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Ivanshal [37]
3 years ago
5

1. Write a two-column proof for the following conjecture. Given: K is the midpoint of . Prove: is an isosceles triangle.

Mathematics
1 answer:
Over [174]3 years ago
7 0
1. Angle LKM and Angle JKM are supplementary 1. LJ is a straight line, since K is a midpoint
2. LK = KJ 2. K is a midpoint
3. Angle L + Angle M + Angle LKM = Angle LKM + Angle MJK 3. A triangle's total degrees is equal to the total degrees of two supplementary angles.
4. Angle JKM = Angle L + Angle M 4. Simple Algebra: Subtract Angle 
5. LK = KM 5. MK is intersecting LJ at a midpoint.
6. Triangle LKM is isoceles 6. Two of three sides and angles are equal.
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Answer:

t=\frac{\bar d -0}{\frac{s_d}{\sqrt{n}}}=\frac{0.983 -0}{\frac{1.685}{\sqrt{12}}}=2.021

df=n-1=12-1=11

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If we compare the the p value with the significance level provided \alpha=0.1, we see that p_v < \alpha, so then we can reject the null hypothesis. and there is a significant increase in the miles per gallon from 2017 to 2019 at 10% of significance.

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Previous concepts

A paired t-test is used to compare two population means where you have two samples in  which observations in one sample can be paired with observations in the other sample. For example  if we have Before-and-after observations (This problem) we can use it.  

Let put some notation :

x=test value 2017 , y = test value 2019

x: 28.7 32.1 29.6 30.5 31.9 30.9 32.3 33.1 29.6 30.8 31.1 31.6

y: 31.1 32.4 31.3 33.5 31.7 32.0 31.8 29.9 31.0 32.8 32.7 33.8

Solution to the problem

The system of hypothesis for this case are:

Null hypothesis: \mu_y- \mu_x \leq 0

Alternative hypothesis: \mu_y -\mu_x >0

Because if we have an improvement we expect that the values for 2019 would be higher compared with the values for 2017

The first step is calculate the difference d_i=y_i-x_i and we obtain this:

d: 2.4, 0.3, 1.7,3,-0.2, 1.1, -0.5, -3.2, 1.4, 2, 1.6, 2.2

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\bar d= \frac{\sum_{i=1}^n d_i}{n}= \frac{11.8}{12}=0.983

The third step would be calculate the standard deviation for the differences, and we got:

s_d =\frac{\sum_{i=1}^n (d_i -\bar d)^2}{n-1} =1.685

We assume that the true difference follows a normal distribution. The 4th step is calculate the statistic given by :

t=\frac{\bar d -0}{\frac{s_d}{\sqrt{n}}}=\frac{0.983 -0}{\frac{1.685}{\sqrt{12}}}=2.021

The next step is calculate the degrees of freedom given by:

df=n-1=12-1=11

Now we can calculate the p value, since we have a right tailed test the p value is given by:

p_v =P(t_{(11)}>2.021) =0.0342

If we compare the the p value with the significance level provided \alpha=0.1, we see that p_v < \alpha, so then we can reject the null hypothesis. and there is a significant increase in the miles per gallon from 2017 to 2019 at 10% of significance.

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