When σ is unknown, we use t distributions. When the sampling distribution of x̄ is close to normal, we can find the probabilities involving x̄ by standardizing:
Z = (x̄-U) / (σ/ √(n))
Remember that the sampling distribution of x̄ has mean μ and standard deviation σ/√(n).
When we don’t know σ, we estimate it using the sample standard deviation Sx. What happens now when we standardize?
?? = (x̄-U) / (Sx / √(n))
When we use Sx instead of σ, we get values that have much greater spread than z. This statistic has a new distribution that is new to us, called a t distribution.
It has a different shape than the standard normal curve: still symmetric with a single peak at 0, but with much more area in the tails.
The statistic t has the same interpretation as any standardized statistic: it says how far x̄ is from its meanμin standard deviation units. There is a different t distribution for each sample size.
We specify a particular t distribution by giving degrees of freedom (df). When we perform inference about a population meanμusing a t distribution, the appropriate degrees of freedom are found by subtracting 1 from the sample size n, making df = n-1.
We can also write the t distribution with n-1 degrees of freedom as Tn-1 for short.
The t distribution: Degrees of Freedom
Draw an SRS of size n from a large population that has a Normal distribution with mean μ and standard deviation σ. The statistic t = (x̄-U) / (Sx / √(n)) has the t distribution with degrees of freedom, df = n-1. When the population isn’t Normal, this statistic will have approximately a Tn-1 distribution if the sample size is large enough.
Facts about t distributions
- The density curves of the two distributions are similar in shape to the standard normal curve. They are symmetric about zero, single peaked, and bell shaped.
- The spread of the t distribution is a bit greater than that of the standard normal distribution. The two distributions have more probability in the tails and less in the center than does the standard normal. This is true because substituting the estimate Sx for the fixed parameter σ introduces more variation into the statistic.
- As the degrees of freedom increase, the t density curve approaches the standard normal density curve even more closely. This happens because Sx estimates σ more accurately as the sample size increases. So using Sx in place of a causes a little extra variation on the sample is large.
To find the critical value for a t-distribution using your calculator: 2nd -> invT() then enter the area (1-C%/2) and degrees of freedom (n-1).
Conditions for estimating µ
As with proportions, you should check some important conditions before constructing a confidence interval for population mean. Two of these should be familiar by now: the random condition, and when sampling without replacement, the 10% condition. Now instead of the large counts condition we use with proportions, we will use the CLT to confirm that the sampling distribution of x̄ will be approximately normal if the sample size is large enough (n ≥ 30).
Conditions for constructing a confidence interval about a mean
- Random: the data come from a well designed random sample or randomized experiment.
- 10%: when assembling without replacement check that n≤ 1/10N.
- Normal/Large sample: the population has a normal distribution or the sample is large. If the population distribution has an unknown shape and n < 30, use a graph of the sample data to assess the normality of the population. Do not use T procedures if the graph shows strong skewness or outliers.
Larger samples improve the accuracy of critical values from the two distributions when the population is not normal. This is true for two reasons:
- The sampling distribution of X-bar for large sample size is close to normal
- As the sample size n grows, the sample standard deviation Sx will give a more accurate estimate of σ.
It is very difficult to use a graph of sample data to assess the normality of a population distribution. If the graph has a skewed shape or if there are outliers present, it could be because the population distribution isn't normal. Skewness or outliers can also occur naturally in a random sample from the normal population. To be safe, you should only use a T distribution for small samples with no outliers or strong skewness.
On the AP exam, you will not be asked to construct a confidence interval if one of the conditions for inference are clearly violated. If you think that there's a problem with one of the conditions when asked to construct a confidence interval, you should state your concerns and proceed with calculating and interpreting the interval.
Constructing a confidence interval for µ
When the conditions are met, the sampling distribution of x̄ has roughly a normal distribution with meanμand standard deviation σ/√(n). Because we don’t know σ, we estimate it by the sample standard deviation Sx. We then estimate the standard deviation of the sampling distribution with Sx̄ = Sx/√(n). This is called the standard error of the sample mean x̄, or just the standard error of the mean.
The standard error of the sample mean x̄ is SEx = Sx/√(n), were Sx is the sample standard deviation. It describes how far the x̄ will typically be fromμin repeated SRSs of size n.
To construct a confidence interval for µ: statistic±(critical value)*(standard deviation of statistic)
This one-sample t interval for a population mean is similar to the one-sample z-interval for population.
When the conditions are met, a C% confidence interval for the unknown mean μ is X̄ += t*(Sx/√(n))
Where t* is the critical value for the Tn-1 distribution, with C% of the area between -1* and t*
On a calculator, you can find the t interval using: STAT -> TESTS -> T Interval
You can arrange to have both high confidence and small margin of error by taking enough observations. The margin of error ME of the confidence interval is ME = t*(Sx/√(n)). To determine the sample size for desired margin of error, it made sense to set the expression for ME less than or equal to the specified value for the inequality for n. There are two problems with this approach:
- We don't know the sample standard deviation Sx because we haven't produced the data yet.
- The critical value t* depends on the sample size and that we chose. This second problem is more serious. To get the correct value of t*, we need to know the sample size. But that's what we're trying to find. There's no easy solution to this problem, so the way we will approach it is this: come up with a reasonable estimate for the population standard deviation σ from a similar study that was done in the past or from a small scale pilot study. By pretending thatσwas known, we can use the one sample Z interval for U: x̄ ± z* (σ/√(n))
Using the appropriate standard normal critical value z* for confidence level C, we can then solve for the margin of error.
Choosing sample size for a desired margin of error when estimating µ:
To determine a sample size that will yield a C% confidence interval for a population mean with a specific margin of error ME:
- Get a reasonable value for the population standard deviation oh from an earlier or pilot study.
- Find the critical value z* from a standard normal curve for confidence level C.
- Set the expression for the margin of error to be less than or equal to ME in solve for n: z*(σ/√(n)) ≤ ME
Confidence intervals are one of the two most common types of statistical inference. We use confidence intervals and we want to estimate a population parameter. The second common type of inference, called significance tests, has a different goal: to assess the evidence provided by data about some claim concerning a parameter.