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Advanced Placement Statistics and Probability Name:__________________________ Chapter 11 Inference for the Mean of a Population In the previous chapter we made an unrealistic assumption that we knew the value of ____________. In reality the ______________ is unknown and in this chapter we will proceed with out knowing the population ________________. There are some conditions that need to be set concerning the inference about the mean. The first one is that the sample needs to be obtained by a __________. This is a very important condition that needs to be met. The second condition is that observations from a given population need to be _______________. Since we do not know the population _______________ we must change our standard deviation. Now we use the formula _______________ to find the standard error of the sample ____________. In the previous chapter we knew the standard deviation, but now we use ____________ instead of the population standard deviation divided by the square root of n. The result of this substitution is that now our distribution will not be _____________. In stead of using a normal distribution (z-distribution) we will now use the ___________________. Since we no longer have a normal distribution, the z-score formula will not work. We must use the one sample ________________ which is equal to the formula _______________________. This formula uses the idea that we have _______________ degrees of freedom. As you look in the book, the graph on page 618 has something interesting going on. As the degrees of ____________ increase, the curve starts to approach the ____________________ distribution. Now turn to table C (the back of the book). You will notice that as the degrees of ___________ increase the tscore starts to approach the _______________ when n is really large. Our first inference, just like in the previous chapter, will be confidence intervals. The formula doesn’t change that much. The original formula, _________________ will now become __________________. There are two major differences with the new formula for finding the confidence interval. The first is the replacement of the z*. The z* has been replaced with the ___________. This _________ is similar to the z of alpha divided by two (my equation). You now must refer to Table _____ to find all the appropriate t-scores. When looking up a certain t-score, the numbers at the top of the table represent _________________ and the numbers at the side represent _______________________. It is important to know the sample size, or the ______________________ cannot be found, which is needed to find the corresponding tscore. The second major change to the formula is the ___________________. Instead of using the population ________________, which we no longer are given, we must use ______________________. To test the hypothesis, nothing has changed that much. The null and _______________ hypothesis are still stated. The only major difference is the ________________ which is now __________________. When conducting a test there are important steps that one must take…(see example 11.2) 1. _________________________________________________ 2. _________________________________________________ you should check to see if your sample is _______________ by using the ______________ probability plot. It should be in a __________________. 3. _________________________________________________ 4. _________________________________________________ These steps are very important. One thing you must pay attention to is that you must never ever _______________ the null or alternative hypothesis. Rejecting a hypothesis is okay, but you must have _______________________ evidence. Finding the p-value for a t-score is a little harder than using the normal distribution table. You can estimate it by finding where your _____________ lies. If it is between two t-scores, you can get a rough idea for the ____________. Your calculators have the ability to find the actual ____________ by using the ONE-SAMPLE _________________. Under a match pair design you are comparing the differences between two items. The null hypothesis for this type of test is always going to be equal to ____________. Thus we are saying that the ______________ between the two items will be _________. When conducting a hypothesis test for match pair be sure to follow all ________steps. The formula used in this test is basically the same as a t-test, but mu will be equal to _________. When finding the confidence interval for a match pair, the result will be the _________________ between the two items. Thus the interval contains the true ______________ with some level of confidence. The significance test is considered ___________ if the confidence level or pvalue is not changed even if we assume the procedure is _______________. Unlike the previous chapter, all variables such as the sample mean, standard deviation, and the tscores is highly influenced by ___________________. Due to this we have set some parameters to go by based upon the sample size. The t procedures are… 1. ___________________________________________________ 2. ___________________________________________________ 3. ___________________________________________________ 4. ___________________________________________________ So, if you have a large sample that contains ___________________ the t-procedures could be used. In practice we hope to show that the null hypothesis is _____________. So, a _____________ power is very important. When conducting a test to disprove the null hypothesis we assume that the fixed level of alpha to be _______________. Finding the power of a test is just like the previous chapter, except now we use a ________________ and the ___________________ in the formula. Remember that power is equal to ______________________. Comparing Two Means We are now going to compare two different populations by obtaining two ____________________ from each population. Remember that each sample obtained is ____________________, ______________________, and has been produced by a ______________________. The conditions for comparing the two different populations are that… 1. ___________________________________________________ 2. ___________________________________________________ When we actually do a hypothesis test, we will always set the null hypothesis to be __________________________________. The alternative has three possible outcomes and they are _______________, ________________ or ________________. Before we get into the actual test, we need to discuss the sampling distribution of difference of the two means. The mean of the difference of the two sample means is equal to _______________________. That is the difference of the sample means is and ____________________ estimator for _______________________________________. The variance is a bit different than before. Now the variance will be ________________ if the population standard deviation is known. The SE, which stands for _____________________________________________, of the sampling distribution where the standard deviation is unknown will be ________________________________. This will then give us the new t-score formula with is… ________________________________ If the standard deviation for the population is known, then we use…. ________________________________ There are two options when dealing with the degrees of freedom. The TI 83/89 or other graphing calculators automatically calculate the ___________________ based upon the data. If a calculator is not available and tables are the only things a student has to rely upon then the degree of freedom is equal to ________________ of the _____________ or _________________. Now that we have the SE, we can actually test a hypothesis about the differences between two populations or even calculate a confidence interval concerning the ______________ between two populations. The new equation for confidence interval with an unknown standard deviation is… _________________________________ where the degree of freedom is found by using the _____________ of ___________ or _____________. If you have a confidence interval concerning the difference and the interval covers 0 then you cannot __________________ any ___________________ concerning the difference between the two populations. (see example 11.12). The robustness of the two sample t-test depends on more than one sample. So, for the sake of the values given on page 636 we will use the _____________ of the two samples instead of using one sample. Remember to check the normality of your sample one at a time. To check for normality you can use the __________________ and check to see if it is ________________________. The formula for the degrees of freedom for a two sample t-statistics is already in the calculators. The formula is… ___________________________________ only when n is __________ or larger. This value is the actual ______________. The method used when a calculator is not available nor is this equation is to ___________________________________________________.