So the probability that the sample mean is greater than 22 is between 0.005 and 0.025 (or between 0.5% and 2.5%)Įxercise. To obtain the one-tailed probability, divide the two-tailed probability by 2. Then we calculate t, which follows a t-distribution with df = (n-1) = 24.įrom the tables we see that the two-tailed probability is between 0.01 and 0.05. Suppose that is unknown and we need to use s to estimate it. We found that the probability that the sample mean is greater than 22 is P( > 22) = 0.0548. In the previous example we drew a sample of n=16 from a population with μ=20 and σ=5. So, all we can say is that P(|Z| > 2.00) is between 2% and 5%, probably closer to 5%! Using the z-table, we found that it was exactly 4.56%. Now, suppose that we want to know the probability that Z is more extreme than 2.00. So, if we look at the last row for z=1.96 and follow up to the top row, we find thatĮxercise: What is the critical value associated with a two-tailed probability of 0.01? The t-table also provides much less detail all the information in the z-table is summarized in the last row of the t-table, indexed by df = ∞. The t-table is presented differently, with separate rows for each df, with columns representing the two-tailed probability, and with the critical value in the inside of the table. The (one-tailed) probabilities are inside the table, and the critical values of z are in the first column and top row. The z table gives detailed correspondences of P(Z>z) for values of z from 0 to 3, by. Note: If n is large, then t is approximately normally distributed. Has a t distribution with (n-1) degrees of freedom (df)
![expected value of xbar expected value of xbar](http://image.sciencenet.cn/album/201511/09/092219c49i49n4lyjydgjg.jpg)
If X is approximately normally distributed, then As the degrees of freedom increase, the t distribution approaches the standard normal distribution. There are actually many t distributions, indexed by degrees of freedom (df). However, we can estimate σ using the sample standard deviation, s, and transform to a variable with a similar distribution, the t distribution. If the standard deviation, σ, is unknown, we cannot transform to standard normal.
![expected value of xbar expected value of xbar](http://image.sciencenet.cn/album/201511/09/091950q34qpulp8pp1f218.jpg)
How will this affect the standard error of the mean? How do you think this will affect the probability that the sample mean will be >22? Use the Z table to determine the probability. So the probability that the sample mean will be >22 is the probability that Z is > 1.6 We use the Z table to determine this:Įxercise: Suppose we were to select a sample of size 49 in the example above instead of n=16. Suppose we draw a sample of size n=16 from this population and want to know how likely we are to see a sample average greater than 22, that is P( > 22)? For continuous random variable with mean value and probability density. 2 Var (X ) E (X - ) 2 From the definition of the variance we can get. If the standard deviation, σ, is known, we can transform to an approximately standard normal variable, Z:įrom the previous example, μ=20, and σ=5. The variance of random variable X is the expected value of squares of difference of X and the expected value. If X has a distribution with mean μ, and standard deviation σ, and is approximately normally distributed or n is large, then is approximately normally distributed with mean μ and standard error. The statistic used to estimate the mean of a population, μ, is the sample mean.