Answer 4 b. $$ n=25 $$ scores Answer 4 22. For a population with $$ \sigma=16 $$, how large a sample is necessary to have a standard error that is a. equal to 8 points? b. equal to 4 points? Success Tips c. equal to 2 points? 23. If the poputation standard deviation is $$ \sigma=24 $$, how large a sample is necessary to have a standard error that is SIEN OUT a. equal to 6 points? Arsuar 9 b. equal to 3 points?

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To solve the problem, we need to use the formula for the standard error (SE) of the mean, which is given by:

$$
SE = \frac{\sigma}{\sqrt{n}}
$$

where:
- $$ SE $$ is the standard error,
- $$ \sigma $$ is the population standard deviation,
- $$ n $$ is the sample size.

We can rearrange this formula to solve for $$ n $$:

$$
n = \left(\frac{\sigma}{SE}\right)^2
$$

Now, let's solve each part step by step.

### Problem 22
Given $$ \sigma = 16 $$:

#### a. For $$ SE = 8 $$:
$$
n = \left(\frac{16}{8}\right)^2
$$

#### b. For $$ SE = 4 $$:
$$
n = \left(\frac{16}{4}\right)^2
$$

#### c. For $$ SE = 2 $$:
$$
n = \left(\frac{16}{2}\right)^2
$$

### Problem 23
Given $$ \sigma = 24 $$:

#### a. For $$ SE = 6 $$:
$$
n = \left(\frac{24}{6}\right)^2
$$

#### b. For $$ SE = 3 $$:
$$
n = \left(\frac{24}{3}\right)^2
$$

Now, let's calculate the values for $$ n $$ for each case.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{16}{8}\right)^{2}$$
- step1: Divide the terms:
  $$2^{2}$$
- step2: Evaluate the power:
  $$4$$
Calculate or simplify the expression $$ (16/2)^2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{16}{2}\right)^{2}$$
- step1: Divide the terms:
  $$8^{2}$$
- step2: Evaluate the power:
  $$64$$
Calculate or simplify the expression $$ (24/3)^2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{24}{3}\right)^{2}$$
- step1: Divide the terms:
  $$8^{2}$$
- step2: Evaluate the power:
  $$64$$
Calculate or simplify the expression $$ (24/6)^2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{24}{6}\right)^{2}$$
- step1: Divide the terms:
  $$4^{2}$$
- step2: Evaluate the power:
  $$16$$
Calculate or simplify the expression $$ (16/4)^2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{16}{4}\right)^{2}$$
- step1: Divide the terms:
  $$4^{2}$$
- step2: Evaluate the power:
  $$16$$
Here are the calculated sample sizes for each part of the problem:

### Problem 22
Given $$ \sigma = 16 $$:

#### a. For $$ SE = 8 $$:
$$
n = \left(\frac{16}{8}\right)^2 = 4
$$

#### b. For $$ SE = 4 $$:
$$
n = \left(\frac{16}{4}\right)^2 = 16
$$

#### c. For $$ SE = 2 $$:
$$
n = \left(\frac{16}{2}\right)^2 = 64
$$

### Problem 23
Given $$ \sigma = 24 $$:

#### a. For $$ SE = 6 $$:
$$
n = \left(\frac{24}{6}\right)^2 = 16
$$

#### b. For $$ SE = 3 $$:
$$
n = \left(\frac{24}{3}\right)^2 = 64
$$

### Summary of Results
- For Problem 22:
  - a. $$ n = 4 $$
  - b. $$ n = 16 $$
  - c. $$ n = 64 $$

- For Problem 23:
  - a. $$ n = 16 $$
  - b. $$ n = 64 $$
To determine the necessary sample size for a given standard error (SE), use the formula: $$ n = \left(\frac{\sigma}{SE}\right)^2 $$ **Problem 22:** - Given $$ \sigma = 16 $$ - a. For $$ SE = 8 $$: $$ n = 4 $$ - b. For $$ SE = 4 $$: $$ n = 16 $$ - c. For $$ SE = 2 $$: $$ n = 64 $$ **Problem 23:** - Given $$ \sigma = 24 $$ - a. For $$ SE = 6 $$: $$ n = 16 $$ - b. For $$ SE = 3 $$: $$ n = 64 $$ So, the required sample sizes are: - 4, 16, 64 for Problem 22 - 16, 64 for Problem 23

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