WARM UP ONE SAMPLE T-Interval The average life of an Energy Saving Light bulb, in hours, is unknown. You investigate by randomly testing 9 bulbs and record their lives in the following hours: 3675 3597 3412 3976 2831 3597 3211 3725 3222 . Estimate the true mean life of these bulbs using 95% C.I. ONE SAMPLE T-Interval SRS – Stated Approximately Normal Distribution – SHOW Graph We can be 95% confident that the true mean hours of bulb life is between 3208.9 and 3734.6 hours.

Chapter 23 T- Distribution: z n = ∞ n = 30 n = 10 n = 2 The following are statements that compare t-distributions to the normal distribution. I. t distributions are also mound shaped and symmetric. II. t distributions have more spread than the normal distribution. III. As degrees of freedom increase, the variance of t distributions becomes smaller.

Performing a Test for ONE SAMPLE T-TEST for Means: 1. Define your Parameter μ = true mean… 2. State the Hypothesis; H0: μ = # Ha: μ < > or ≠ # 3. Name Test and Check assumptions 4. …PHANTOMS ONE SAMPLE t-TEST

Example1: (continued) The manufacturer of an Energy Saving Light bulb claims that their bulbs have an average life of 3800 hours . To verify this you randomly test 9 bulbs and record their lives in the following hours: 3675 3597 3412 3976 2831 3597 3211 3725 3222 Is there sufficient evidence at the 0.05 level to contradict the manufacturer’s claim? ONE SAMPLE t-TEST μ = The true population mean hours of life for the energy saving light bulbs. SRS – Stated Approximately Normal Distribution – SHOW Graph H0: μ = 3800 Ha: μ ≠ 3800 Since the P-Value < 0.05 REJECT H0. There is evidence that the Bulbs do NOT last 3800 hours as claimed.

Example 2: As a Quality Control Specialist for Doritos you feel that the filling machine has malfunctioned and that the average weight is now different. You know that the population mean weight follows a normal distribution with µ = 30g After opening and weighing 12 random bags you find a sample mean of 28.883 grams with s = 3.6. Is there evidence that the bags are underfilled? μ = The true mean weight for a bag of Doritos. ONE SAMPLE t-TEST 1- SRS -stated 2- Appr. Norm - stated Since the P-value of 0.1527 > 0.05, we fail to REJECT H0. There is no evidence that the true mean weight of the Doritos Bags is less than 30g.

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WARM – UP In an attempt you estimate the average Baseball Player salary in 2011 you randomly sample 12 players: (All in Millions of Dollars) $2.7, 2.9, 1.5, 2.2, 2.5, 2.0, 1.7, 2.9, 2.8, 2.6, 0.8, 2.7 1. Have all the assumptions been met for an Inference? 2. What would be the critical value of t for a 98% Conf. Int.? 3. What is the Standard Error for this sample? 4. What is the Margin of Error for a 98% Confidence Int.? NO! SRS √’s out, but a histogram of the data reveals gross skewness. (Not Appr. Normal) | invT (1 – .98)/2 , 11 | = 2.718 Standard Error = = 0.1899 Margin of Error = = 0.5161

In 2007 baseball players made on average $1. 8 million In 2007 baseball players made on average $1.8 million. Using these randomly sample 12 players (All in Millions of Dollars) from 2011 , Is there evidence that salaries are now different? $2.7, 2.9, 1.5, 2.2, 2.5, 2.0, 1.7, 2.9, 2.8, 2.6, 0.8, 2.7 μ = The true mean salary for a 2011 baseball player ONE SAMPLE t-TEST H0: μ = 1.8 Ha: μ ≠ 1.8 Since the P-Value < 0.05 we REJECT H0 . There is evidence that Baseball salaries have changed. PWC SRS – Stated Approximately Normal Distribution – NO, the shown Graph depicts skewness