Concept explainers
Recovering small quantities of calcium in the presence of magnesium can be a difficult problem for an analytical chemist. Suppose the amount of calcium
where the error term
A second procedure has random variable
where the error term
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An Introduction to Mathematical Statistics and Its Applications (6th Edition)
- Two samples each have n = 4 scores. The first sample has a variance of s2 = 10 and the second sample has a variance of s2 = 6. What is the estimated standard error for the sample mean difference? A. standard error = 1 B. standard error = 2 C. standard error = 4 D. standard error = 8 E. cannot be determined from the information givenarrow_forwardSay that we have two datapoints x and y. Both x and y are independently normally distributed with variance 1. The mean of x is µx and the mean of y is µy. 1. You have a null hypothesis of µx= µy. You will reject if |x − y| > 2. What is the type I error rate of this test? 2. You have a null hypothesis of µx= µy. You will reject if |x − y| > 2. If µx - µy = 10, what is the type II error rate of this test?arrow_forwardIn order for a new drug to be sold on the market, the variance of the active ingredient in each dose should be 0.02mg. A random sample of 12 tablets with a dosage strength of 56.02mg is taken. The variance of the active ingredient from this sample is found to be 0.0055. Does the data suggests at α=0.05 that the variance of the drug in the tablets is less than the desired amount? Assume the population is normally distributed. Step 1: State the null and alternative hypotheses. Round to four decimal places when necessary. Step 2: Determine the critical value(s) of the test statistic. If the test is two-tailed, separate the values with a comma. Round your answer to three decimal places. Step 3: Determine the value of the test statistic. Round your answer to three decimal places. Step 4: Make the decision and conclusionarrow_forward
- To test for the difference in means, Px - Hy, from two independent samples with the variance unknown but assumed equal, calculate the relevant test statistic given that = 54.8, s = 85.76, s = 71.38, 1x = 14, ny = 16, x = 555 and y = 575. %3D %3D %3D %3D O a. -6.47 O b. -6.57 O c. 6.47 O d. -6.31arrow_forwardLet X be a random variable with mean u= E(X) = 10.33 and variance o = 12.81. Determine: E{10- E 10- 4arrow_forwardAtandom variable X is normally distributed with a mean of 121 and a variance of 121, and a random variable Y is normally distributed with a mean of 170 and a variance of 289. The random variables have a correlation coefficient equal to 05. Find the mean and variance of the random varlable below W= 4X+5Y ked: Pw (Do not round) Scorearrow_forward
- Let X be a random variable with a mean equal to 100 and a standard deviation equal to 15. Let W=3x-2. Find the mean and variance of W.arrow_forwardIf X is a Gaussian random variable with mean zero and variance oʻ, find the pdf of Y = \X\.arrow_forwardA researcher tests their new treatment for sleep disorder among a sample of 16 individuals who have been diagnosed with a sleep disorder. The researcher knows that individuals diagnosed with a sleep disorder tend to have sleep quality scores that form a normal distribution with a µ = 50 and predicts that the new treatment will improve their sleep (higher scores mean higher improvement). After the treatment is administered to individuals in the sample, the mean is found to be M = 55 and the variance is s2 = 64. Conduct all the steps of hypothesis testing using an alpha = .05. -use formulas not excelarrow_forward
- An engineer wants to know if producing metal bars using a new experimental treatment rather than the conventional treatment makes a difference in the tensile strength of the bars (the ability to resist tearing when pulled lengthwise). At α=0.02, answer parts (a) through (e). Assume the population variances are equal and the samples are random. If convenient, use technology to solve the problem. Treatment Tensile strengths (newtons per square millimeter) Experimental 400 413 434 409 420 377 392 Conventional 381 446 436 350 404 354 375 361 355 386 (a) Identify the claim and state H0 and Ha. The claim is "The new treatment ▼ makes a difference does not make a difference in the tensile strength of the bars." What are H0 and Ha? The null hypothesis, H0, is ▼ mu 1 equals mu 2μ1=μ2 mu 1 less than or equals mu 2μ1≤μ2 mu 1 greater than or equals mu 2μ1≥μ2 . The alternative…arrow_forwardThe maximum patent life for a drug is 17 years. Subtracting the length of time required by the FDA for testing and approval of the drug provides the actual patent life for the drug that is, the length of time that the company has to recover research and development costs and to make a profit. The distribution of the lengths of actual patent lives for new drugs is given below, where Y is a random variable representing actual patent life of a drug (in years): y P(Y=y) F(y) 4 5 6 7 8 9 10 11 0.06 0.08 0.11 0.15 0.22 0.18 0.12 0.08 a) Complete the table above with the cumulative probability distribution for actual patent life (F(y) = P(Y ≤ y)). b) What is the probability that the actual patent life of a random drug is 6 years or less? c) What is the probability that the actual patent life of a random drug is more than 8 years? Show this in two ways: i. using the probability distribution for Y ii. using the cumulative probability distribution for Y, and applying our probability of comple-…arrow_forwardAn engineer wants to know if producing metal bars using a new experimental treatment rather than the conventional treatment makes a difference in the tensile strength of the bars (the ability to resist tearing when pulled lengthwise). At α=0.05, answer parts (a) through (e). Assume the population variances are equal and the samples are random. If convenient, use technology to solve the problem. Treatment Tensile strengths (newtons per square millimeter) Experimental 443 376 431 439 398 368 360 Conventional 392 392 400 425 372 370 439 366 392 381 (a) Identify the claim and state H0 and Ha. The claim is "The new treatment makes a difference makes a difference does not make a difference in the tensile strength of the bars." What are H0 and Ha? The null hypothesis, H0, is mu 1 equals mu 2μ1=μ2 mu 1 equals mu 2μ1=μ2 mu 1 less than or equals mu 2μ1≤μ2 mu 1 greater than or equals mu 2μ1≥μ2 . The…arrow_forward
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill