Comparison of Two Means
There are only two different formulae for degrees of freedom here: right or wrong one to use; it's simply that the Welch-Satterthwaite version is. Chi-square distribution for different number of degrees of freedom. both the calculation of confidence intervals and the workings of hypothesis tests. The pairing is carried out typically due to a connection between the first. A confidence interval gives an estimated range of values which is likely to include an . (1-C)/2 critical value for the t distribution with n-1 degrees of freedom, t(n-1). Wunderlich," Journal of the American Medical Association, ,
Because the t distribution is symmetric about a mean of zero, the following is true.
The CEO claims that an average Acme light bulb lasts days. A researcher randomly selects 15 bulbs for testing. The sampled bulbs last an average of days, with a standard deviation of 50 days. If the CEO's claim were true, what is the probability that 15 randomly selected bulbs would have an average life of no more than days? There are two ways to solve this problem, using the T Distribution Calculator. Both approaches are presented below. Solution A is the traditional approach.
It requires you to compute the t statistic, based on data presented in the problem description.
Comparison of Two Means
Then, you use the T Distribution Calculator to find the probability. Solution B is easier. You simply enter the problem data into the T Distribution Calculator.
The calculator computes a t statistic "behind the scenes", and displays the probability. Both approaches come up with exactly the same answer. Solution A The first thing we need to do is compute the t statistic, based on the following equation: Now, we are ready to use the T Distribution Calculator. Since we know the t statistic, we select "T score" from the Random Variable dropdown box. Then, we enter the following data: The t statistic is equal to - 0.
The calculator displays the cumulative probability: Hence, if the true bulb life were days, there is a This time, we will work directly with the raw data from the problem. We will not compute the t statistic; the T Distribution Calculator will do that work for us. Since we will work with the raw data, we select "Sample mean" from the Random Variable dropdown box. Assuming the CEO's claim is true, the population mean equals Since this estimate is based on two independent pieces of information, it has two degrees of freedom.
The two estimates are independent because they are based on two independently and randomly selected Martians. The estimates would not be independent if after sampling one Martian, we decided to choose its brother as our second Martian.
Degrees of Freedom
As you are probably thinking, it is pretty rare that we know the population mean when we are estimating the variance. The process of estimating the mean affects our degrees of freedom as shown below. Returning to our problem of estimating the variance in Martian heights, let's assume we do not know the population mean and therefore we have to estimate it from the sample.
We have sampled two Martians and found that their heights are 8 and 5. We can now compute two estimates of variance: Are these two estimates independent? The answer is no because each height contributed to the calculation of M. Suppose in the example above, the student wishes to have a margin of error equal to 0.
Confidence Intervals for Unknown Mean and Unknown Standard Deviation In most practical research, the standard deviation for the population of interest is not known. In this case, the standard deviation is replaced by the estimated standard deviation salso known as the standard error. Since the standard error is an estimate for the true value of the standard deviation, the distribution of the sample mean is no longer normal with mean and standard deviation.
Student's t Distribution
Instead, the sample mean follows the t distribution with mean and standard deviation. The t distribution is also described by its degrees of freedom.
For a sample of size n, the t distribution will have n-1 degrees of freedom. The notation for a t distribution with k degrees of freedom is t k.
As the sample size n increases, the t distribution becomes closer to the normal distribution, since the standard error approaches the true standard deviation for large n. Example The dataset "Normal Body Temperature, Gender, and Heart Rate" contains observations of body temperature, along with the gender of each individual and his or her heart rate.
This value is approximately 1. The estimated standard deviation for the sample mean is 0.