Can you please finish all three questions on this assignment?
Is California Warming? is california warming? 7 Assignment 6 - Hypothesis Testing Student’s t test is commonly used to compare a single sample mean to an expected value of the population mean. When population statistics are not known, the sample standard deviation s is your best and only estimate of s for the population from which it has been taken. You can still use the 95% confidence interval of the mean, estimated from the sample standard deviation, and the t statistic de- scribed in class to predict the range around an expected value of µ within which 95% of the means of samples of size n taken from that population will occur. Here too, once the sample mean lies outside the 95% confidence interval, the probability of it being from a popu- lation with a mean of µexpected is less than 0.05. Expressed as a formula, if the ratio t = X�µexpecteds/pn is less than the critical 5% value of �t or greater than +t, then the sample mean is considered to have come from a population with a mean significantly different to µexpected. The appropriate critical value of t for a sample is easily found in tables of this statistic that are in most statistical texts or online. It depends on the probability level, the number of degrees of freedom, and whether the test is one- or two-tailed: Probability level We use a 5% probability level, which is the probabil- ity than many researchers use as a standard "statistically signifi- cant level." Degrees of freedom If you have a sample of size n and the mean of the sample is a specified value, then all of the data within the sample except one are free to be any number at all, but the final one is fixed because the sum of the data in the sample, divided by n must equal the mean. For a one-sample t test, if your sample size is n, then you need to use the t value that has n � 1 degrees of freedom. One-tailed and two-tailed tests Our alternative hypothesis does not specify anything other than "The mean of the population from which the sample has been drawn is different to an expected value." Therefore, these are two-tailed hypotheses because nothing is specified about the direction of the difference. The null hypothe- sis could be rejected by a difference in either a positive or negative direction. Sometimes, however, you may have an alternative hy- pothesis that specifies a direction. For instance, "The mean of the population from which the sample has been taken is greater than an expected value." Or "The mean of the population from which the sample A has been taken is less than the mean of the pop- is california warming? 8 ulation from which sample B has been taken." These are called one-tailed hypotheses. If you have an alternative hypothesis that is directional, the null hypothesis will not just be one of no difference. For instance, if the alternative hypothesis states that the mean of the population from which the sample has been taken will be less than an expected value, then the null hypothesis should state, "The mean of the pop- ulation from which the sample has been taken will be no different to, or more, than the expected value." You need to be cautious, however, because a directional hypothesis will affect the location of the region where the most extreme 5% of outcomes will occur. For any two-tailed hypothesis the 5% rejec- tion region is split equally into two areas of 2.5% on the negative and positive sides of µ. Student’s t test may also be used to test for the significance of the slope of a regression line. We will do this in our next assignment. In preparation: 1. State the null and alternative two-tailed hypotheses for our analy- sis of the California temperature trend. 2. The degrees of freedom for the residual (error) variation in a re- gression analysis are always n � 2. In this case, what is the critical t value for a two-tailed test using the 5% probability level? 3. State the condition for rejecting the null hypothesis. In other words, would your calculated t statistic need to be greater or less than the critical value? For this assignment, include on a single doc: For (1) and (3) short answers of a few sentences and for (2) the numeric value. Assignment 1 - Collecting and Displaying Data Assignment 4 - Data, Populations, and Statistics Assignment 5 - Linear Regression Assignment 6 - Hypothesis Testing Assignment 7 - Testing the Significance of the Slope of the Regression