12. Suppose (1, 2, ..., n) is a random sample from a normal distribution, find the absolute efficiency of the maximum likelihood estimator of 0 if, x; ~ N(0, 1).
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- REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults†. Assume that REM sleep time is normally distributed for both children and adults. A random sample of n1 = 9 children (9 years old) showed that they had an average REM sleep time of x1 = 2.5 hours per night. From previous studies, it is known that ?1 = 0.6 hour. Another random sample of n2 = 9 adults showed that they had an average REM sleep time of x2 = 2.10 hours per night. Previous studies show that ?2 = 0.5 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 10% level of significance. Solve the problem using both the traditional method and the P-value method. (Test the difference ?1 − ?2. Round the test statistic and critical value to two decimal places. Round the P-value to four decimal places.) test statistic critical value…REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults†. Assume that REM sleep time is normally distributed for both children and adults. A random sample of n1 = 11 children (9 years old) showed that they had an average REM sleep time of x1 = 2.5 hours per night. From previous studies, it is known that σ1 = 0.7 hour. Another random sample of n2 = 11 adults showed that they had an average REM sleep time of x2 = 2.00 hours per night. Previous studies show that σ2 = 0.6 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 10% level of significance. Solve the problem using both the traditional method and the P-value method. (Test the difference μ1 − μ2. Round the test statistic and critical value to two decimal places. Round the P-value to four decimal places.) test statistic critical value…Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
- REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults†. Assume that REM sleep time is normally distributed for both children and adults. A random sample of n1 = 8 children (9 years old) showed that they had an average REM sleep time of x1 = 2.9 hours per night. From previous studies, it is known that ?1 = 0.9 hour. Another random sample of n2 = 8 adults showed that they had an average REM sleep time of x2 = 2.00 hours per night. Previous studies show that ?2 = 0.6 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance. (a) What is the level of significance?State the null and alternate hypotheses. H0: ?1 = ?2; H1: ?1 ≠ ?2H0: ?1 = ?2; H1: ?1 < ?2 H0: ?1 = ?2; H1: ?1 > ?2H0: ?1 < ?2; H1: ?1 = ?2 (b) What sampling distribution will you use? What assumptions are you…REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults†. Assume that REM sleep time is normally distributed for both children and adults. A random sample of n1 = 11 children (9 years old) showed that they had an average REM sleep time of x1 = 2.9 hours per night. From previous studies, it is known that ?1 = 0.6 hour. Another random sample of n2 = 11 adults showed that they had an average REM sleep time of x2 = 2.20 hours per night. Previous studies show that ?2 = 0.7 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance. State the null and alternate hypotheses. H0: ?1 = ?2; H1: ?1 ≠ ?2 H0: ?1 < ?2; H1: ?1 = ?2 H0: ?1 = ?2; H1: ?1 > ?2 H0: ?1 = ?2; H1: ?1 < ?2 (b) What sampling distribution will you use? What assumptions are you making? The Student's t. We assume…TRUE OR FALSEa. When the assumptions of distributions are met, the parametric tests are more powerful than non-parametric testsb. In Poisson distribution, the mean is equal to its variance c. The Main objective of Statistical inference is to make inferences about the population-based n the whole set of population values d. The Variable X defined X~N(0,1) is called the Standard Normal Distribution e. Welch Test for testing two population means does not assume that the variances are equal
- Estimate the likelihood function for the following random samples: (a) The observation values (x¡,X2,X3, X, ) = (1, 3, 2, 2) with are х, ~ Binomial(3,0). The observation values are(x ,X2, X3, X4 ) = (1.12, 3.32, 1.98, 2.12) with (b) х ~ Ехponential(0).If X is a continuous random variable, what is the value of X that will be exceeded 30% of the time, if X has a gamma distribution with a mean of 2 and a variance of 2? (just set up the equation in itssimplest form)A researcher that wanted to estimate the expectation AY of a random variable Y got three independent observations, Y, Y, Y The researcher knows the value o, of the variance of Y and is considering the following estimators: Pi = 4) Yi + (+) ¥ Py = () Yn + (;) ¥½ + () Y½ in (}) Yi + (}) ¥z + (() %D Which of the following is correct? ONone of the above Ois an unbiased estimator of l and it has the smallest variance of the three estimators. Ois an unbiased estimator of µ and it has the smallest variance of the three estimators. O and i, are both unbiased estimators of fl and Var (ſîz) < Var (îì3). is an unbiased estimator of µ and it has the smallest variance of the three estimators.
- derive maximum likelihood estimator(s) of the underlying parameter(s) *In detailedREM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults†. Assume that REM sleep time is normally distributed for both children and adults. A random sample of n1 = 8 children (9 years old) showed that they had an average REM sleep time of x1 = 2.5 hours per night. From previous studies, it is known that ?1 = 0.9 hour. Another random sample of n2 = 8 adults showed that they had an average REM sleep time of x2 = 1.60 hours per night. Previous studies show that ?2 = 0.6 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance. what is the sample test statistic find or estimate p valueFor each random variable defined, describe the set of possible values for the variable, and state whether the variable is discrete or continuous. (a) U = number of times a surfer has to paddle in front of a wave before catching one (b) X = length of a randomly selected angelfish