Answer:
Let p = "<em>I will hang out with my dad on Sunday</em>"
Let q = "<em>I will go snowboarding on Sunday</em>"
Let r = "<em>I will go to Kirkwood</em>"
i) p v q
i) p v qii) q --> r
i) p v qii) q --> riii) ~p
i) p v qii) q --> riii) ~piv) q ^ r
Answer:
The test statistic for Norah's test is ![t = -2](https://tex.z-dn.net/?f=t%20%3D%20-2)
Step-by-step explanation:
Norah is in charge of a quality control test that involves measuring the amounts in a sample of bottles to see if the sample mean amount is significantly different than 500 ml.
This means that at the null hypothesis we test if the sample mean is 500 ml, that is:
![H_0: \mu = 500](https://tex.z-dn.net/?f=H_0%3A%20%5Cmu%20%3D%20500)
At the alternate hypothesis, we test if it is differente than 500 ml, that is:
![H_a: \mu \neq 500](https://tex.z-dn.net/?f=H_a%3A%20%5Cmu%20%5Cneq%20500)
The test statistic is:
![t = \frac{X - \mu}{\frac{s}{\sqrt{n}}}](https://tex.z-dn.net/?f=t%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Cfrac%7Bs%7D%7B%5Csqrt%7Bn%7D%7D%7D)
In which X is the sample mean,
is the value tested at the null hypothesis, s is the standard deviation of the sample and n is the size of the sample.
500 is tested at the null hypothesis:
This means that ![\mu = 500](https://tex.z-dn.net/?f=%5Cmu%20%3D%20500)
She takes a random sample of 16 bottles and finds a mean amount of 497ml, and a sample standard deviation of 6ml.
This means that ![n = 16, X = 497, s = 6](https://tex.z-dn.net/?f=n%20%3D%2016%2C%20X%20%3D%20497%2C%20s%20%3D%206)
Calculate the test statistic for Norah's test.
![t = \frac{X - \mu}{\frac{s}{\sqrt{n}}}](https://tex.z-dn.net/?f=t%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Cfrac%7Bs%7D%7B%5Csqrt%7Bn%7D%7D%7D)
![t = \frac{497 - 500}{\frac{6}{\sqrt{16}}}](https://tex.z-dn.net/?f=t%20%3D%20%5Cfrac%7B497%20-%20500%7D%7B%5Cfrac%7B6%7D%7B%5Csqrt%7B16%7D%7D%7D)
![t = -2](https://tex.z-dn.net/?f=t%20%3D%20-2)
The test statistic for Norah's test is ![t = -2](https://tex.z-dn.net/?f=t%20%3D%20-2)
Answer:
D. x= 4, y = 2
Step-by-step explanation:
Since it's multiple choice, you can plug in the values
So we are trying to find this red line's length.
We can either find it directly, or use the blue line firs, and then use it as a leg for the green triangle.
So the blue leg is a hypotenuse for two of the edges. So:
![blue^2 = leg^2 + leg^2](https://tex.z-dn.net/?f=blue%5E2%20%3D%20leg%5E2%20%2B%20leg%5E2)
from the Pythagorean Theorem
OR
![blue = \sqrt{leg^2 + leg^2}](https://tex.z-dn.net/?f=blue%20%3D%20%20%5Csqrt%7Bleg%5E2%20%2B%20leg%5E2%7D%20)
Which works out to:
![blue = \sqrt{10^2 + 10^2} = \sqrt{100+100} = \sqrt{200} = \sqrt{(100)(2)}=10 \sqrt{2}](https://tex.z-dn.net/?f=blue%20%3D%20%20%5Csqrt%7B10%5E2%20%2B%2010%5E2%7D%20%3D%20%20%5Csqrt%7B100%2B100%7D%20%3D%20%5Csqrt%7B200%7D%20%3D%20%20%5Csqrt%7B%28100%29%282%29%7D%3D10%20%5Csqrt%7B2%7D%20%20%20%20)
So now that we have that, using the Pythagorean Theorem again gives:
![red = \sqrt{blue^2 + 10^2} = \sqrt{(10 \sqrt{2})^2+10^2}= \sqrt{200+100}= \sqrt{300}](https://tex.z-dn.net/?f=red%20%3D%20%20%5Csqrt%7Bblue%5E2%20%2B%2010%5E2%7D%20%3D%20%20%5Csqrt%7B%2810%20%5Csqrt%7B2%7D%29%5E2%2B10%5E2%7D%3D%20%5Csqrt%7B200%2B100%7D%3D%20%5Csqrt%7B300%7D)
![\sqrt{300}= \sqrt{100*3}=10 \sqrt{3}](https://tex.z-dn.net/?f=%20%5Csqrt%7B300%7D%3D%20%5Csqrt%7B100%2A3%7D%3D10%20%5Csqrt%7B3%7D%20%20%20)
So the length of the red line is found that way.
But wait! There's more!
As it turns out, the red line can be found with an easier way that works with cubes and boxes (cuboids). It's really easy:
![a^2 + b^2+c^2=d^2](https://tex.z-dn.net/?f=a%5E2%20%2B%20b%5E2%2Bc%5E2%3Dd%5E2)
Where a, b, and c are all 10m, and d is the red line. This greatly reduces the math:
![d = \sqrt{10^2+10^2+10^2} = \sqrt{100+100+100} = \sqrt{300}](https://tex.z-dn.net/?f=d%20%3D%20%20%5Csqrt%7B10%5E2%2B10%5E2%2B10%5E2%7D%20%3D%20%5Csqrt%7B100%2B100%2B100%7D%20%3D%20%20%5Csqrt%7B300%7D%20)
which gives the same answer as above, which you can see.