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Course: Statistics and probability > Unit 11
Lesson 3: Estimating a population mean- Introduction to t statistics
- Simulation showing value of t statistic
- Conditions for valid t intervals
- Reference: Conditions for inference on a mean
- Conditions for a t interval for a mean
- Example finding critical t value
- Finding the critical value t* for a desired confidence level
- Example constructing a t interval for a mean
- Calculating a t interval for a mean
- Confidence interval for a mean with paired data
- Making a t interval for paired data
- Interpreting a confidence interval for a mean
- Sample size for a given margin of error for a mean
- Sample size and margin of error in a confidence interval for a mean
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Example finding critical t value
Given a confidence level, we can calculate a critical value (t*) in a t-distribution with n-1 degrees of freedom.
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- I believe that Sal accidentally selected the wrong option at the end: instead of 2.624, he chooses 2.264.(30 votes)
- Whenever an error like this occurs in a video, a comment will appear to correct it; However, these comments are only visible if you are not watching in full-screen mode.(1 vote)
- I have a question....why is CC only in Bulgarian?(7 votes)
- Huh... for me I can see closed captions in English, it also says that English is the default language. But other than that, I don't know why it's not there in any languages other than English and Bulgarian(1 vote)
- Could anyone please tell me, where are we gonna study about degrees of freedom (D.F) in-depth ?
Thanks in advance! :)(5 votes) - Please fix the Bulgarian-->English translation issue so the English reads correctly. Also cute mistake at3:54(2.264/2.624) I also made this mistake at first.(5 votes)
- Whenever an error like3:54occurs in a video, a comment will appear to correct it; However, these comments are only visible if you are not watching in full-screen mode.(1 vote)
- before calculations, do we need to first check the conditions needed for using a t-test? ie random, Normal and independence(1 vote)
- I think it is needed, but in this question, we don't have to check these conditions because we do not have a choice about that. eg,It is impposible to find a critical t value, for the conditions are not satisfied.(3 votes)
- Does anyone know where I can invT in TI-83 calculator? I couldn't seem to find it.(2 votes)
- I'm using a Ti-84, so I have no idea whether it's going to be different for a 83 but:
press 2nd > vars(dist). It should be under the distribution tab.(1 vote)
- How to download t distribute table?(2 votes)
- You can just google t-value table and it should be there. You can take a screenshot of it or save it to your computer.(1 vote)
- Find the value of t for student's distribution that satisfy the following condition: the combined area to the right of t and the left of -t is 0.01 and v=5?(1 vote)
- To find the critical value of t for Student's t-distribution that satisfies the condition where the combined area to the right of t and to the left of -t is 0.01 and the degrees of freedom (v) are 5, you can use a t-distribution table or an appropriate statistical software. Here's how you would generally approach finding this t-value:
Identify the significance level for each tail: Since the combined area in the tails is 0.01, and it's split equally between the two tails, the area in each tail is 0.005. This is because you're looking for the critical values at both the positive and negative ends of the t-distribution.
Look up the critical t-value:
- If using a t-distribution table, find the row corresponding to 5 degrees of freedom.
- Find the column that represents an area (in the upper tail) of 0.005. Tables may be organized by the middle area (e.g., 0.99, 0.98, etc.), so you might need to calculate the complementary cumulative area (1 - 0.995 = 0.005).
- Read off the critical t-value. This value will be your t-value for t_0.005 with 5 degrees of freedom.
Using statistical software or an online calculator:
- You can input the degrees of freedom (5) and the cumulative probability for the upper tail (0.995 for the positive critical value since you want the area beyond it to be 0.005).
- The software or calculator will provide the critical t-value, which is typically given as a positive number. Due to the symmetry of the t-distribution, the negative of this value represents the critical value on the left side.
For example, using a standard statistical calculator or software like R, you could use a command like qt(0.995, 5) to get the positive critical value, which should return approximately 3.365. Thus, your critical t-values would be approximately ±3.365 for df = 5 with 0.01 combined area in the tails.
