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Hypothesis Testing for the Mean: 𝜎 not known Testing a Claim about a Mean: 𝜎 not Known We first need to make sure we meet the requirements. 1. The sample observations are a simple random sample. 2. The value of the population standard deviation 𝜎 is not known 3. Either the Population is normal, or 𝑛 > 30 Test Statistic for Testing a Claim about a Proportion 𝑥 − 𝜇𝑥 𝑡= 𝑠 𝑛 Testing a Claim about a Mean: 𝜎 Known P-value method in 5 Steps 1. State the hypothesis and state the claim. 2. Compute the test value. (Involves find the sample statistic). 3. Draw a picture and find the P-value. 4. Make the decision to reject 𝐻0 or not. (compare P-value and 𝛼) 5. Summarize the results. Testing a Claim about a Mean: 𝜎 Known A simple random sample of 40 recorded speeds is obtained from cars traveling on a section of Highway 405 in Los Angeles. The sample has a mean of 68.4 mi/h and a standard deviation of 5.7 mi/h Use a 0.05 significance level to test the claim that the mean speed of all cars is greater than the posted speed limit of 65mi/h. 1. 1. 𝐻𝑎 : 𝜇 > 65 claim and 𝐻0 : 𝜇 = 65 2. 𝑡 = 𝑥−𝜇𝑥 𝑠 𝑛 = 68.4−65 5.7/ 40 = 3.77 3. P-value = 0.000537 4. 0.000537 < 0.05 so we reject the null. 5. There is sufficient evidence to support the claim that the mean is greater than 65 mi/hr. Or use [Stat]→ Test → TTest Testing a Claim about a Mean: 𝜎 Known • In an analysis investigation the usefulness of pennies, the cents portions of 100 randomly selected credit card charges are recorded. The sample has a mean of 47.6 cents and a standard deviation of 33.5 cents. If the amounts from 0 cents to 99 cents are all equally likely, the mean is expected to 49.5 cents. Use a 0.01 significance level to test the claim that the sample is from a population with a mean equal to 49.5 cents. What does the result suggest about the cents portions of credit charge charges? Testing a Claim about a Mean: 𝜎 Known • Homework!! 8-5: 13-27 odd