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## 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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# Using a table to estimate P-value from t statistic

AP.STATS:

VAR‑7 (EU)

, VAR‑7.E (LO)

, VAR‑7.E.1 (EK)

In a significance test about a population mean, we first calculate a test statistic based on our sample results. We can then use a table to estimate the p-value based on that test statistic using a t distribution with n-1 degrees of freedom.

## Want to join the conversation?

- Why should we include the area above and below the 2.75 t value when t = a positive value?(6 votes)
- This is because our alternative hypothesis is that mu is not equal to 0, so we need to look at cases at both extremes from 0.

If the Ha were stating that mu>0, then we would only be concerned with the upper end.

If the Ha were stated that mu<0, then we would be concerned with the lower end.(6 votes)

- Anywhere I can get this t* table?(5 votes)
- Here is a link to the t-table I commonly use:

https://t-tables.net/(1 vote)

- According to my statistic book reference tables.... In this lesson he is using the t-table for a one-tailed test, but according to this question he should be using the t-values needed for rejection of the null hypothesis for the two-tailed test.

right?(2 votes)- Yes, but he multiplied the probability of getting the sample mean or higher by two to get the p-value by using the fact that the t-distribution is symmetric, giving the same result.(4 votes)

- what do you do when a t-value is negative? would you look it up the same way, but find the p-value using 1-P-value(t)?(2 votes)
- You use the closest number to the value on the table.(2 votes)

- The critical calues - t-statistictable in my textbook doesn't look like this. Instead it has a 95% and 99% significnce level column. Can anyone tell me why?(2 votes)
- in need of some help, how do i work out P= 2320(1.2)^x? they say the answer is 11,971 but I'm not sure how they got the answer, could anyone help?(1 vote)
- what should I do when my value is not on the table(1 vote)
- Fewer than 97% of adults have cell phones. A poll of 1069 adults found 91% had cell phones. What is the value of this he test statistic?(0 votes)

## Video transcript

- [Instructor] Caterina was testing her null hypothesis is that the true population mean of some
data set is equal to zero versus her alternative hypothesis, is that it's not equal to zero and then she takes a sample of six observations and then using that sample her test statistic, I can never say that, test statistic was T is equal to 2.75. Assume that the conditions
for inference were met. What is the approximate
P-value for Caterina's test? Like always, pause this video and see if you can figure it out. I just always like to remind ourselves what's going on here, so
there's some population here. She has a null hypothesis that the mean is equal to
zero, or the alternative is that it's not equal to zero. She wants to test her null hypothesis so she takes a sample of size six. From that, since the population parameter we care about is a population mean, she would calculate the sample mean in order to estimate that and the sample standard deviation. From that, we can calculate this T value. The T value is going to be equal to the difference between her sample mean and the assumed population mean from the null hypothesis, that's what this little sub zero means, it means it's the assumed mean from the null hypothesis, divided by our estimate of the standard deviation of the sampling and distribution. I say estimate because unlike when we were dealing with proportions, with proportions we can actually calculate the assumed, based on the null hypothesis, sampling distribution standard deviation, but
here we have to estimate it. It's going to be our
sample standard deviation divided by the square root of N. In this example, they
calculated all of this for us. They said hey, this is going to be equal to 2.75 and so we can just use that to figure out our P-value. Let's just think about what
that is asking us to do. The null hypothesis is
that the mean is zero. The alternative is is that
it is not equal to zero. This is a situation where, if we're looking at the T
distribution right over here, my quick drawing of a T distribution. If this is the mean of our T distribution, what we care about is things that are at least 2.75 above the mean and at least 2.75 below the mean because we care about things that are
different from the mean, not just things that are greater than the mean or less than the mean. We would look at, we would say what's the probability of getting a T value that is 2.75
or more above the mean, and similarly, what's the probability of getting a T value that is
2.75 or more below the mean? This is negative 2.75 right over there. What we have here is a T table and a T table is a little bit different than a Z table because there's several things going on. First of all, you have
your degrees of freedom. That's just going to be
your sample size minus one. In this example, our sample size is six, so six minus one is five, and so we are going to be in
this row right over here. Then what you want to do is, you want to look up your T value. This is T distribution critical values, so we want to look up 2.75 on this row. We see 2.75, it's a little bit less than that but that's the closest value. It's a good bit more than this right over here, so it's a little bit closer to this value than this value. Our tail probability, and remember, this is only giving us this probability right over here, our tail probability is going to be between 0.025 and 0.02 and it's going to be closer to this one. It's gonna be approximately this. It'll actually be a little bit greater because we're gonna go a little bit in that direction because
we are less than 2.757. We can say this is approximately 0.02. That's 0.02 approximately,
the T distribution is symmetric, this is going
to be approximately 0.02. Our P-value, which is going to be the probability of getting a T value that is at least 2.75 above the mean or 2.75 below the mean, the P-value is going to be approximately the sum of these areas, which is 0.04. Then of course, Caterina would want to compare that to her significance level that she set ahead of time, and if this is lower than that, then she would reject the null hypothesis and that would suggest the alternative. If this is not lower than her significance level, well then she would not be able to reject her null hypothesis.