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STA220- 5.3 Guided Notes Confidence Interval for a Population Mean: Studentβs t-Statistic Small Sample ο³ Unknown t-statistic is used when the sample size is small. (π₯Μ β π) π‘ = (π /βπ) in which the sample standard deviation, s, replaces the population standard deviation, ο³. The actual amount of variability in the sampling distribution of t depends on the sample size n. A convenient way of expressing this dependence is to say that the t-statistic has (n β 1) __________________________________ (df). Small-Sample Confidence Interval for µ Μ ± ππΆ\π ( π where ta/2 is based on (n β 1) degrees of freedom. π ) βπ Conditions Required for a Valid Small-Sample Confidence Interval for µ 1. A random sample is selected from the target population. 2. The population has a relative frequency distribution that is approximately normal. Example: Let π‘0 be a specific value of t. Use the table of critical values of t to find π‘0 - values such that following statements are true. π. π(π‘ β₯ π‘0 ) = 0.025, where df = 11 π. π(βπ‘0 β€ π‘ β€ π‘0 ) = .99), where df = 10 Estimation Example Mean (s Unknown) A random sample of n = 25 has π₯Μ = 50 and s = 8. Set up a 95% confidence interval estimate for π. Example: Consider the pharmaceutical company that desires an estimate of the mean increase in blood pressure of patients who take a new drug. The blood pressure increases (measured in points) for the n = 6 patients in the human testing phase are shown below. Use this information to construct a 95% confidence interval for π, the mean increase in blood pressure associated with the new drug for all patients in the population. Interpret the interval in the words of the problem. Blood Pressure Increase (Points) for Six Patients 1.7 3.0 .8 3.4 2.7 2.1 Example: Youβre a time study analyst in manufacturing. Youβve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1. What is the 90% confidence interval estimate of the population mean task time?