Find two $$ z $$ values so that $$ 42 \% $$ of the middle area is bounded by them. Use 3 The Standard Normal Distribution Table and enter the answers rounded to 2 decimal places, separated by a comma if needed. $$ z=\text {. } $$

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We want the central $$42\%$$ of the standard normal distribution to lie between two numbers $$-z$$ and $$z$$. In other words, we need

$$
P(-z \le Z \le z) = 0.42.
$$

Because the standard normal distribution is symmetric, we can write

$$
P(-z \le Z \le z) = 2\Phi(z)-1 = 0.42,
$$

where $$\Phi(z)$$ is the cumulative distribution function (CDF) of the standard normal distribution.

1. Solve for $$\Phi(z)$$:

$$
2\Phi(z)-1=0.42 \quad \Longrightarrow \quad 2\Phi(z)=1.42 \quad \Longrightarrow \quad \Phi(z)=0.71.
$$

2. Now, we must find the value $$z$$ such that

$$
\Phi(z)=0.71.
$$

Using the Standard Normal Distribution Table, look up the value corresponding to a cumulative probability of $$0.71$$. Typically, these tables give values for $$z=0.55$$ and $$z=0.56$$:
- For $$z = 0.55$$, $$\Phi(0.55) \approx 0.7088$$.
- For $$z = 0.56$$, $$\Phi(0.56) \approx 0.7123$$.

Our target value ($$0.71$$) lies between these two. Interpolating (or choosing the value that is closest when rounded to two decimals), we take

$$
z \approx 0.56.
$$

3. Since the distribution is symmetric, the two $$z$$ values are

$$
z = -0.56 \quad \text{and} \quad z = 0.56.
$$

Thus, the values are

$$
z = -0.56, \; 0.56.
$$
The two $$ z $$ values are -0.56 and 0.56.

Find two values so that of the middle area is bounded by them. Use 3 The Standard Normal Distribution Table and enter the answers rounded to 2
decimal places, separated by a comma if needed.

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Find two values so that of the middle area is bounded by them. Use 3 The Standard Normal Distribution Table and enter the answers rounded to 2 decimal places, separated by a comma if needed.