### Testing the Standard Deviation of Heights in Major-League Baseball Players**Introduction**Data show that men between the ages of 20 and 29 in a general population have a mean height of 69.3 inches, with a standard deviation of 2.6 inches. A baseball analyst wonders whether the standard deviation of heights of major-league baseball players is less than 2.6 inches. The heights (in inches) of 20 randomly selected players are shown in the table.**Hypothesis Testing**We will test this notion at the $\alpha = 0.10$ level of significance.**Hypotheses**- **Null Hypothesis ($H_0$)**: $\sigma = 2.6$- **Alternative Hypothesis ($H_1$)**: $\sigma < 2.6$**Steps to Follow**1. **Data Collection**: Click on the icon to view the data table containing the heights of 20 randomly selected major-league baseball players.2. **Calculate the Value of the Test Statistic**: To compare the calculated standard deviation of the sample heights to the population standard deviation.**Graphical Representation**- There are no specific graphs or diagrams included in this example, but typically, you would see a data table with the recorded heights and a graph displaying the distribution of heights if needed.**Conclusion**By conducting this hypothesis test, you can determine whether the variation in heights of major-league baseball players significantly deviates from that of the general population, which can provide insights into the physical characteristics unique to professional athletes.For further clarification or interactive tools, educators and learners can typically use the given data link to explore deeper into the calculations and visual representations. ### Data Table Representation of Numerical ValuesBelow is a grid that displays numerical values. Each cell in the grid contains a specific number:| 72 | 74 | 71 | 71 | 76 ||----|----|----|----|----|| 70 | 77 | 76 | 72 | 72 || 77 | 73 | 75 | 70 | 73 || 74 | 75 | 73 | 74 | 74 |This grid can be utilized in various educational contexts, such as teaching statistical analysis, demonstrating the calculation of measures of central tendency (mean, median, mode), or understanding data distribution.#### Key Points for Analysis:1. **Identification of Patterns**: - Look for any discernible patterns within the rows and columns. - Note any repeating numbers and their frequency.2. **Central Tendency**: - Calculate the mean, median, and mode for each row and column. - Understand how these statistical measures differ across the table.3. **Range and Variability**: - Determine the range of values in each row and column by subtracting the smallest value from the largest value. - Assess the variability within the data set.4. **Practical Applications**: - Use the data to simulate real-world scenarios like temperature readings, scoring data, or other quantifiable metrics.#### Example Calculations:- **Row 1**: Mean = (72 + 74 + 71 + 71 + 76) / 5- **Column 1**: Mean = (72 + 70 + 77 + 74) / 4- **Overall Median**: Arrange all individual values in ascending order and find the middle value(s).Understanding how to interpret and manipulate this type of data is a fundamental skill in various fields including statistics, mathematics, and data science.

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Description: ### Testing the Standard Deviation of Heights in Major-League Baseball Players**Introduction**Data show that men between the ages of 20 and 29 in a general population have a mean height of 69.3 inches, with a standard deviation of 2.6 inches. A base

State the hypotheses. That is, there is no evidence to conclude that the standard deviation of…

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Data show that men between the ages of 20 and 29 in a general population have a mean height of 69.3 inches, with a standard deviation of 2.6 inches. A baseball analyst wonders whether the standard deviation of heights of major-eague baseball players is less than 2.6 inches. The heights (in inches) of 20 randomly selected players are shown in the table.

Data show that men between the ages of 20 and 29 in a general population have a mean height of 69.3 inches, with a standard deviation of 2.6 inches. A baseball analyst wonders whether the standard deviation of heights of major-eague baseball players is less than 2.6 inches. The heights (in inches) of 20 randomly selected players are shown in the table.

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Concept explainers

Inverse Normal Distribution

The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.

Mean, Median, Mode

It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.

Z-Scores

A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.

Topic Video

Question

### Testing the Standard Deviation of Heights in Major-League Baseball Players

**Introduction**

Data show that men between the ages of 20 and 29 in a general population have a mean height of 69.3 inches, with a standard deviation of 2.6 inches. A baseball analyst wonders whether the standard deviation of heights of major-league baseball players is less than 2.6 inches. The heights (in inches) of 20 randomly selected players are shown in the table.

**Hypothesis Testing**

We will test this notion at the $\alpha = 0.10$ level of significance.

**Hypotheses**

- **Null Hypothesis ($H_0$)**: $\sigma = 2.6$
- **Alternative Hypothesis ($H_1$)**: $\sigma < 2.6$

**Steps to Follow**

1. **Data Collection**: Click on the icon to view the data table containing the heights of 20 randomly selected major-league baseball players.

2. **Calculate the Value of the Test Statistic**: To compare the calculated standard deviation of the sample heights to the population standard deviation.

**Graphical Representation**

- There are no specific graphs or diagrams included in this example, but typically, you would see a data table with the recorded heights and a graph displaying the distribution of heights if needed.

**Conclusion**

By conducting this hypothesis test, you can determine whether the variation in heights of major-league baseball players significantly deviates from that of the general population, which can provide insights into the physical characteristics unique to professional athletes.

For further clarification or interactive tools, educators and learners can typically use the given data link to explore deeper into the calculations and visual representations.

### Data Table Representation of Numerical Values

Below is a grid that displays numerical values. Each cell in the grid contains a specific number:

| 72 | 74 | 71 | 71 | 76 |
|----|----|----|----|----|
| 70 | 77 | 76 | 72 | 72 |
| 77 | 73 | 75 | 70 | 73 |
| 74 | 75 | 73 | 74 | 74 |

This grid can be utilized in various educational contexts, such as teaching statistical analysis, demonstrating the calculation of measures of central tendency (mean, median, mode), or understanding data distribution.

#### Key Points for Analysis:
1. **Identification of Patterns**:
- Look for any discernible patterns within the rows and columns.
- Note any repeating numbers and their frequency.

2. **Central Tendency**:
- Calculate the mean, median, and mode for each row and column.
- Understand how these statistical measures differ across the table.

3. **Range and Variability**:
- Determine the range of values in each row and column by subtracting the smallest value from the largest value.
- Assess the variability within the data set.

4. **Practical Applications**:
- Use the data to simulate real-world scenarios like temperature readings, scoring data, or other quantifiable metrics.

#### Example Calculations:

- **Row 1**: Mean = (72 + 74 + 71 + 71 + 76) / 5
- **Column 1**: Mean = (72 + 70 + 77 + 74) / 4
- **Overall Median**: Arrange all individual values in ascending order and find the middle value(s).

Understanding how to interpret and manipulate this type of data is a fundamental skill in various fields including statistics, mathematics, and data science.

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