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## Statistics and probability

### Course: Statistics and probability > Unit 12

Lesson 4: Tests about a population mean- Writing hypotheses for a significance test about a mean
- Writing hypotheses for a test about a mean
- Conditions for a t test about a mean
- Reference: Conditions for inference on a mean
- Conditions for a t test about a mean
- When to use z or t statistics in significance tests
- Example calculating t statistic for a test about a mean
- Calculating the test statistic in a t test for a mean
- Using TI calculator for P-value from t statistic
- Using a table to estimate P-value from t statistic
- Calculating the P-value in a t test for a mean
- Comparing P-value from t statistic to significance level
- Making conclusions in a t test for a mean
- Free response example: Significance test for a mean

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# When to use z or t statistics in significance tests

AP.STATS:

VAR‑7 (EU)

, VAR‑7.E (LO)

, VAR‑7.E.1 (EK)

When to use z or t statistics in significance tests.

## Want to join the conversation?

- when n (sample size) is greater or equal to 30, can we use use z statistics because the sampling distribution of the sample mean is approximately normal, right? if this is the case, then why does t table contain rows where the degree of freedom is 100, 1000 etc (i.e. degree of freedom = n - 1)? if n is greater or equal to 30, we would be using a z table anyway, so is the rows in t table that have degree of freedom greater than 30 redundant?(14 votes)
- I guess it's to show that the t-table approixmates to a normal z-table when n i large. In those t-tables I've seen they jump/skip more and more degrees after 30 and ends with the z-table value.(9 votes)

- This video would have been so helpful waaay back when we were first introduced to sampling distributions.(8 votes)
- I agree. He calls this a "primer", which Webster's defines as an introduction. Shouldn't this have come first?(4 votes)

- But when Sal used
**simulation technique**for calculating p value, the answer is very different.

In this previous video- *"Estimating a P-value from a simulation"* https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/idea-of-significance-tests/v/estimating-p-value-from-simulation

I calculated the p value by the formula which Sal just described. However, the answer is very very different.

Simulation p value- 7.5%

formula p value - 0.16%

I have calculated it three times. Can anyone explain this difference in p values ?(3 votes)- Yes, I also get a p-value of 0.16% with the formula.

I think the problem here is that we do not meet all the conditions for inference. We do not meet the normal condition. n < 30. and n * p is way less than 10. This problem with normality we can also see in the simulation that is performed in the problem, where the distribution of dots is very skewed to the right, being very far from being normal.(1 vote)

- Does anyone know at3:53what a standard deviation of the sampling distribution of the sample mean is? If so, any chance can you explain it?(2 votes)
- Say we have a population P.

We are interested in knowing the population parameter mean, but we cant access all the population.

We take a sample called p1 and find its sample mean.

We take a sample again say p2 and find its sample mean.

We do this severally for n samples and so we have n means.

We then plot the means we have a sampling distribution of means.

By the Central Limit Theorem the means would have a normal distribution...just an aside.

Now, finding the standard deviation of that sampling distribution we just plotted is what Sal referred to as the standard deviation of the sampling distribution of the sample mean. It's quite a mouthful. Hopefully you can wrap your head around it now.

H_L.(1 vote)

- Why does Sal introduce another variable of p_0, when he could just use p_1 in calculation Z statistic?(2 votes)
- Let's say 𝐻₀: 𝑝 < 𝑝₁

Then there is no assumed population proportion, we just assume that the true population proportion is less than whatever value 𝑝₁ is, and 𝑝₀ is the true population proportion given that 𝐻₀ is true.

By convention we always treat 𝑝₀ and 𝑝₁ as separate quantities regardless of what 𝐻₀ says.(1 vote)

- This whole video is great for the difference between when we use z-statistics and when we use t-statistics. How do we know when to use Chi-square-statistics versus f-statistics versus ANOVA versus Linear Regression versus Multiple Regression?

