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Advanced Placement Statistics and Probability Name:________________________ Chapter 10 _______________________ is a method of drawing a conclusion about a certain population derived from a sample. Several assumptions are made about the sample and they are that the sample is _____________________, ________________________, and the sample was obtained by a ____________________________. Estimating with Confidence As a sample is obtained, the sample mean should start to get closer to the true population mean. This idea is a product of ______________________________ numbers. As we collect more samples from the same population and find the averages for those samples, the standard deviation of the sample means is now the population _______________________________ divided by the __________________. Once the sample means have been calculated we can make a prediction for the ________________ mean. We will first make a prediction with some level of confidence. If we obtain a sample from a population and have calculated the sample mean we can make some prediction for the population mean. Suppose we obtain a sample mean equal to 415. A 95% confidence interval produces the numbers ______________ and ____________ with a margin of error equal to 22.3. To say that ______________ and _____________ is a 95% confidence interval for the population means that ______________ of the repeated ___________ trials will capture the ______________________ mean. Our sample mean is the estimate, and the margin of error is what we ___________ and ______________ to our estimate to get our interval. There are two conditions that must be met in order to predict the population mean based upon the sample. Number one is ________________________________________ and the second one is _____________________________________________________. Some of the more common confidence intervals have z-scores related to the error allowed. The 90%, 95%, and the 99% confidence level have corresponding z-scores which are _________, ___________, and ___________ respectfully. In my notes I have denoted that z* in the book is the same as _________ in my equation, where _______ is the amount of error allowed. The reason we divide ________ by two is because ____________________________________________________________________. Once you have obtained a sample that meets the two conditions then we can use the formula ___________________ or my formula __________________ to create the confidence interval. If your sample size increases then the M.E., which stands for __________________________, will _________________. This again is a product of the ___________________________________ numbers. Therefore when the sample size increases the intervals will start to _____________________________________. When all the calculations have been made and the intervals found then our conclusion about the population mean is that __________________________________________ _____________________________________________________________________. Remember that in our conclusion, our opinions should not be noted in conclusion, unless it is asked. There are three things that will make the ME get smaller and they are ________ __________________, ________________________, and _____________________. We can always find out how many items we need in a sample to estimate the population mean. The ______________________________ is equal to z* times the standard deviation divided by the square root of n. Therefore we are solving for n given a certain margin of error. There are some cautions in confidence intervals. The data must be a ____________ from the population. If the data produced is biased then _____________ formulas will not save the data. If there are _______________ in your data, it may influence your sample mean. The sample size may be ______________ and if the population is not normal then errors could be produced. If ______ is bigger than or equal to ______ then the confidence level is not greatly disturbed by the nonnormal population. We must know ______. If the sample size is large enough then we can use the sample ________________ instead of _______. Test of Significance The idea behind the test of significance is to assess the ____________ provided by the data about a claim made concerning the population. In testing for significance a _________________________ is set up which is usually a claim that is to be tested. The _________________________ is usually what we suspect to happen in the data collected. Once the data is collected then the data is compared to the claim. A number is assigned to our sample mean to tell us how unlikely our observation would be if the _________________ is true. The larger the z-score, the stronger the evidence is ____________ the ______________________________. We usually measure the strength of the evidence by the ________________. This probability is the likelihood that our sample should produce its results. Again, when we collect data we must make sure that the data is __________________________, ________________________, and was obtained by a _________________________________. Conducting a test for the population mean, you must do the following: 1. _________________________________________________ 2. _________________________________________________ 3. _________________________________________________ 4. _________________________________________________ We should never ___________________ the null hypothesis. We must say that we have sufficient ____________________. We should also never say that we _______________ the alternative hypothesis if there is significant ________________________. The p-value for a two-sided test is found by ________ both sides of the normal distribution that falls into error. Testing against a fixed level of significance is comparing the ___________ vs. ______________. If your _____________ is lower than the ______ then we have sufficient evidence against the null hypothesis. Another method of testing the mean is by using the confidence interval. If the ____________ falls outside the confidence interval produced by data, then there is ______________________ evidence against the _________________________. Making Sense of Statistical Significance Setting a level of significance is deciding how much information is required to have sufficient evidence to _________________ the null hypothesis. If you want strong evidence then __________ needs to be small. In addition to attempting to gather evidence to ____________ the null hypothesis there may be consequences to rejecting the ______________________. Statistical significance doesn’t mean that it is _______________ significant. The sample obtained must still be __________________, _____________________________, and generated by _______________________. Inference as Decision A Type I error occurs when we ________________ the null hypothesis and it turns out to be _________________. A Type II error occurs when we ________________ the null hypothesis and it turns out to be ________________. _______ is assigned to a Type I error which is called the significance level for a Type I error. To produce a Type II error you must ____________ the null hypothesis. Once you have stated the null and alternative hypothesis you must find the corresponding sample mean value that will create a ______________ error. As soon as you find this value, you then compare it to what the alternative believes to be true. From there you find a z-score related to the sample mean value necessary to create a Type I error to the alternative hypothesis. The probability that a fixed level alpha significance test will reject the null hypothesis, when a particular alternative value of the parameter is called the power of the test against the alternative. The power of a test against any alternative is _____________ _______________________________________. In order to find the power of the test you must complete the following steps: 1. __________________________________________________ 2. __________________________________________________ 3. __________________________________________________ In order to increase the power of a test you can do several things. You can ______________ alpha, consider a an alternative ____________________________, increase the ___________________, and decrease the ________________________________. If we do not have sufficient evidence to reject the null hypothesis, we cannot ____________ the null hypothesis. We can only say that we do not have __________________ against the null hypothesis. When testing the hypothesis you must: 1. __________________________________________________ 2. __________________________________________________ 3. __________________________________________________ 4. __________________________________________________