I only need D answered ### Central Limit Theorem ApplicationA population of values presents a distribution with a mean (μ) of 169.7 and a standard deviation (σ) of 15.3. You intend to draw a random sample of size n = 127.According to the Central Limit Theorem:**(a) What is the mean of the distribution of sample means?** \[ \mu_{\bar{X}} = \_\_\_\_ \]**(b) What is the standard deviation of the distribution of sample means?** (Report answer accurate to 2 decimal places.) \[ \sigma_{\bar{X}} = \_\_\_\_ \]**(c) In a random sample of n = 127, what is the probability that its sample mean is more than 170.4?** (Round to three decimal places.) \[ \Pr(\bar{X} > 170.4) = \_\_\_\_ \]**(d) In a random sample of n = 127, what is the probability that its sample mean is less than 170.5?** (Give your answer to three decimal places.) \[ \Pr(\bar{X} < 170.5) = \_\_\_\_ \]\[ \rightarrow \text{Next Question} \]### Explanation of Concepts**Central Limit Theorem (CLT):**The Central Limit Theorem states that the distribution of the sample mean will tend to be normal or nearly normal if the sample size is large enough, typically n > 30, regardless of the shape of the population distribution. **Mean of Sample Distribution ( $\mu_{\bar{X}} $ ):**The mean of the distribution of sample means is equal to the population mean ( $\mu$ ).**Standard Deviation of Sample Distribution ( $\sigma_{\bar{X}} $ ):**The standard deviation of the distribution of sample means (also known as the standard error) is calculated using the formula:\[ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \]where $\sigma$ is the population standard deviation and n is the sample size.**Probability Calculations:**To calculate the probability that a sample mean is greater than or less than a specified value, you typically use the z-score formula:\[ z = \frac{X - \mu_{\bar

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A population of values has a distribution with a 169.7 and a = 15.3. You intend to draw a random sample of size n= 127. According to the Central Limit Theorem: (a) What is the mean of the distribution of sample means? %3D (b) What is the standard deviation of the distribution of sample means? (Report answer accurate to 2 decimal places.) (c) in a random sample of n127, what is the probability that its sample mean is more than 170.47 Round to three decimal places. (d) In a random sample of ne127, what is the probability that its sample mean is less than 170.5? Give your answer to three decimal places. Next Questio

A population of values has a distribution with a 169.7 and a = 15.3. You intend to draw a random sample of size n= 127. According to the Central Limit Theorem: (a) What is the mean of the distribution of sample means? %3D (b) What is the standard deviation of the distribution of sample means? (Report answer accurate to 2 decimal places.) (c) in a random sample of n127, what is the probability that its sample mean is more than 170.47 Round to three decimal places. (d) In a random sample of ne127, what is the probability that its sample mean is less than 170.5? Give your answer to three decimal places. Next Questio

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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

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I only need D answered

### Central Limit Theorem Application

A population of values presents a distribution with a mean (μ) of 169.7 and a standard deviation (σ) of 15.3. You intend to draw a random sample of size n = 127.

According to the Central Limit Theorem:

**(a) What is the mean of the distribution of sample means?**
\[ \mu_{\bar{X}} = \_\_\_\_ \]

**(b) What is the standard deviation of the distribution of sample means?**
(Report answer accurate to 2 decimal places.)
\[ \sigma_{\bar{X}} = \_\_\_\_ \]

**(c) In a random sample of n = 127, what is the probability that its sample mean is more than 170.4?**
(Round to three decimal places.)
\[ \Pr(\bar{X} > 170.4) = \_\_\_\_ \]

**(d) In a random sample of n = 127, what is the probability that its sample mean is less than 170.5?**
(Give your answer to three decimal places.)
\[ \Pr(\bar{X} < 170.5) = \_\_\_\_ \]

\[ \rightarrow \text{Next Question} \]

### Explanation of Concepts

**Central Limit Theorem (CLT):**
The Central Limit Theorem states that the distribution of the sample mean will tend to be normal or nearly normal if the sample size is large enough, typically n > 30, regardless of the shape of the population distribution.

**Mean of Sample Distribution ( $\mu_{\bar{X}} $ ):**
The mean of the distribution of sample means is equal to the population mean ( $\mu$ ).

**Standard Deviation of Sample Distribution ( $\sigma_{\bar{X}} $ ):**
The standard deviation of the distribution of sample means (also known as the standard error) is calculated using the formula:
\[ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \]
where $\sigma$ is the population standard deviation and n is the sample size.

**Probability Calculations:**
To calculate the probability that a sample mean is greater than or less than a specified value, you typically use the z-score formula:
\[ z = \frac{X - \mu_{\bar

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