I guess I need a chart that compares all the requirements for each type of evaluation similar to the two columns we end up with at4:40!(2 votes) - When using sample standard deviations, Why is the biased n taken instead of the unbiased estimator (n-1)? Thanks!(1 vote)
- 4:00

I've seen this description before and it confuses me greatly. That dividing by the square-root of the population is going to change the standard deviation tremendously. Shouldn't the standard deviation of a sample group be more similar to the standard deviation of the overall population?(1 vote) - 2:44

I don't understand why should we look at the propabilities values that far above or below the mean(1 vote)- because he used inequal sign at the Ha, which means greater than nad less than at the same time(1 vote)

- When finding the Z for proportions, At2:22Sal Khan divides by the standard deviation of the sample. Shouldnt this be the standard deviation of the sampling distribution since we are looking at how many SDs away from the mean our proportion is ,
**within**the sampling distribution?(0 votes)- When he writes it, he explains that it
*is*the SD of the sampling distribution. I think the little p hat might have confused you, but both you and Sal are right.(1 vote)

## Video transcript

- [Tutor] What I wanna do in
this video is give a primer, I'm thinking about when to use a z statistic versus a t statistic, when we are doing significance tests. So there's two major scenarios that we will see in an
introductory statistics class, one is when we are
dealing with proportions, so I'll write that on the
left side right over here and the other is when we
are dealing with means. In the proportion case, when we are doing our significance test, we will set up some null hypothesis, that usually deals with
the population proportion, we might say it is equal to some value, let's just call that P sub one and then maybe you have
an alternative hypothesis, that, well, no, the population proportion is greater than that or less than that or it's just not equal to that, so let me just go with that one, it's not equal to P sub one and then what we do to actually test, to actually do the significance test is we take a sample from the population, it's going to have a sample size of n, we need to make sure that we feel good about making the inference, we've talked about the
conditions for inference in previous videos multiple times, but from this we calculate
the sample proportion and then from this, we
calculate the P value and the way that we do the P value, remember the P value is the probability of getting a sample proportion
at least this extreme and if it's below some threshold, we reject the null hypothesis
and suggest the alternative and over here the way we do that is well, we find an associated z value for that P for that sample proportion and the way that we calculate it, we say, okay look, our z is going to be, how many of the sampling distributions standard deviations are
we away from the mean and remember the mean of
the sampling distribution is going to be the population proportion, so here we've got this sample statistic, this sample proportion, the difference between that
and the assumed proportion, remember when we do
these significance tests, we try to figure out the probability assuming the null hypothesis is true and so when we see this P sub zero, this is the assumed proportion
from the null hypothesis, so that's the difference
between these two, the sample proportion and
the assumed proportion and then you'd wanna divide
it by what's often known as the standard error of the statistic, which is just the standard deviation of the sampling distribution
of the sample proportion and this works out well
for our proportions, because in proportions, I
can figure out what this is, this is going to be
equal to the square root of the assumed population
proportion times one minus the assumed population
proportion, all of that over n and then I would use this z statistic to figure out the P
value and in this case, I would look at both
tails of the distribution, because I care about how far I am either above or below the
assumed population proportion. Now with means, there's
definitely some similarities here, you will make a null hypothesis, maybe you assume the population
mean is equal to mu one and then there's going to be
an alternative hypothesis, that maybe your population
mean is not equal to mu one and you're gonna do something very simple, you take your population,
you take a sample of size n and instead of calculating
a sample proportion, you calculate a sample mean and actually you can
calculate other things, like a sample standard deviation, but now you have an issue, you say, well ideally I
would use a z statistic and you could, if you were able to say, well I could take the difference
between my sample mean and the assumed mean
in the null hypothesis, so that would be this right over here, that's what that zero means, the assumed mean from the null hypothesis and I would then divide by the
standard error of the mean, which is another way of
saying the standard deviation of the sampling distribution
of the sample mean, but this is not so easy to figure out, in order to figure out this, this is going to be the standard deviation of the underlying population divided by the square root of n. We know what n is going to
be, if we conducted a sample, but we don't know what
the standard deviation is, so instead what we do is we estimate this and so we'll take the sample mean, we subtract from that the
assumed population mean from the null hypothesis and we divide by an estimate of this, which is going to be our
sample standard deviation divided by the square root of n, but because this is an estimate, we actually get a better result, instead of saying, hey,
this is an estimate of our z statistic, we will
call this our t statistic and as we will see, we will
then look this up in a t table and this will give us a better
sense of the probability.