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Universidad de Costa Rica Programa de Posgrado en Computación e Informática Doctorado en Computación e Informática Curso Estadística 18 de febrero 2013 Nombre:______________________________________________________________ Segundo examen corto de Probabilidad Pregunta 1 Suponga que una muestra de 35 observaciones es obtenida de una población con media y varianza . Entonces la ∑ se calcula como a. b. c. d. ̅ ̅ ̅ ̅ Pregunta 2 1 Pregunta 3 (Puede usar la figura 3-13 arriba) Pregunta 4 2 Pregunta 5 Entonces ̅ se distribuye como _________________________________________, ̅ Nota. Para las preguntas 6 y 7 use alguno de los siguientes resultados, si así lo requiere: ( ) ( ) . Pregunta 6 Pregunta 7 3 Pregunta 8 Si dos v.a. X e Y no están correlacionadas, entonces a. Son independientes b. No tienen por qué ser independientes c. siempre d. Ninguna de las anteriores es cierta Pregunta 9 Entonces, ⁄ a) ( ⁄ ) ∫ b) ( ⁄ ) ∫ c) ( ⁄ ) ∫⁄ ∫ d) ( ⁄ ) ∫⁄ ∫ ⁄ ∫ ∫ Pregunta 10 Ourtown Health Department reported that the height of women in the city is approximately normally distributed with a mean of 5 feet, 4 inches (i.e., 64 inches) and a standard deviation of 3 inches. Suppose we select a random sample of five women from our school, measure the height of each, and calculate the sample mean. If we wished to know whether the height of women at our school is typical of the height of women in Ourtown, how should we compare our sample data to information we have about the Ourtown population distribution? We should compare the sample mean to the population distribution of Ourtown women as we do with an individual score. 4 We should compare each individual score in our sample, one at a time, to the population of individual scores. There really isn't a way to make any worthwhile comparison. We should compare this sample mean to a sampling distribution of all possible means for samples of 5 women from the population of women in Ourtown. Pregunta 11 In what way does the sampling distribution of means for a sample with n = 5 differ from a sampling distribution for a sample with n = 100? How do these sampling distributions compare to the population distribution? There is more variance in the sampling distribution with n = 5 than when n = 100, and both sampling distributions have more variance than the population. There is less variance in the sampling distribution with n = 5 than when n = 100, and the sampling distribution with n = 100 has a variance similar to the population. The variance is much smaller for the n = 100 sampling distribution than for the n = 5, and both sampling distributions have less variance than the population. Pregunta 12 Recall that the population mean for the height of Ourtown women is 5 feet, 4 inches (64 inches), the population standard deviation is 3 inches, and the distribution is normal. Now, try ordering the likelihood of the following four instances from the most likely to least likely. (It may be helpful to draw pictures.) 1. You select one woman from Ourtown at random and she is 5 feet, 4 inches (64 inches) tall to the nearest inch. 2. You select one woman from Ourtown at random and she is 6 feet, 1 inch (73 inches) tall to the nearest inch. 3. You select a random sample of 10 women from Ourtown, and the mean of their heights, to the nearest inch, is 5 feet, 4 inches. 4. You select a random sample of 10 women from Ourtown, and the mean of their heights, to the nearest inch, is 6 feet, 1 inch. Which of the following is the correct order? Why? Consider the reasons before you check the answer. 5 1, 3, 2, 4 3, 1, 2, 4 1, 3, 4, 2 3, 1, 4, 2 Pregunta 13 Imagine you draw a sample of 100 cases from a population with a mean of 57 and a standard deviation of 20. What are the mean ( ̅ ̅ = 57, ̅ = 20 ̅ = 57, ̅ = 200 ̅ = 57, ̅ =4 ̅ = 57, ̅ =2 and the standard error ( ̅) for the sampling distribution? Pregunta 14 Imagine that you select a sample of n = 25 women from Ourtown and find that the mean height of the sample is 5 feet, 7 inches (67 inches). The population mean is known to be 5 feet, 4 inches (64 inches) with standard deviation = 3 inches. Would a sample mean this large be more likely or less likely if n = 5? Why? Equally likely. Sample means are randomly distributed. More likely. The sampling distribution of the mean would be wider if the sample were smaller, so sample means would be more likely to be farther from the population mean. More likely. The sampling distribution is wider for larger samples resulting in more extreme obtained means. Less likely. The mean of the sample will increase as the sample size increases. Cada pregunta vale un punto. Se evaluará sobre 12. Prof. Ileana Castillo Arias, Ph.D. 6