1. Student’s t-distributions

2. Student’s t-Model: Family of distributions similar to the Normal model but changes based on degrees-of- freedom. Degrees-of-freedom = sample size (n) – 1 Differs because we need to estimate the mean and standard deviation of a population using our sample data The graph of Student’s t is generally wider than the normal model. The larger the sample size the closer Student’s t- model approach the Normal model. http://www.stat.tamu.edu/~jhardin/applets/signed/T.html

3. One Sample t-interval Conditions: Randomization 10% Condition Nearly Normal – Two possible graphs to check: 1. Histogram: unimodal and symmetric, with no outliers 2. Normal Probability Plot: a straight line, no curve **It can be skewed if the sample size is large. Generally n > 30. CLT kicks in at that point. All conditions have been met to use Student’s t-model for a one sample t-interval

4. Normal Probability Plot Finds the z-score of each value compared to the sample mean and standard deviation. Plots these against the values themselves. No curve in the plot shows a unimodal and symmetric distribution

5. Mechanics: Calculate sample mean,, and sample standard deviation, s Find the degrees of freedom, df = n – 1 Calculate critical value using given confidence level and degrees of freedom, One Sample t-interval

6. Example: Coffee Machine A coffee vending machine dispenses coffee into a paper cup. You’re supposed to get 10 ounces of coffee, but the amount varies slightly from cup to cup. Below are the amounts measured in a random sample of 20 cups. Is there evidence that the machine is shortchanging customers? Construct a 95% confidence interval. 9.99.710.010.19.9 9.69.8 10.09.5 9.710.19.99.610.2 9.810.09.99.59.9

7. Conditions: Randomization: Stated as a random sample 10% Condition: 20 cups is less than 10% of all cups dispensed by the coffee machine Nearly Normal Condition: The histogram is unimodal and approximately symmetric. Or The normal probability plot shows an approximately linear pattern. All conditions have been met to use Student’s t-model for a one sample t-interval.

8. Mechanics: (9.752, 9.938) We are 95% confident that the true mean amount of coffee from the coffee machine is between 9.752 to 9.938 ounces.

9. 1 Sample t-Test Inference about the mean of a population, µ. Hypotheses: H 0 : µ = µ 0 The population mean is _____ H A : µ ;≠ µ 0 Conditions: 1. Randomization 2. 10% Condition 3. Nearly Normal (draw a picture)

10. Mechanics: Calculate sample mean,, and sample standard deviation, s Find the degrees of freedom, df = n – 1 State alpha value Calculate test statistic, Find P-Value Conclusion: Compare P-Value to alpha. State conclusion in context

11. Hypothesis: H 0 : µ = 10; The mean amount of coffee dispensed by the coffee machine is 10 ounces. H A : µ < 10; The mean amount of coffee dispensed by the coffee machine is less than 10 ounces. Conditions: 1. Randomization: Random sample 2. 10% Condition: 20 cups is less than 10% of all cups dispensed. 3. Nearly Normal: The histogram is unimodal and approximately symmetric with no outliers. Example: Coffee Machine

12. Mechanics: Conclusion: Since the P-Value is less than alpha (0.0012 < 0.05) we reject the null hypothesis. There is statistically significant evidence that the mean amount of coffee from the coffee machine is less than 10 ounces.