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INFERENCES ABOUT MEANS CHAPTER 23 When dealing with quantitative data we are talking about means instead of proportions. We should always start our analysis with a histogram of the data. The Central Limit Theorem still holds and our sampling distribution will still be approximately normal as long as our sample size is large enough. The standard deviation of our π sample is : π We canβt calculate SE from quantitative data like we could for π proportions, so we estimate it with: . π We canβt use the Normal model for this, since the sample size affects the model, so instead we will use the Studentβs t-model, which requires naming the degrees of freedom (df) which is n-1. T models are unimodal, symmetric, and bell shaped, but have fatter tails than the normal model. The more degrees of freedom, the more a t model approaches a normal model. Sampling Distribution Model for Means: Instead of calculating z as our sample statistic, we use t. πβπ π= πΊπ¬(π) With n-1 degrees of freedom. π πΊπ¬ π = π One-sample t-interval: π ± πβ πβπ β πΊπ¬ π Assumptions & Conditions Independence: Randomization and 10% Normal Population Assumption: β’ Nearly Normal condition β you MUST make a histogram of the data. If n<15 if must be clearly normal. Between 15 and 40 the data needs to be unimodal and reasonably symmetric. Above 40 t model is good unless the data is extremely skewed. Hypothesis testing for means One sample t-test for the mean: ππβπ π β ππ = πΊπ¬(π) A coffee machine dispenses coffee into paper cups. Youβre supposed to get 10 ounces of coffee, but the amount varies slightly from cup to cup. Here are the amounts measured in a random sample of 20 cups. Is there evidence that the machine is shortchanging customers? 9.9 9.7 10.0 10.1 9.9 9.6 9.8 9.8 10.0 9.5 9.7 10.1 9.9 9.6 10.2 9.8 10.0 9.9 9.5 9.9 A company has set a goal of developing a battery that lasts over 5 hours (300 minutes) in continuous use. A first test of 12 of these batteries measured the following lifespans (in minutes): 321, 295, 332, 351, 281, 336, 311, 253, 270, 326, 311, and 288. β’ Is there evidence that the company has met its goal?