribution? • (5 votes) robshowsides 9 years agoPosted 9 years ago. Direct link to robshowsides's post “Good question! They are ...” Good question! They are TOTALLY DIFFERENT! A normal distribution just means the good old bell curve that you know and love. The "standard" normal distribution being the bell curve with mean 0 and stdev 1, which lets you use your Z-table. A normalized distribution means any distribution which has a total area (or probability) under it equal to 1. So of course every probability density function (PDF) should be normalized, but sometimes you make up some new shape for a PDF (say, some function f(x)), and you are happy with the shape, but then you calculate the total area under the curve and it's, say, 13. Well, then you have to take the additional step of dividing your new function by 13, so your normalized PDF would be f(x)/13, which would now have a total area of 1 underneath. Just to be clear, the standard normal distribution is, of course, normalized. (10 votes) Edward Malthouse 13 years agoPosted 13 years ago. Direct link to Edward Malthouse's post “The comment that for n>30...” The comment that for n>30 (xbar-mu)/(s/sqrt(n)) is normal is not correct. The convergence of the CLT depends on how non-normal the population distribution is. For example, consider a Bernoulli trial. The rule of thumb to use the normal approximation is that n*pi>5 and n(1-pi)>5. If pi=1%, then n must exceed 500. n=30 is not large enough. When n>30 or so the t and the z distribution are approximately equal and textbooks stop giving percentiles of the t distribution in the tables. • (6 votes) vokal9031 12 years agoPosted 12 years ago. Direct link to vokal9031's post “I think that for a non-bi...” I think that for a non-binomial setting, which has more than two outcomes, and deals with averages, you can attain a probability through a Z-statistic so long as n>30. For a binomial setting, like a Bernoulli trial you state as an example, with only two outcomes, and deals with proportions, then the rule of thumb to use normal approximation is indeed np>5 and n(1-p)>5 (or in other sources they use np>10 and n(1-p)>10). So the difference in the "rules of thumb" for normal approximation depends (5 votes) Adam Vitale 11 years agoPosted 11 years ago. Direct link to Adam Vitale's post “I thought z= (x bar) - (m...” I thought z= (x bar) - (mu) / (Standard dev) Can you please explain? • (4 votes) dysmnemonic 11 years agoPosted 11 years ago. Direct link to dysmnemonic's post “X and X̅ are standardised...” X and X̅ are standardised slightly differently. In both cases, the denominator is the square root of the variance, like so: This fits with what we know about the central limit theorem. For X, the variance is σ². For X̅, however, the variance is σ²/n, because we expect that X̅ will have a smaller variance (or tend to be closer to the mean) as n increases. (4 votes) Adolf Gore 9 years agoPosted 9 years ago. Direct link to Adolf Gore's post “at 5:41 Sal mentions a "r...” at • (3 votes) eigenface 9 years agoPosted 9 years ago. Direct link to eigenface's post “"Rule of thumb" is just a...” "Rule of thumb" is just an expression, it means a good generalization or a simple way to remember something. There is no single "Rule of Thumb." :P (3 votes) Rahul Singh 10 years agoPosted 10 years ago. Direct link to Rahul Singh's post “What do you do if populat...” What do you do if population variance is known but the sample is less than 30? • (3 votes) Rochelle Hall 9 years agoPosted 9 years ago. Direct link to Rochelle Hall's post “What are some other ways ...” What are some other ways to tell whether a z-statistic or a t-statistic should be used? • (1 vote) Dr C 9 years agoPosted 9 years ago. Direct link to Dr C's post “Whether you know the popu...” Whether you know the population standard deviation, or only have the sample standard deviation. That is actually the only thing to consider when choosing between and t and z statistics. (5 votes)Want to join the conversation?
4:43
On a test whose distribution is approximately normal with a mean of 50 and a standard deviation of 10, the results for three students were reported as follows:
Student Paul has a z-score of -1.00.
Student Quincy has a z-score of +2.00.
Show your calculations.
Who did better on the test?
How many standard deviation units is each score from the mean? Compare the results of the three students.
