4.2.8: Let i denote the mean of a random sample of size n from a distribution that has mean u and variance o2 = 10. Findn so that the probability is approximately 0.954 that the random %3D interval (i -,x +;) includes u.
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- Q3. Suppose that X;i, i = 1,2, .,n have an Exponential distribution with parameter 2. a) Find maximum likelihood estimator (Â) for 2. b) Is â is an unbiased estimator?The average wait time to get seated at a popular restaurant in the city on a Friday night is 12 minutes. Is the mean wait time greater for men who wear a tie? Wait times for 13 randomly selected men who were wearing a tie are shown below. Assume that the distribution of the population is normal. 10, 11, 10, 14, 12, 14, 12, 10, 12, 13, 14, 13, 12 What can be concluded at the the αα = 0.10 level of significance level of significance? For this study, we should use (t-test for population mean, z-test for population proportion) The null and alternative hypotheses would be: H0: (symbol) (symbol) _____ H1: (symbol) (symbol) _____ The test statistic (symbol) = _____(please show your answer to 3 decimal places.) The p-value = _____ (Please show your answer to 4 decimal places.) The p-value is (symbol) αα Based on this, we should the null hypothesis. Thus, the final conclusion is that ... The data suggest that the population mean wait time for…Suppose that Y, Y2,.-.,Y, are independent random variables from a gamma distribution of the parameter (2,6+4). a) Use the method of moment to obtain estimator of B. b) Show that the estimator B, obtained in part a) is unbiased. c) Show that the estimator B is a minimum variance unbiased estimator of ß.
- * Let X1, X2, ..., X, be an independent sam- ple from a normal distribution with unknown mean u and variance o?. Show that the pair (X,5), where X=-Ex,. 5 =(x, - X)'. i=1 is a sufficient statistic for (u, o²). Given A> 0, consider AS as an estimator of o². For what values of A is AS (i) (ii) unbiased? maximum likelihood, Which value of A minimises the mean square error E(A5³ – o*)?Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find the probability of obtaining a reading between -2.307°C and 2.708°C. P(- 2.307 < Z < 2.708)=The average wait time to get seated at a popular restaurant in the city on a Friday night is 9 minutes. Is the mean wait time different for men who wear a tie? Wait times for 13 randomly selected men who were wearing a tie are shown below. Assume that the distribution of the population is normal. 7, 8, 9, 10, 10, 8, 10, 9, 10, 8, 11, 11, 9 What can be concluded at the the αα = 0.01 level of significance level of significance? For this study, we should use The null and alternative hypotheses would be: H0:H0: H1:H1: The test statistic = (please show your answer to 3 decimal places.) The p-value = (Please show your answer to 4 decimal places.) The p-value is αα Based on this, we should the null hypothesis. Thus, the final conclusion is that ... The data suggest the population mean is not significantly different from 9 at αα = 0.01, so there is statistically insignificant evidence to conclude that the population mean wait time for…
- Assume that adults have IQ scores that are normally distributed with a mean of mμ=100 and a standard deviation σ= 15. Find the probability that a randomly selected adult has an IQ less than 127. The probability that a randomly selected adult has an IQ less tha 127 isA company manufactures tonnis balls. When its tennis balls are dropped onto a concrete surface from a height of 100 inches, the company wants the mean height the balls bounce upward to be 54.7 inches. This average is maintained by periodically testing random samples of 25 tennis balls. If the t-value falls between -to ak and t as then the company will be satisfied that it is manufacturing acceptable tennis balls. A sample of 25 balls is randomly selected and tested. The mean bounce height of the sample is 56.8 inches and the standard deviation is 0.25 inch. Assume the bounce heights are approximately normally distributed. Is the company making acceptable tennis balls? Find -th os and to 96- -to.96 = 0.96 = (Round to three decimal places as needed.)A professor grades on a curve by assigning C's to all scores from u- ; to u+%, B's to all scores from i+% to µ+ , and D's to all scores from u – * to µ – 5. Everyone who scores below a D gets an F, and everyone who scores above a B gets an A. a. If the scores are normally distributed with mean u and variance o², what percentage of students will receive each grade? Hint: convert to z-scores and use pnorm() to find probabilities (percentages) b. If the scores are uniformly distributed (continuous) from 0 to 100 what percentage will receive each grade? Hint: first determine the values of u and o for this distribution