Using the p-value method, the decision rule is:
- |z| < 1.645: do not reject the null hypothesis.
- |z| > 1.645: reject the null hypothesis.
<h3>What is the relation between the p-value and the test hypothesis?</h3>
Depends on if the p-value is less or more than the significance level:
- If it is more, the null hypothesis is not rejected.
- If it is less, it is rejected.
In this problem, we have a two-tailed test, as we are testing if the mean is different of a value. For a significance level of 0.1, the critical value of z(when a p-value of 0.1 is obtained) is of |z| = 1.645, hence the decision rule is:
- |z| < 1.645: do not reject the null hypothesis.
- |z| > 1.645: reject the null hypothesis.
More can be learned about p-values at brainly.com/question/13873630
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For (2), start with the base case. When n = 2, we have
(n + 1)! = (2 + 1)! = 3! = 6
2ⁿ = 2² = 4
6 > 4, so the case of n = 2 is true.
Now assume the inequality holds for n = k, so that
(k + 1)! > 2ᵏ
Under this hypothesis, we want to show the inequality holds for n = k + 1. By definition of factorial, we have
((k + 1) + 1)! = (k + 2)! = (k + 2) (k + 1)!
Then by our hypothesis,
(k + 2) (k + 1)! > (k + 2) 2ᵏ = k•2ᵏ + 2ᵏ⁺¹
and k•2ᵏ ≥ 2•2² = 8, so
k•2ᵏ + 2ᵏ⁺¹ ≥ 8 + 2ᵏ⁺¹ > 2ᵏ⁺¹
which proves the claim.
Unfortunately, I can't help you with (3). Sorry!
Odd integers are 2 apart from each other (in between them is an even number)
x and x+2 is answer
