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* * * Shape: When the population distributions are Normal, the sampling distribution of 𝑥1 − 𝑥2 is Normal. In other cases, 𝑥1 − 𝑥2 will be Normal if both sample sizes are ≥ 30. * Center: The mean of the sampling distribution is 𝜇1 − 𝜇2 * Spread: The standard deviation of the sampling distribution is 𝜎2 1 𝑛1 + 𝜎2 2 𝑛2 as long as each sample is no more than 10% of the population. * Confidence Interval: (𝑥1 −𝑥2 ) ± 𝑡∗ 𝑠21 𝑛1 + 𝑠22 𝑛2 * Still have to check conditions first! * * *𝑧 = 𝑥1 −𝑥2 −(𝜇1 −𝜇2 ) *𝑡 = 𝑥1 −𝑥2 −(𝜇1 −𝜇2 ) 𝜎2 1 𝜎2 2 + 𝑛1 𝑛2 𝑠2 1 𝑠2 2 + 𝑛1 𝑛2 * “How do the sizes of longleaf pine trees in the northern and southern halves of the forest compare?” To find out, researchers took random samples of 30 trees from each half and measured the diameter in centimeters. Comparative boxplots of the data and summary statistics from Minitab are found below. Construct and interpret a 90% confidence interval for the difference in the mean diameter of longleaf pines in the northern and southern halves of the forest. * State: * Plan: * Do: * Conclude: * Look at pg 637 in your book…look at the formula for df * If you use your calculator…this is what they are doing, so your df will most likely be a decimal. If you are using your calculator to give you your answers – you must report what they report as df or you will get it wrong. If you do what we normally do (n-1) then you have to keep what the formula gives you…not what the calc. gives! * * Does increasing the amount of calcium in our diet reduce blood pressure? The subjects were 21 healthy men who volunteered to take part in the experiment. They were randomly assigned to two groups: 10 of the men received a calcium supplement for 12 weeks, while the control group of 11 men received a placebo pill that look identical. The experiment was doubleblind. The response variable is the decrease in systolic blood pressure for a subject after 12 weeks, in millimeters of mercury. An increase appears as a negative response. Do the data provide sufficient evidence to conclude that a calcium supplement reduces blood pressure more than a placebo? * State: * Plan: * Do: * Conclude: * The heights of ten-year-old girls follows a Normal distribution with mean 𝜇𝑓 = 56.4 inches and standard deviation 𝜎𝑓 = 2.7 inches. The heights of ten-year-old boys follows a Normal distribution with mean 𝜇𝑚 = 55.7 inches and standard deviation 𝜎𝑚 = 3.8 inches. A researcher takes a random sample of 12 ten-year-old girls and 8 ten-year-old boys in the US. After analyzing the data, the researcher reports that the mean height 𝑥𝑚 of the boys is larger than the mean height 𝑥𝑓 of the girls. Describe the shape, center, and spread of the sampling distribution 𝑥𝑓 − 𝑥𝑚 . * Find the probability of getting a difference in sample means 𝑥𝑓 − 𝑥𝑚 that is less than 0. 𝑧= 𝑥1 − 𝑥2 − (𝜇1 − 𝜇2 ) 𝜎 21 𝜎 2 2 𝑛1 + 𝑛2 * Pg 652 (35, 39, 40, 45, 49, 54**, 61-66) **For #54 – Use the “unequal” – we DO NOT pool means problems! *