2. A hospital was concerned about reducing its wait time. A targeted wait time goal of 25 minutes was set. After implementing an improvement framework and process, a sample of 342 patients showed the mean wait time was 23.25 minutes, with a standard deviation of 16.37 minutes. Complete parts (a) and (b) below. a. If you test the null hypothesis at the 0.01 level of significance, is there evidence that the population mean wait time is less than 25 minutes? State the null and alternative hypotheses. A. H0: µ < 25, H1: µ > 25 B. H0: µ ? 25, H1: µ < 25 C. H0: µ > 25, H1: µ < 25 D. H0: µ < 25, H1: µ ? 25 E. H0: µ = 25, H1: µ ? 25 F. H0: µ ? 25, H1: µ ? 25 Find the test statistic for this hypothesis test. test statistic = ______ (Type an integer or a decimal. Round to two decimal places as needed.) Find the p-value. The p-value is ______. (Type an integer or a decimal. Round to three decimal places as needed.) Is there sufficient evidence to reject the null hypothesis? (Use a 0.01 level of significance.) A. Reject the null hypothesis. There is sufficient evidence at the 0.01 level of significance that the population mean wait time is greater than 25 minutes. B. Reject the null hypothesis. There is sufficient evidence at the 0.01 level of significance that the population mean wait time is less than 25 minutes. C. Do not reject the null hypothesis. There is insufficient evidence at the 0.01 level of significance that the population mean wait time is greater than 25 minutes. D. Do not reject the null hypothesis. There is insufficient evidence at the 0.01 level of significance that the population mean wait time is less than 25 minutes. b. Interpret the meaning of the p-value in this problem. Choose the correct answer below. A. The p-value is the probability that the actual mean wait time is 25 minutes given the sample mean wait time is 23.25 minutes. B. The p-value is the probability that the actual mean wait time is more than 23.25 minutes. C. The p-value is the probability of getting a sample mean wait time of 23.25 minutes or less if the actual mean wait time is 25 minutes. D. The p-value is the probability that the actual mean wait time is 23.25 minutes or less.
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A sample of n sludge specimens is selected and the pH of each one is determined. The one-sample t test will then be used to see if there is compelling evidence for concluding that true average pH is less than 7.0. What conclusion is appropriate in each of the following situations? (a) n = 7, t = -2.8, ̑ = 0.05 Reject the null hypothesis. There is sufficient evidence that the true average pH is less than 7.0. Reject the null hypothesis. There is not sufficient evidence that the true average pH is less than 7.0. Do not reject the null hypothesis. There is sufficient evidence that the true average pH is less than 7.0. Do not reject the null hypothesis. There is not sufficient evidence that the true average pH is less than 7.0. (b) n = 16, t = -3.1, ̑ = 0.01 Reject the null hypothesis. There is sufficient evidence that the true average pH is less than 7.0. Reject the null hypothesis. There is not sufficient evidence that the true average pH is less than 7.0. Do not reject the null hypothesis. There is sufficient evidence that the true average pH is less than 7.0. Do not reject the null hypothesis. There is not sufficient evidence that the true average pH is less than 7.0. (c) n = 14, t = -1.1, ̑ = 0.05 Reject the null hypothesis. There is sufficient evidence that the true average pH is less than 7.0. Reject the null hypothesis. There is not sufficient evidence that the true average pH is less than 7.0. Do not reject the null hypothesis. There is sufficient evidence that the true average pH is less than 7.0. Do not reject the null hypothesis. There is not sufficient evidence that the true average pH is less than 7.0. (d) n = 7, t = 0.6, ̑ = 0.05 Reject the null hypothesis. There is sufficient evidence that the true average pH is less than 7.0. Reject the null hypothesis. There is not sufficient evidence that the true average pH is less than 7.0. Do not reject the null hypothesis. There is sufficient evidence that the true average pH is less than 7.0. Do not reject the null hypothesis. There is not sufficient evidence that the true average pH is less than 7.0. (e) n = 7, x̄ = 6.68, s/∑n = 0.0820 We would reject the null hypothesis for any significance level at or above 0.004. We would fail to reject the null hypothesis for any significance level at or below 0.996. We would reject the null hypothesis for any significance level at or below 0.004. We would fail to reject the null hypothesis for any significance level at or above 0.996. You may need to use the appropriate table in the Appendix of Tables to answer this question.
