Statistics Recitation Assignment: Sampling Distributions and Confidence Intervals, Assignments of Probability and Statistics

Download Statistics Recitation Assignment: Sampling Distributions and Confidence Intervals and more Assignments Probability and Statistics in PDF only on Docsity! Recitation assignment due 2-16-06. Always show your method! 1. The profit from each sale made by a company is regarded as a random sample made with-replacement from a distribution having mean $47.51 and sd 31.22. a. The above reflects wide sampling variation, why? b. If a sample of 50 sales is selected with- replacement what are the expected value and sd of the profit from the first sale (sample of one)? c. Refer to (b). What are the expected value and sd of the total profit from 50 such sales? d. Refer to (c). Sketch the approximate normal distribution of total profit suggested by the CLT. e. Refer to (b). What are the expected value and sd of the sample average profit from 50 such sales? f. Refer to (e). Sketch the approximate normal distribution of sample average profit. 2. A company has 10,678 employees whose projected benefit score x can only be determined by careful health audit. From them an equal probability without-replacement sample of 70 will be selected for this audit. Suppose (unknown to the company) the scores x have a population average of $634 with sd $244. a. Do we know that the population distribution of health score x is normal? Sketch what the population distribution would look like if it were normal. enclosing the population mean. does it do so in this case? c. Determine a 92% confidence interval for the population mean. You will need to consult the z-table. 4. Suppose that a population of account balances has been brought under statistical control with mean $73.22 and s.d. $5.74. a. Sketch the form of the population distribution. b. Sketch the form of the distribution of the sample mean of a random with-replacement sample of 100 accounts. Do so in the same scale a used for (a) above in order that the two may be compared. What is the comparison in respect of the means and sd’s? c. A sample of n = 5 accounts is selected with- replacement. Suppose that the sample mean is $70.52 with sample sd s = $6.57. Give a t- based 95% confidence interval for the population mean based upon this sample. d. Refer to (c) above. There is 95% chance that such a confidence interval will contain the true population mean. Has this one done so? e. Why would it have been wrong to use 1.96 to form this interval? What did we assume, not assumed in (1), that makes the t-method applicable to this problem? 4. An advertiser wishes to estimate the proportion p of consumers who will favor a new package over the existing one. A with- replacement sample of 40 consumers will be panel-tested to determine their preferences. a. Suppose (unknown to the company) that 32% of consumers favor the new package. Sketch the approximate distribution of the sample fraction pHAT who will favor the new package. b. Suppose that of the 40 sample consumers there are 17 who favor the new package. What are your estimates of p and the sd of the population? c. Refer to (b). Give the standard error of pHAT for this sample. d. Refer to (a, b). Give a 97% confidence interval for p based upon the CLT. Around 97%