1. Chapter 23 Inference for One- Sample Means

2. Steps for doing a confidence interval: 1)State the parameter 2)Conditions 1) The sample should be chosen randomly 2) The sample distribution should be approximately normal - the population is known to be normal, or - the sample size is large (n  30), or - graph data to show approximately normal 3) 10% rule – The sample should be less than 10% of the population 4) σ will almost always be unknown 3) Calculate the interval If σ is unknown we perform a t-interval…the t distribution is based on an unknown standard deviation and different sample sizes (known as degrees of freedom) 4) Write a statement about the interval in the context of the problem.

3. Formula for a t-confidence interval: estimate Critical value Standard deviation of statistic Margin of error Degrees of freedom

4. In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups, takes calcium or placebo. The paper reports a mean seated systolic blood pressure of 114.9 with standard deviation of 9.3 for the placebo group. Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group.

5. μ = the true mean systolic blood pressure of healthy white males State the parameters Justify the confidence interval needed (state assumptions) 3) The sample should be less than 10% of the population. The population should be at least 540 healthy white males, which I will assume. 2) The sample distribution should be approximately normal. Since n = 54 >30, by the CLT we can assume the sample distribution is approximately normal. Since the conditions are satisfied a t – interval for means is appropriate. 1) The sample must be random which is stated in the problem. 4)  is unknown

6. We are 95% confident that the true mean systolic blood pressure for healthy white males is between 112.36 and 117.44. Calculate the confidence interval. 95% CI Explain the interval in the context of the problem.

7. Steps for a hypothesis test : 1)Define the parameter 2)Hypothesis statements 3)Assumptions 4)Calculations (Find the p-value) 5)Decision and Conclusion in context

8. 1) The sample should be chosen randomly 2) The sample distribution should be approximately normal - the population is known to be normal, or - the sample size is large (n  30), or - graph data to show approximately normal 3) 10% rule – The sample should be less than 10% of the population 4) σ will almost certainly be unknown, we perform t-test Conditions for one-sample means

9. Formulas:  unknown: t =

10. Example In 1998, as an advertising campaign, the Nabisco Company announced a “1000 Chips Challenge,” claiming that every 19-ounce bag of their Chips Ahoy cookies contained at least 1000 chocolate chips. Dedicated Statistics students at the Air Force Academy (no kidding) purchased some randomly selected bags of cookies, and counted the chocolate chips. Some of their data are given below. What does this say about Nabisco’s claim? Test an appropriate hypothesis at 5% significance. 12191214108712001419112113251345 12441258135611321191127012951135

11. Parameters and Hypotheses μ = the true mean number of chocolate chips in each bag of Chips Ahoy H 0 : μ = 1000 H a : μ > 1000

12. Assumptions (Conditions) Since the conditions are met, a t-test for the one-sample means is appropriate. 1) The sample must be random which is stated in the problem. 3) The sample should be less than 10% of the population. The population should be at least 160 bags of Chips Ahoy, which we will assume. 2) The sample distribution should be approximately normal. (Check with an appropriate graphical display) 4)  is unknown The boxplot shows no outliers, so assume that the sample distribution is approx. normal.

13. Calculations  = 0.05 1238.1875 94.282 1238.1875 94.282

14. Conclusion: Decision: Since p-value < , I reject the null hypothesis at the.05 level. There is sufficient evidence to suggest that the true mean number of chocolate chips in each bag of Chips Ahoy is greater than 1000.