Two statistically independent ran dom have variances of Variables X and new Man- Y o² = 9 and of dom variables are T + = 25. TWO defined by U = 3x + 4 Y V = 5X-2Y
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- A light bulb manufacturer wants to compare the mean lifetimes of two of its light bulbs, model A and model B. Independent random samples of the two models were taken. Analysis of 13 bulbs of model A showed a mean lifetime of 1358 hours and a standard deviation of 105 hours. Analysis of 12 bulbs of model B showed a mean lifetime of 1357 hours and a standard deviation of 103 hours. Assume that the populations of lifetimes for each model are normally distributed and that the variances of these populations are equal. Construct a 90% confidence interval for the difference µ, –, between the mean lifetime lj of model A bulbs and the mean lifetime µ, of model B bulbs. Then find the lower limit and upper limit of the 90% confidence interval. Carry your intermediate computations to at least three decimal places. Round your responses to at least two decimal places. (If necessary, consult a list of formulas.) Lower limit:|| Upper limit:IVitamin B6 is one of the vitamins in a multiple vitamin pill manufactured by a phar- maceutical company. The pills are produced with a mean of 50 mg of vitamin B6 per pill. The company believes that there is a deterioration of 1 mg/month, so that after 4 months it expects that µ = 46. A consumer group suspects that < 46 after 4 months. Assume the amount of vitamin B6 per pill is normally distributed with unknown variance. : µ (i) Define a test for Ho = 46 against H₁ μ< 46 at a significance level a = 0.01 based : µ on a random sample of size n. (ii) If a sample of 20 pills yielded a mean of x = 45.88 with a (sample) standard deviation of s = 0.17, what is the conclusion based on your test defined in part (i)? Also, compute the associated p-value. (iii) Find the smallest significance level a under which we can reject Ho based on the data given in part (ii). Does it equal to the p-value found in part (ii)? Explain why.If B, is an OLS estimator of a regression coefficient 'j associated with one of the explanatory variables, such that j=1,2,..,n, asymptotic standard error of B, ill refer to the: Select one: O a. estimated variance of B, when the error term is normally distributed O b. estimated variance of a given coefficient when the error term is not normally distributed O C. square root of the estimated variance of B. j when the error term is normally distributed. d. O d. square root of the estimated variance of B; when the error term is not normally distributed
- A set of solar batteries is used in a research satellite. The satellite can run on only one battery, but it runs best if more than one battery is used. The variance o of lifetimes of these batteries affects the useful lifetime of the satellite before it goes dead. If the variance is too small, all the batteries will tend to die at once. Why? If the variance is too large, the batteries are simply not dependable. Why? Engineers have determined that a variance of o = 23 months (squared) is most desirable for these batteries. A random sample of 30 batteries gave a sample variance of 15.4 months %3D (squared). Using a 0.05 level of significance, test the claim that o? = 23 against the claim that oʻ is different from 23. (b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.) What are the degrees of freedom?You have been asked to determine if two different production processes have different mean numbers of units produced per hour. Process 1 has a mean defined as μ₁ and process 2 has a mean defined as µ₂. The null and alternative hypotheses are H₂: μ₁ −μ₂ ≤0 and H₁ : µ₁ − µ₂ > 0. The process variances are unknown but assumed to be equal. Using random samples of 36 observations from process 1 and 49 observations from process 2, the sample means are 60 and 50 for populations 1 and 2 respectively. Complete parts a through d below. Click the icon to view a table of critical values for the Student's t-distribution. THIV IVVI VIMUVUV IVA Since the test statistic is IV The test statistic is t = MITM IV ܝ The critical value(s) is(are) 1.663. (Round to three decimal places as needed. Use a comma to separate answers as needed.) greater than t + ny – 2,0 ² reject Ho. c. Can you reject the null hypothesis, using a probability of Type I error x = 0.05, if the sample standard deviation from process 1…12. The OLS estimators of the coefficients in a multivariate linear regression model will be unbiased and consistent if the following set of conditions holds A. expected value of the error term given the independent variables is zero, the variance of the error term does not depend on the values of the independent variables, the observations (Yi,Xi,1,X₁,2,...,Xik) are i.i.d., large outliers are unlikely B. expected value of the error term given the independent variables is zero, the variance of the error term does not depend on the values of the independent variables, there is no perfect multicollinearity between the independent variables, the observations (Y₁X₁,1X₁,2,...,Xik) are i.i.d. C. expected value of the error term given the independent variables is zero, there is no perfect multicollinearity between the independent variables, the observations (Yi,Xi,1,Xi,2,...,Xik) are i.i.d., large outliers are unlikely D. the variance of the error term does not depend on the values of the…
