PLEASE HELP ANSWER.(correctly)
2 answers:
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In a flowchart proof, <u>statements</u> and <u>conclusions</u> are connected with arrows.
In terms of mathematics, a statement is simply any sentence in which it can be verifiably true or false. A statement cannot be a subjective opinion. It must be an objective fact and there must not be any ambiguity involved. A conclusion is also a statement that derives from the first statement made.
As an example, you can have the simple argument "if it rains, then it gets wet outside". So the box on the left would be "it rains" and the box on the right would be "it gets wet outside". An arrow connecting the two shows the logical flow of how the argument is set up.
See the diagram below.
Side note: the box on the left is also considered the antecedent because it comes before the conclusion.
Step-by-step explanation:
To show that a statement is a tautology using truth table - is to show that all the entries in the expression are truths T.
We can do this by taking each statement, expression by expression. For example, to show that
[~p ∧ (p ∨ q)] → q
is a tautology, knowing we have 3 columns, we have 2^3 = 8 rows. We start by putting putting truth values for p, q, and r respectively
Next, we find ~p, then find (p ∨ q), then find ~p ∧ (p ∨ q), before finally arriving at the required [~p ∧ (p ∨ q)] → q
TERMINOLOGIES AND SYMBOLS
- T means Truth
- F means False
- ~p means negation of p.
~p is F if p is T, and vice versa
- ∧ means conjunction.
p ∧ q is T only if p is T and q is T.
- ∨ means disjunction.
p ∨ q is T if either p or q is T.
- → is for conditional 'if then'
p → q is T if both p and q are T, or both p and q are F, or p is F and q is T, otherwise, it is F.
THE STEP BY STEP WORKINGS FOR THE STATEMENTS GIVEN ARE IN THE ATTACHMENT.
Nice job with part A on getting those correct answers.
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Part B
- mu = mean = 43
- sigma = standard deviation = 3
Let's calculate the z score for the raw score x = 37
z = (x - mu)/sigma
z = (37 - 43)/3
z = -6/3
z = -2
A negative z score tells us we're below the mean. Specifically, we are exactly 2 standard deviations under the mean.
If you repeat those steps for x = 46 (not changing mu or sigma), then you should get z = 1. So we're now 1 standard deviation above the mean.
Ultimately, we want to know the area under the standard normal curve (aka z curve) from z = -2 to z = 1.
Refer to the chart below which is a breakdown of the Empirical Rule. The pink areas are within 1 standard deviation of the mean. We have two regions taking up roughly 34% of the full area under the curve, so they combine to 34+34 = 68%
Then we add on another 13.5% which is the blue area on the left (between z = -2 and z = -1)
So 13.5+68 = 81.5% of the area under the curve is between z = -2 and z = 1.
<h3>Answer: 81.5</h3>Note: the percent sign is already taken care of for us, so there's no need to type it in. But the answer above of course means 81.5% and that answer is approximate.
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Part C
Let's find the z score for x = 34. The values of mu and sigma are the same as before.
z = (x - mu)/sigma
z = (34-43)/3
z = -9/3
z = -3
So computing P(X > 34) is the same as P(Z > -3)
Notice how adding up all of the values in the chart mentioned gets us:
2.35+13.5+34+34+13.5+2.35 = 99.7
which means 99.7% of the distribution is within 3 standard deviations of the mean. The remaining 100% - 99.7% = 0.3% is the combined area of both tails. So each tail is (0.3%)/2 = 0.15%
In short, 0.15% of the area under the curve is to the left of z = -3
Therefore, 100% - 0.15% = 99.85% of the widgets are going to have z values larger than -3, which in turn means they have values larger than 34.
<h3>Answer: 99.85</h3>Again you don't have to worry about the percent sign. Like before, the answer is approximate.