These t-values are what you would use to establish thresholds for hypothesis testing or confidence intervals where the total significance level (alpha) is split equally between the upper and lower tails, corresponding to a critical area of 0.01.(2 votes)
- How do I do this on a Ti-84?(1 vote)
- Finding the critical t-value on a TI-84 calculator:
To find the critical t-value for constructing a confidence interval on a TI-84 calculator, follow these steps:
- Turn on your calculator and access the distribution menu: Press the2nd
key and thenVARS
to access theDISTR
menu.
- SelectinvT
: Scroll down to option4:invT(
or you may need to directly input4
if your calculator shows a numbered list.
- Enter the area to the right of the t-value: Since you need the critical t-value for a 98% confidence interval, you need the t-value that leaves 1% in the upper tail (as the confidence interval is two-tailed). Thus, enter0.01
(1 - 0.99) as the area. Input this value.
- Enter the degrees of freedom: For a sample size of 15, you have 14 degrees of freedom (n - 1
). Input14
after the area, separated by a comma.
- Close the function and calculate: Your input should look something like this:invT(.01, 14)
. PressENTER
to calculate.
- Read the t-value: The calculator will display the t-value, which corresponds to the critical t-value for your specified confidence level and degrees of freedom.(2 votes)
- I think it should be D(2.624)not B(2.264)?(1 vote)
- Whenever an error like this occurs in a video, a comment will appear to correct it; However, these comments are only visible if you are not watching in full-screen mode.(1 vote)
Video transcript
- [Instructor] We are asked
what is the critical value, t star or t asterisk, for constructing a 98% confidence interval for a mean from a sample size of n is
equal to 15 observations? So just as a reminder
of what's going on here, you have some population. There's a parameter here, let's say it's the population mean. We do not know what this
is, so we take a sample. Here we're going to take a sample of 15, so n is equal to 15, and from that sample we can calculate a sample mean. But we also want to construct
a 98% confidence interval about that sample mean. So we're going to go take that sample mean and we're going to go plus or
minus some margin of error. Now in other videos we have talked about that we want to use
the t distribution here because we don't want to
underestimate the margin of error, so it's going to be t star times the sample standard deviation divided by the square root of our sample
size, which in this case is going to be 15, so
the square root of n. What they're asking us is what is the appropriate critical value? What is the t star that we
should use in this situation? We're about to look at, I
guess we call it a t table instead of a z table, but
the key thing to realize is there's one extra variable
to take into consideration when we're looking up the
appropriate critical value on a t table, and that's this
notion of degree of freedom. Sometimes it's abbreviated df. I'm not going in depth
on degrees of freedom. It's actually a pretty deep concept, but it's this idea that you
actually have a different t distribution depending on
the different sample sizes, depending on the degrees of freedom, and your degree of freedom is going to be your sample size minus one. In this situation, our degree
of freedom is going to be 15 minus one, so in this
situation our degree of freedom is going to be equal to 14. This isn't the first time
that we have seen this. We talked a little bit
about degrees of freedom when we first talked about
sample standard deviations and how to have an unbiased estimate for the population standard deviation. In future videos we'll go into
more advanced conversations about degrees of freedom,
but for the purposes of this example, you need to know that when you're looking at the t distribution for a given degree of freedom,
your degree of freedom is based on the sample
size and it's going to be your sample size minus one
when we're thinking about a confidence interval for your mean. Now let's look at the t table. We want a 98% confidence interval and we want a degree of freedom of 14. Let's get our t table out, and I actually copied and pasted this
bottom part and moved it up so you could see the whole thing here. What's useful about this t table is they actually give
our confidence levels right over here, so if you
want a confidence level of 98%, you're going to look at this column, you're going to look at
this column right over here. Another way of thinking about
a confidence level of 98%, if you have a confidence level of 98%, that means you're leaving 1% unfilled in at either end of the
tail, so if you're looking at your t distribution,
everything up to and including that top 1%, you would
look for a tail probability of 0.01, which is, you
can't see right over there. Let me do it in a slightly brighter color, which would be that tail
probability to the right. Either way, we're in this
column right over here. We have a confidence level of 98%. Remember, our degrees of freedom, our degree of freedom here,
we have 14 degrees of freedom, so we'll look at this row right over here. So there you have it. This is our critical t value, 2.624. So let's just go back here. 2.264 is this choice right
over here, and we're done.