Answer:
- x = 81 and 3/4
Step-by-step explanation:
8 × 2 – 4x = 363
16 - 4x = 363
Subtract 16 from each side:
16 - 16 - 4x = 363 - 16
- 4x = 327
Divide each side by 4:
- 4x ÷ 4 = 327 ÷ 4
- x = 81 and 3/4
The nearest tenth would be the first decimal place, so the answer would be 23.8
Answer:
Slope is 3/2
Step-by-step explanation:
3x-2y=6
-2y=-3x+6
y=3/2-3
Complete Question
A random sample of 300 circuits generated 13 defectives. a. Use the data to test
![H_o : p = 0.05](https://tex.z-dn.net/?f=H_o%20%3A%20p%20%3D%200.05)
Versus
![H_1 : p \ne 0.05](https://tex.z-dn.net/?f=H_1%20%3A%20p%20%5Cne%200.05)
Use α = 0.05. Find the P-value for the test
Answer:
The p-value is
Step-by-step explanation:
From the question we are told that
The sample size is n = 300
The number of defective circuits is k = 13
Generally the sample proportion of defective circuits is mathematically represented as
![\^ p = \frac{k}{n}](https://tex.z-dn.net/?f=%5C%5E%20p%20%3D%20%5Cfrac%7Bk%7D%7Bn%7D)
=> ![\^ p = \frac{13}{300}](https://tex.z-dn.net/?f=%5C%5E%20p%20%3D%20%5Cfrac%7B13%7D%7B300%7D)
=> ![\^ p = 0.0433](https://tex.z-dn.net/?f=%5C%5E%20p%20%3D%200.0433)
Generally the standard Error is mathematically represented as
![SE = \sqrt{\frac{p(1- p)}{n} }](https://tex.z-dn.net/?f=SE%20%20%3D%20%5Csqrt%7B%5Cfrac%7Bp%281-%20p%29%7D%7Bn%7D%20%7D)
=> ![SE = \sqrt{\frac{0.05(1- 0.05)}{300} }](https://tex.z-dn.net/?f=SE%20%20%3D%20%5Csqrt%7B%5Cfrac%7B0.05%281-%200.05%29%7D%7B300%7D%20%7D)
=> ![SE = 0.0126](https://tex.z-dn.net/?f=SE%20%20%3D%200.0126)
Generally the test statistics is mathematically represented as
![z = \frac{\^ p - p }{SE}](https://tex.z-dn.net/?f=z%20%3D%20%20%5Cfrac%7B%5C%5E%20p%20-%20p%20%7D%7BSE%7D)
=> ![z = \frac{0.0433 - 0.05 }{0.0126}](https://tex.z-dn.net/?f=z%20%3D%20%20%5Cfrac%7B0.0433%20-%200.05%20%7D%7B0.0126%7D)
=> ![z = -0.5317](https://tex.z-dn.net/?f=z%20%3D%20%20-0.5317)
From the z table the area under the normal curve to the left corresponding to -0.5317 is
![(P < -0.5317 ) = 0.29747](https://tex.z-dn.net/?f=%28P%20%3C%20-0.5317%20%29%20%3D%200.29747)
Generally the p-value is mathematically represented as
![p-value = 2 * P(Z < -0.5317 )](https://tex.z-dn.net/?f=p-value%20%3D%20%202%20%2A%20P%28Z%20%3C%20%20-0.5317%20%29)
=> ![p-value = 2 * 0.29747](https://tex.z-dn.net/?f=p-value%20%3D%20%202%20%2A%200.29747)
=>
Answer:
We want to find:
![\lim_{n \to \infty} \frac{\sqrt[n]{n!} }{n}](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B%5Csqrt%5Bn%5D%7Bn%21%7D%20%7D%7Bn%7D)
Here we can use Stirling's approximation, which says that for large values of n, we get:
![n! = \sqrt{2*\pi*n} *(\frac{n}{e} )^n](https://tex.z-dn.net/?f=n%21%20%3D%20%5Csqrt%7B2%2A%5Cpi%2An%7D%20%2A%28%5Cfrac%7Bn%7D%7Be%7D%20%29%5En)
Because here we are taking the limit when n tends to infinity, we can use this approximation.
Then we get.
![\lim_{n \to \infty} \frac{\sqrt[n]{n!} }{n} = \lim_{n \to \infty} \frac{\sqrt[n]{\sqrt{2*\pi*n} *(\frac{n}{e} )^n} }{n} = \lim_{n \to \infty} \frac{n}{e*n} *\sqrt[2*n]{2*\pi*n}](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B%5Csqrt%5Bn%5D%7Bn%21%7D%20%7D%7Bn%7D%20%3D%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B%5Csqrt%5Bn%5D%7B%5Csqrt%7B2%2A%5Cpi%2An%7D%20%2A%28%5Cfrac%7Bn%7D%7Be%7D%20%29%5En%7D%20%7D%7Bn%7D%20%3D%20%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7Bn%7D%7Be%2An%7D%20%2A%5Csqrt%5B2%2An%5D%7B2%2A%5Cpi%2An%7D)
Now we can just simplify this, so we get:
![\lim_{n \to \infty} \frac{1}{e} *\sqrt[2*n]{2*\pi*n} \\](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B1%7D%7Be%7D%20%2A%5Csqrt%5B2%2An%5D%7B2%2A%5Cpi%2An%7D%20%5C%5C)
And we can rewrite it as:
![\lim_{n \to \infty} \frac{1}{e} *(2*\pi*n)^{1/2n}](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B1%7D%7Be%7D%20%2A%282%2A%5Cpi%2An%29%5E%7B1%2F2n%7D)
The important part here is the exponent, as n tends to infinite, the exponent tends to zero.
Thus:
![\lim_{n \to \infty} \frac{1}{e} *(2*\pi*n)^{1/2n} = \frac{1}{e}*1 = \frac{1}{e}](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B1%7D%7Be%7D%20%2A%282%2A%5Cpi%2An%29%5E%7B1%2F2n%7D%20%3D%20%5Cfrac%7B1%7D%7Be%7D%2A1%20%3D%20%5Cfrac%7B1%7D%7Be%7D)