You said z= (x bar) - (mu) / (Standard dev/ square root of n)
For X, Z = (X-μ) / σ
For X̅, Z = (X̅ - μ) / (σ / √n) 5:41
Video transcript
I want to use this videoto kind of make sure we intuitively and otherwise andunderstand the difference between a Z-statistic--something I have trouble saying-- and a T-statistic. So in a lot of what we're doingin this inferential statistics, we're trying tofigure out what is the probability of getting acertain sample mean. So what we've been doing,especially when we have a large sample size-- so letme just draw a sampling distribution here. So let's say we have a samplingdistribution of the sample mean right here. It has some assumed mean valueand some standard deviation. What we want to do is any resultthat we get, let's say we get some samplemean out here. We want to figure out theprobability of getting a result at least asextreme as this. So you can either figure out theprobability of getting a result below this and subtractedthat from 1, or just figure out this arearight over there. And to do that we've beenfiguring out how many standard deviations above the meanwe actually are. The way we figured that out iswe take our sample mean, we subtract from that our meanitself, we subtract from that what we assume the mean shouldbe, or maybe we don't know what this is. And then we divide that by thestandard deviation of the sampling distribution. This is how many standarddeviations we are above the mean. That is that distanceright over there. Now, we usually don't knowwhat this is either. We normally don't knowwhat that is either. And the central limit theoremtold us that assuming that we have a sufficient sample size,this thing right here, this thing is going to be the samething as-- the sample is going to be the same thing as thestandard deviation of our population divided bythe square root of our sample size. So this thing right over herecan be re-written as our sample mean minus the mean ofour sampling distribution of the sample mean divided bythis thing right here-- divided by our population mean,divided by the square root of our sample size. And this is essentially ourbest sense of how many standard deviations away fromthe actual mean we are. And this thing right here, we'velearned it before, is a Z-score, or when we're dealingwith an actual statistic when it's derived from the samplemean statistic, we call this a Z-statistic. And then we could look it upin a Z-table or in a normal distribution table to say what'sthe probability of getting a value of thisZ or greater. So that would give usthat probability. So what's the probabilityof getting that extreme of a result? Now normally when we've donethis in the last few videos, we also do not know what thestandard deviation of the population is. So in order to approximate thatwe say that the Z-score is approximately, or theZ-statistic, is approximately going to be-- so let me justwrite the numerator over again-- over, we estimate thisusing our sample standard deviation-- let me do this ina new color-- with using our sample standard deviation. And this is OK if our samplesize is greater than 30. Or another way to think aboutit is this will be normally distributed if our samplesize is greater than 30. Even this approximation willbe approximately normally distributed. Now, if your sample size is lessthan 30, especially if it's a good bit less than30, all of a sudden this expression will not benormally distributed. So let me re-write theexpression over here. Sample mean minus the mean ofyour sampling distribution of the sample mean divided by yoursample standard deviation over the square root ofyour sample size. We just said if this thing iswell over 30, or at least 30, then this value right here, thisstatistic, is going to be normally distributed. If it's not, if this is small,then this is going to have a T-distribution. And then you're going to do theexact same thing you did here, but now you would assumethat the bell is no longer a normal distribution, so thisexample it was normal. All of Z's are normallydistributed. Over here in a T-distribution,and this will actually be a normalized T-distributionright here because we subtracted out the mean. So in a normalizedT-distribution, you're going to have a mean of 0. And what you're going to do isyou want to figure out the probability of getting a T-valueat least this extreme. So this is your T-value youwould get, and then you essentially figure out the areaunder the curve right over there. So a very easy rule of thumbis calculate this quantity either way. Calculate this quantityeither way. If you will have more than 30samples, if your sample size is more than 30, your samplestandard deviation is going to be a good approximator for your population standard deviation. And so this whole thing isgoing to be approximately normally distributed, and soyou can use a Z-table to figure out the probabilityof getting a result at least that extreme. If your sample size is small,then this statistic, this quantity, is going to have aT-distribution, and then you're going to have to use aT-table to figure out the probability of getting a T-valueat least this extreme. And we're going to see thisin an example a couple of videos from now. Anyway, hopefully that helpedclarify some things in your head about when to use aZ-statistic or when to use a T-statistic.