- A light bulb manufacturer wants to compare the mean lifetimes of two of its light bulbs, model A and model B. Independent random samples of the two models were taken. Analysis of 15 bulbs of model A showed a mean lifetime of 1348 hours and a standard deviation of 81 hours. Analysis of 11 bulbs of model B showed a mean lifetime of 1351 hours and a standard deviation of 109 hours. Assume that the populations of lifetimes for each model are normally distributed and that the variances of these populations are equal. Construct a 99% confidence interval for the difference H-, between the mean lifetime u, of model A bulbs and the mean lifetime u, of model B bulbs. Then find the lower limit and upper limit of the 99% confidence interval, Carry your intermediate computations to at least three decimal places. Round your responses to at least two decimal places. (If necessary, consult a list of formulas.) Lower limit: Upper limit: Explanation Check 2021 McGraw-Hill Education. Al Rights Reserved…Two groups of mice were injected with a measured amount of tumor pulp. The first group of 27 mice was given a high dosage of chemotherapy while the second group of 30 mice was given a low dosage of chemotherapy. After forty days, the first group had an average tumor size of 0.51cc with a variance of 0.10; the second group had an average tumor size of 0.64cc with a variance of 0.045. Were the tumor sizes of group 1 significantly smaller than those of group 2? What is the observed test statistic?You have been asked to determine if two different production processes have different mean numbers of units produced per hour. Process 1 has a mean defined as µ₁ and process 2 has a mean defined as µ2. The null and alternative hypotheses are Ho: μ₁ −μ₂ ≤0 and H₁ : µ₁ −µ₂ > 0. The process variances are unknown but assumed to be equal. Using random samples of 36 observations from process 1 and 49 observations from process 2, the sample means are 60 and 50 for populations 1 and 2 respectively. Complete parts a through d below. Click the icon to view a table of critical values for the Student's t-distribution. a. Can you reject the null hypothesis, using a probability of Type I error α = 0.05, if the sample standard deviation from process 1 is 28 and from process 2 is 23? (Round to three decimal places as needed.) The test statistic is t =
- Which of the following is true of fixed effect estimators A. The fixed effects estimator is equal to the instrumental variable estimator if R^2 is equal to 1. B. The fixed effects estimators are biased if the regression model exhibits multicollinearity. C. The fixed effects estimators have lower variance than the ordinary least squares estimators. D. The fixed effects estimators have large standard errors when R^2 lies close to 0.Suppose we have two SRSS from two distinct populations and the samples are independent. We measure the same variable for both samples. Suppose both populations of the values of these variables are normally distributed but the means and standard deviations are unknown. For purposes of comparing the two means, we use (a) Two-sample t procedures (b) Matched pairs t procedures (c) z procedures (d) The least-squares regression line (e) None of the above. The answer isA weight-loss program wants to test how well their program is working. The company selects a simple random sample of 51 individual that have been using their program for 15 months. For each individual person, the company records the individual's weight when they started the program 15 months ago as an x-value. The subject's current weight is recorded as a y-value. Therefore, a data point such as (205, 190) would be for a specific person and it would indicate that the individual started the program weighing 205 pounds and currently weighs 190 pounds. In other words, they lost 15 pounds. When the company performed a regression analysis, they found a correlation coefficient of r = 0.707. This clearly shows there is strong correlation, which got the company excited. However, when they showed their data to a statistics professor, the professor pointed out that correlation was not the right tool to show that their program was effective. Correlation will NOT show whether or not there is…
