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Assumptions and Conditions • Independence Assumption: – Randomization Condition: The data arise from a random sample or suitably randomized experiment. Randomly sampled data (particularly from an SRS) are ideal. – 10% Condition: When a sample is drawn without replacement, the sample should be no more than 10% of the population. Assumptions and Conditions (cont.) – Nearly Normal Condition: • Normal Population Assumption: – We can never be certain that the data are from a population that follows a Normal model, but we can check the – Nearly Normal Condition: The data come from a distribution that is unimodal and symmetric. • Check this condition by making a histogram or Normal probability plot. One-Sample t-test for the Mean • The conditions for the one-sample t-test for the mean are the same as for the one-sample t-interval. • We test the hypothesis H0: μ = μ0 using the statistic x − μ0 tn −1 = SE ( x ) • The standard error of the sample mean is Assumptions and Conditions (cont.) • The smaller the sample size (n < 15 or so), the more closely the data should follow a Normal model. • For moderate sample sizes (n between 15 and 40 or so), the t works well as long as the data are unimodal and reasonably symmetric. • For larger sample sizes, the t methods are safe to use even if the data are skewed. Example: Is the mean weight of female college students still 132 pounds? To test this, you take a random sample of 20 students, finding a mean of 137 pounds with a standard deviation of 14.2 pounds. Use a significance level of 0.1. 1. Hypothesis μ = population mean weight of female college students SE ( x ) = s n Ho: μ = 132 Ha: μ ≠ 132 • When the conditions are met and the null hypothesis is true, this statistic follows a Student’s t model with n – 1 df. We use that model to obtain a P-value. 1 Example: Is the mean weight of female college students still 132 pounds? To test this, you take a random sample of 20 students, finding a mean of 137 pounds with a standard deviation of 14.2 pounds. Use a significance level of 0.1. Example: Is the mean weight of female college students still 132 pounds? To test this, you take a random sample of 20 students, finding a mean of 137 pounds with a standard deviation of 14.2 pounds. Use a significance level of 0.1. 2. Check Assumptions/Conditions 3. Calculate Test t= • SRS is stated • σ is unknown, use t-distribution • We will assume an approximately normal distribution x − μ0 137 − 132 = = 1.575 14.2 SE ( x ) 20 0.132 P ( x = 137 μ = 132 ) = 0.264 Example: Is the mean weight of female college students still 132 pounds? To test this, you take a random sample of 20 students, finding a mean of 137 pounds with a standard deviation of 14.2 pounds. Use a significance level of 0.1. 4. Conclusion Since P-value is greater than alpha, we fail to reject the population mean weight of female students is 132 pounds. Example: A father is concerned that his teenage son is watching too much television each day, since his son watches an average of 2 hours per day. His son says that his TV habits are no different than those of his friends. Since this father has taken a stats class, he knows that he can actually test to see whether or not his son is watching more TV than his peers. The father collects a random sample of television watching times from boys at his son's high school and gets the following data 1.9 2.3 2.2 1.9 1.6 2.6 1.4 2.0 2.0 2.2 Is the father right? That is, is there evidence that other boys average less than 2 hours of television per day? 1. Hypothesis μ = population mean number of hours watching TV Ho : μ = 2 Ha : μ < 2 Example: Example: A father is concerned that his teenage son is watching too much television each day, since his son watches an average of 2 hours per day. His son says that his TV habits are no different than those of his friends. Since this father has taken a stats class, he knows that he can actually test to see whether or not his son is watching more TV than his peers. The father collects a random sample of television watching times from boys at his son's high school and gets the following data 1.9 2.3 2.2 1.9 1.6 2.6 1.4 2.0 2.0 2.2 Is the father right? That is, is there evidence that other boys average less than 2 hours of television per day? A father is concerned that his teenage son is watching too much television each day, since his son watches an average of 2 hours per day. His son says that his TV habits are no different than those of his friends. Since this father has taken a stats class, he knows that he can actually test to see whether or not his son is watching more TV than his peers. The father collects a random sample of television watching times from boys at his son's high school and gets the following data 1.9 2.3 2.2 1.9 1.6 2.6 1.4 2.0 2.0 2.2 Is the father right? That is, is there evidence that other boys average less than 2 hours of television per day? 2. Check Assumptions/Conditions 3. Calculate Test SRS is stated σ is unknown, use t-distribution Based on the linearity of the normal probability plot, we have an approximately normal distribution. t= x − μ0 2.01 − 2 = = 0.0918 0.345 SE ( x ) 10 0.5356 P ( x = 2.01 μ = 2 ) = 0.5356 2 Example: A father is concerned that his teenage son is watching too much television each day, since his son watches an average of 2 hours per day. His son says that his TV habits are no different than those of his friends. Since this father has taken a stats class, he knows that he can actually test to see whether or not his son is watching more TV than his peers. The father collects a random sample of television watching times from boys at his son's high school and gets the following data 1.9 2.3 2.2 1.9 1.6 2.6 1.4 2.0 2.0 2.2 Is the father right? That is, is there evidence that other boys average less than 2 hours of television per day? 4. Conclusion α = 0.05 Since the P-value is greater than alpha, we fail to reject the population mean number of hours watching TV by the other boys is 2. 3