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AP Statistics Section 12.1 A Now that we have looked at the principles of testing claims, we proceed to practice. We begin by dropping the unrealistic assumption that we know the population standard deviation when testing claims about a population mean. As with confidence intervals, this leads to the use of ___ t distributions when carrying out significance tests about . One Sample t-Test Draw an SRS of size n from the population. The one-sample t statistic: x 0 t s n has the t distribution with n – 1 degrees of freedom. There is a slight change in the procedure for computing the pvalue. The next examples show this change. Between .025 and .05 TI 83 / 84 : 2nd VARS 5 : tcdf(1.81,10000,19) .043 Between .001 and .0025 3.17 P value 2( Between .001 and .0025) P value Between .002 and .005) tcdf (3.17,1000,36) .002 p value 2(.002) .004 These P-values are exact if the population distribution is Normal and are approximately correct for large n in other cases. Example: Is 98.6oF Wrong? From a random sample of 106 people, the mean body temperature was o 98.2 F with a standard deviation of .6229. Test the common belief that the mean body temperature is 98.6oF. Parameter: The population of interest is all people. We wish to test H 0: 98.6 vs H a : 98.6 where mean body temperature Conditions: SRS : Random sample but if not an SRS, results may not generalize to the population. Normality of x dist. : With n 106, CLT gives a dist. that is approx. Normal Independence : Reasonable to expect individual temp. to be independent and since we are sampling w/o replacement N 10n Calculations: t 98.2 98.6 6.611 .6229 106 p value 2(.0005) .001 TI83/84 : STAT TESTS 2 : T - Test Interpretation: Our very low p - value, less than .0005, indicates a very small liklihood of getting a sample with with a mean temp. of 98.2 or smaller when we assume the mean temp. is 98.6. This suggests the mean body temp. is not 98.6 but actually less. If you were to choose a significance level for this test, what would it be? Why? .01 We are trying to disprove a commonly held belief, so we need convincing evidence to disprove the H 0 . Describe a Type I and Type II error in this setting. Type I : Determining that the mean is not 98.6 when in fact it is. Type II : Failing to reject that the mean is not 98.6 when in fact it is lower than 98.6. Construct a 99% confidence interval for the mean body temperature and interpret it. x t s n .6229 98.2 2.626 106 (98.04,98.36) We are 99% confident that the mean body temperature for the population is between 98.04 and 98.36