### Understanding Sampling Distributions1. **Problem 1: Determining Population Parameters** - _Scenario_: Samples of $ n = 36 $ scores are selected from a population. The distribution of sample means has an expected value $ \mu_M = 30 $ and a standard error $ \sigma_M = 4 $. - _Question_: What are the mean and the standard deviation for the population? **Solution**: - The standard error of the mean is given by $ \sigma_M = \frac{\sigma}{\sqrt{n}} $. - Substituting the values: $ 4 = \frac{\sigma}{\sqrt{36}} $. - Solving for $ \sigma $, we get $ \sigma = 4 \times 6 = 24 $. - Therefore, the population mean $ \mu = \mu_M = 30 $ and the population standard deviation $ \sigma = 24 $.---2. **Problem 2: Expected Value for Sample Mean** - _Scenario_: A sample of $ n = 4 $ scores with $ M = 43 $ is selected from a normal population with $ \mu = 40 $ and $ \sigma = 8 $. - _Question_: What is the expected value for the sample mean? **Solution**: - The expected value of the sample mean $ \mu_M $ is equal to the population mean $ \mu $. - Therefore, the expected value for the sample mean is $ \mu_M = 40 $.---3. **Problem 3: Conditions for Normality of Sample Means Distribution** - _Question_: Under which circumstance will the distribution of sample means be normal? - a. It is always normal - b. Only if the population distribution is normal - c. Only if the sample size is greater than 30 - d. If the population is normal or if the sample size is greater than 30 **Solution**: - The Central Limit Theorem states that the distribution of sample means will be approximately normal if the sample size is sufficiently large (usually $ n > 30 $), or if the population from which samples are drawn is normal. - Therefore, the correct choice is: **d. If the population is normal or if the sample size is greater than

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1. Samples of n= 36 scores are selected from a population. If the distribution of sample means has an expected value of µM= 30 and a standard error of ơM= 4, what is the mean and the standard deviation for the population?

1. Samples of n= 36 scores are selected from a population. If the distribution of sample means has an expected value of µM= 30 and a standard error of ơM= 4, what is the mean and the standard deviation for the population?

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### Understanding Sampling Distributions

1. **Problem 1: Determining Population Parameters**
- _Scenario_: Samples of $ n = 36 $ scores are selected from a population. The distribution of sample means has an expected value $ \mu_M = 30 $ and a standard error $ \sigma_M = 4 $.
- _Question_: What are the mean and the standard deviation for the population?

**Solution**:
- The standard error of the mean is given by $ \sigma_M = \frac{\sigma}{\sqrt{n}} $.
- Substituting the values: $ 4 = \frac{\sigma}{\sqrt{36}} $.
- Solving for $ \sigma $, we get $ \sigma = 4 \times 6 = 24 $.
- Therefore, the population mean $ \mu = \mu_M = 30 $ and the population standard deviation $ \sigma = 24 $.

---

2. **Problem 2: Expected Value for Sample Mean**
- _Scenario_: A sample of $ n = 4 $ scores with $ M = 43 $ is selected from a normal population with $ \mu = 40 $ and $ \sigma = 8 $.
- _Question_: What is the expected value for the sample mean?

**Solution**:
- The expected value of the sample mean $ \mu_M $ is equal to the population mean $ \mu $.
- Therefore, the expected value for the sample mean is $ \mu_M = 40 $.

---

3. **Problem 3: Conditions for Normality of Sample Means Distribution**
- _Question_: Under which circumstance will the distribution of sample means be normal?
- a. It is always normal
- b. Only if the population distribution is normal
- c. Only if the sample size is greater than 30
- d. If the population is normal or if the sample size is greater than 30

**Solution**:
- The Central Limit Theorem states that the distribution of sample means will be approximately normal if the sample size is sufficiently large (usually $ n > 30 $), or if the population from which samples are drawn is normal.
- Therefore, the correct choice is: **d. If the population is normal or if the sample size is greater than

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