To solve these problems, we will use properties of the standard normal distribution $Z$, which has a mean $\mu = 0$ and a standard deviation $\sigma = 1$.

We will find the critical value $z_0$ by using a standard normal table (Z-table) or the inverse cumulative distribution function of the standard normal distribution (sometimes written as $\Phi^{-1}(p)$ where $\Phi(z_0)$ is the cumulative probability up to $z_0$).

Let's go through each part:

(a) $P(z \leq z_0) = 0.8472$

Here, you are looking for the value of $z_0$ where the cumulative probability is 0.8472. This is the inverse cumulative distribution function for the standard normal distribution.

From the Z-table or using $\Phi^{-1}(0.8472)$, the value of $z_0$ is approximately: $z_0 \approx 1.02$

(b) $P(-z_0 \leq z \leq z_0) = 0.0484$

This is the probability that the random variable lies between $-z_0$ and $z_0$. First, we need to find the total probability outside this range, which is $1 - 0.0484 = 0.9516$. Since the distribution is symmetric, the area in each tail is $\frac{1 - 0.0484}{2} = 0.4758$.

From the Z-table or using $\Phi^{-1}(0.4758)$, the value of $z_0$ is approximately: $z_0 \approx 0.06$

(c) $P(-z_0 \leq z \leq z_0) = 0.095$

Again, the total area outside this range is $1 - 0.095 = 0.905$. The area in each tail is $\frac{1 - 0.095}{2} = 0.4525$.

From the Z-table or using $\Phi^{-1}(0.4525)$, the value of $z_0$ is approximately: $z_0 \approx 0.11$

(d) $P(z \geq z_0) = 0.1402$

This is the upper tail probability. The cumulative probability up to $z_0$ is $1 - 0.1402 = 0.8598$.

From the Z-table or using $\Phi^{-1}(0.8598)$, the value of $z_0$ is approximately: $z_0 \approx 1.08$

(e) $P(-z_0 \leq z \leq 0) = 0.0434$

This represents the area between $-z_0$ and 0. Since the distribution is symmetric, the cumulative area up to $z_0$ would be $0.5 - 0.0434 = 0.4566$.

From the Z-table or using $\Phi^{-1}(0.4566)$, the value of $z_0$ is approximately: $z_0 \approx 0.11$

(f) $P(-2.01 \leq z \leq z_0) = 0.6121$

First, find the probability that corresponds to $z = -2.01$. From the Z-table, $P(z \leq -2.01) \approx 0.0222$.

Now, the probability up to $z_0$ is $0.0222 + 0.6121 = 0.6343$.

From the Z-table or using $\Phi^{-1}(0.6343)$, the value of $z_0$ is approximately: $z_0 \approx 0.35$

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Let me know if you need any more details on the process! Here are 5 related questions to explore:

1. How is the standard normal distribution used in hypothesis testing?
2. What is the significance of the symmetry of the standard normal distribution?
3. How do you use the Z-table to find probabilities for non-standard normal distributions?
4. How can you transform a normal distribution to a standard normal one?
5. What is the relationship between z-scores and percentiles?

Tip: The area under the standard normal curve is always 1, and it helps to break down problems using the symmetry of the distribution when finding probabilities.