Find the mistake in the following “proof” that for all sets A and B , ( A − B ) ∪ ( A ∩ B ) ⊆ A . Proof; Suppose A and B are sets, and suppose x ∈ ( A − B ) ∪ ( A ∩ B ) . If x ∈ A then x ∈ A − B . Then, by definition of difference, x ∈ A and x ∈ B . Hence x ∈ A , and so ( A − B ) ∪ ( A ∩ B ) ⊆ A by definition of subset.

Question

Problem 3 Ten measurements of an impurity concentration in a process stream have been recorded. The sample mean is 87ppm and the sample standard deviation is ±13 ppm. Consider the null hypothesis that the impurity concentration has a true mean μo. Part A: Desired Probability that the sample mean will satisfy the null hypothesis: P = 0.4 Part B: Using the chart below, determine the4 highest value of the true mean that will lead to the null hypothesis being accepted with the probability assigned in Part A 1.00 0.90 0.80 0.70 0.60 0.50 0.40 Probability of accepting Ho 0.30 0.20 0.10 1 ° 0 30 40 50 75 100 10 0.2 0.4 0.6 0.8 1.0 1.2 =2.5 1.4 1.6 1.8 2.0 2.2 2.4 2.6 d 2.8 3.0 3.2

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Chapter 6.2, Problem 22ES

To determine

Find the mistake in the following “proof” that for all sets A and B, (A−B)∪(A∩B)⊆A.

Proof; Suppose A and B are sets, and suppose x∈(A−B)∪(A∩B). If x∈A then x∈A−B.

Then, by definition of difference, x∈A and x∈B. Hence x∈A, and so (A−B)∪(A∩B)⊆A by definition of subset.

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Problem 3 Ten measurements of an impurity concentration in a process stream have been recorded. The sample mean is 87ppm and the sample standard deviation is ±13 ppm. Consider the null hypothesis that the impurity concentration has a true mean μo. Part A: Desired Probability that the sample mean will satisfy the null hypothesis: P = 0.4 Part B: Using the chart below, determine the4 highest value of the true mean that will lead to the null hypothesis being accepted with the probability assigned in Part A 1.00 0.90 0.80 0.70 0.60 0.50 0.40 Probability of accepting Ho 0.30 0.20 0.10 1 ° 0 30 40 50 75 100 10 0.2 0.4 0.6 0.8 1.0 1.2 =2.5 1.4 1.6 1.8 2.0 2.2 2.4 2.6 d 2.8 3.0 3.2

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