$$\left\{z\left(nf\left(z\right)\right)\right\}=-Z\ \frac{d}{dz}F\left(z\right)$$

$$Z\left\{f(n-k)\right\}=z^{-n}F\left(z\right)$$

$$Z\left\{f\left(n+1\right)\right\}=zF\left(z\right)-zf\left(0\right)$$

$$Z\left\{a^nf\left(n\right)\right\}=F\left(\frac{z}{a}\right)$$

$$\frac{cz\sin\alpha}{z^2-2cz\cos\alpha+c^2}\ ,\ \ z>c$$

$$\frac{cz\sin\alpha}{z^2+2cz\cos\alpha+c^2}\ ,\ \ z>c$$

$$\frac{cz\sin\alpha}{z^2-2cz\cos\alpha+c^2}\ ,\ \ z<c$$

$$\frac{cz\sin\alpha}{z^2-2cz\cos\alpha+c^2}\ ,\ \ z\ge c$$

$$\frac{1}{z-1}$$

$$\frac{z}{z+1}$$

$$\frac{z}{z-1}$$

$$\frac{z}{\left(z-1\right)^2}$$

$$\left(\frac{5z}{5z-1}\right)\left(\frac{7z}{7z-1}\right)\ ,\ \left|z\right|>\frac{1}{5}$$

$$\left(\frac{z}{z-5}\right)\left(\frac{z}{z-7}\right)\ ,\ \left|z\right|>7$$

$$\left(\frac{5z}{5z-1}\right)\left(\frac{7z}{7z-1}\right)\ ,\ \left|z\right|>\frac{1}{7}$$

$$\left(\frac{5z}{5z-1}\right)\left(\frac{7z}{7z-1}\right)\ ,\ \left|z\right|\ge\frac{1}{5}$$

$$2^k+1\ \ \ ,\ k\ge1$$

$$2^k-1\ \ \ ,\ k\ge1$$

$$2^k+1\ \ \ ,\ k>1$$

$$2^k-1\ \ \ ,\ k>1$$

NH Accepted ,the sample is drawn from the population with mean 5.4

AH Accepted ,the sample is drawn from the population with mean 5.4

NH Accepted ,the sample is drawn from the population with mean 6.2

AH Accepted ,the sample is drawn from the population with mean 6.2

$$n\ge30$$

$$n\le30$$

$$n<30$$

$$n>30$$

The 'z' variable cannot be found

When the number of samples is small

When the number of samples is large

When we know the value of the population mean

$$\left|Z\right|=5$$

$$\left|Z\right|=-5$$

$$\left|Z\right|=3$$

$$\left|Z\right|=1.96$$