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Chapter 22 Inference for OneSample Means Steps for doing a confidence interval: 1) 2) • • • • • • • State the parameter Conditions 1) The sample should be chosen randomly or there is random assignment to treatments 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. Formula for a t-confidence interval: Critical value s x t * n estimate Margin of error Standard deviation of statistic df n 1 Degrees of freedom 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 mmHg 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. State the parameters μ = population mean systolic blood pressure of healthy white males Justify the confidence interval needed (state assumptions) 1) The 54 white males were randomly assigned to either the calcium group or the placebo group. 2) The sample distribution should be approximately normal. Since n = 54 >30, by the Central Limit Theorem we can assume the sample distribution is approximately normal. 3) We will assume that this group of white males is representative of the larger population of white males. 4) is unknown Since the conditions are satisfied a t – interval for means is appropriate. Calculate the confidence interval. x 114.9 s xt n 54 n s 9.3 9.3 95% CI df 53 114.9 2.006 54 112.36,117.44 Explain the interval in the context of the problem. We are 95% confident that the true mean systolic blood pressure for healthy white males is between 112.36 and 117.44 mmHg. Conditions for one-sample means 1) The sample should be chosen randomly or there is random assignment to treatment groups 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, which is why we conduct t-test for means. Formula: unknown: statistic - parameter test statistic standard deviation of statistic t= x s n x df n 1 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 purchased some randomly selected bags of cookies, and counted the chocolate chips (no kidding) . Some of their data are given below. 1219 1214 1087 1200 1419 1121 1325 1345 1244 1258 1356 1132 1191 1270 1295 1135 What does this say about Nabisco’s claim? Test an appropriate hypothesis at 5% significance. Parameters and Hypotheses μ = the true mean number of chocolate chips in each bag of Chips Ahoy H0: μ = 1000 Ha: μ > 1000 Assumptions (Conditions) 1) The 16 bags of Chips Ahoy were randomly sampled. 2) The sample distribution should be approximately normal. (Check with an appropriate graphical display, usually a histogram or boxplot) The boxplot shows no outliers and is roughly symmetric, so assume that the sample distribution is approx. normal. 3) The sample should be less than 10% of the population, which would be at least 160 bags of Chips Ahoy. This is a very safe assumption. 4) is unknown Since the conditions are met, a t-test for the one-sample means is appropriate. Calculations x 1238.188 n 16 = 0.05 df 15 x x 1238.188 1000 t 10.105 94.282 s 15 n p value P ( t 10.1053) 2.18 108 2.176 108 .05 Decision and conclusion: Since p-value of .0000000218 is much less than the significance level of .05, I reject the null hypothesis. There is overwhelming evidence to suggest that the true mean number of chocolate chips per bag of Chips Ahoy is greater than 1000.