Adi S.
The set R2×2 of all 2 × 2 matrices form a linear space in an obvious way. For Q90. to Q92., determine whether the given subset of R2×2 is a linear subspace of R2×2. Explain thoroughly. To show it is, you have to demonstrate that all three properties in the definition of linear subspace hold. To show it is not, you only have to show that one property fails. 90. The set of all X ∈ R2×2 that are symmetric, namely XT = X. 91. The set of all X ∈ R2×2 that satisfies X[1 2; 3 4] = [1 0; 0 1]. 92. The set of all X ∈ R2×2 that satisfies AX − XA = [0 0; 0 0], where A is a constant 2 × 2 matrix. (The exact values of the entries of A don't matter.) 93. Let R3×3 be the linear space of all 3 × 3 matrices. Let W be the set of all symmetric 3 × 3 matrices. Then W is a linear subspace of R3×3. Find a basis for W and identify dim(W). For a positive integer n, define Pn as the set of all polynomials in symbol t, of degree at most n. For example, for P2, we have 2 + 5t + t2 ∈ P2, and 2 − t ∈ P2 as well. The addition and scalar multiplication of polynomials make Pn a linear space. The neutral element is the zero polynomial 0. Remark: The zero polynomial 0 is usually not assigned a degree. But sometimes it is considered of degree −∐. With that convention, then it can be thought of as having a degree ≤ n, and so is included in Pn. 94. Every p(t) ∈ P2 can be uniquely written in the form a01 + a1t + a2t2. This statement is announcing a basis A of P2, formed by 1, t, t2. (A) Therefore, what is dim(P2)? (B) What is [2 + 3t − 5t2]A? (C) What is [3 − t]A? 95. [Some Algebra] Every p(t) ∈ P2 can be uniquely written in the form b01 + b1(t − 2) + b2(t − 2)2. In this problem, let's see how this works via a concrete case. Let p(t) = 4 − 5t + 2t2. Rewrite p(t) in the form b01 + b1(t − 2) + b2(t − 2)2. (Suggestion: A convenient way to do this is to introduce the symbol u = t − 2, thus t = u + 2. Replace t by u + 2 in 4 − 5t + 2t2, which you then simplify to the form b01 + b1u + b2u2. Then replace u by t − 2 to get the answer.) 96. Every p(t) ∈ P2 can be uniquely written in the form b01 + b1(t − 2) + b2(t − 2)2. This statement is announcing a basis B of P2. Identify this basis B. 97. Find the 3 × 3 matrices SAB and SBA, where A and B are the two bases of P2 from Q94. and Q96.. 98. Let A and B be the bases of P2 from Q94. and Q96.. Let p(t) = 4 − 5t + 2t2 ∈ P2. (A) What is [p(t)]A? (B) Use SBA found in Q97. and [p(t)]A above to find [p(t)]B. (C) Show that the answer above is consistent with your answer in Q95.. 99. A special case of Rn is R1. Think of R as R1. So R is a 1-dimensional linear space. Define T : P2 → R as T(p(t)) = p(2). For example, T(18 − 17t + 5t2) = 18 − 17 ⋅ 2 + 5 ⋅ 22 = 4. This T is a linear transformation. In our example, if 18 − 17t + 5t2 has been written as 4 + 3(t − 2) + 5(t − 2)2, it would be much easier to see that T maps it to 4. Use the statement that every p(t) ∈ P2 can be uniquely written in the form b01 + b1(t − 2) + b2(t − 2)2 to quickly identify a basis for ker(T). (If you understand what the question is asking, no computation will be needed.) 100. Let A and B be the bases of P2 from Q94. and Q96.. Let D : P2 → P2 be the ‐differential operator‐, i.e. D(p(t)) = p'(t). (E.g., D(t2) = 2t.) (A) Find [D]A. (B) Find [D]B from [D]A via SBADASA B, using the SBA and SAB obtained in Q97.. (C) Find [D]B more directly by finding [D(1)]B, [D(t − 2)]B, [D((t − 2)2)]B, and recall that [D]B = [[D(1)]B [D(t − 2)]B [D((t − 2)2)]B]. (Note: In Calculus I, when taking the derivative of 3(t − 2)2 + 5(t − 2) + 8, simply use the Chain Rule, with t − 2 as the inner function. This would immediately give the answer as 6(t − 2) + 5.)
Sri K.
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