The Accompanying Table Describes Results From Groups Of 8 Births From 8 Different Sets Of Parents. The Random Variable $X$ Represents The Number Of Girls Among 8 Children. Complete Parts (a) Through (d) Below.Click The Icon To View The

by ADMIN 236 views

Understanding the Problem

The accompanying table describes results from groups of 8 births from 8 different sets of parents. The random variable XX represents the number of girls among 8 children. In this problem, we are tasked with completing parts (a) through (d) below.

Table: Results from Groups of 8 Births

Set of Parents Number of Girls
1 4
2 3
3 5
4 2
5 6
6 1
7 7
8 0

Part (a) - Find the Mean of the Random Variable X

The mean of the random variable XX is the average number of girls among the 8 children. To find the mean, we need to calculate the sum of the number of girls in each set of parents and divide it by the total number of sets.

Let's calculate the sum of the number of girls:

4 + 3 + 5 + 2 + 6 + 1 + 7 + 0 = 28

There are 8 sets of parents, so we divide the sum by 8:

28 ÷ 8 = 3.5

Therefore, the mean of the random variable XX is 3.5.

Part (b) - Find the Median of the Random Variable X

The median of the random variable XX is the middle value of the number of girls among the 8 children when they are arranged in order from smallest to largest.

Let's arrange the number of girls in order:

0, 1, 2, 3, 4, 5, 6, 7

Since there are an even number of values, the median is the average of the two middle values. The two middle values are 3 and 4, so the median is:

(3 + 4) ÷ 2 = 3.5

Therefore, the median of the random variable XX is 3.5.

Part (c) - Find the Mode of the Random Variable X

The mode of the random variable XX is the value that appears most frequently among the number of girls.

Let's count the frequency of each value:

  • 0: 1 time
  • 1: 1 time
  • 2: 1 time
  • 3: 1 time
  • 4: 1 time
  • 5: 1 time
  • 6: 1 time
  • 7: 1 time

There is no value that appears more than once, so the mode does not exist.

Part (d) - Find the Standard Deviation of the Random Variable X

The standard deviation of the random variable XX is a measure of the amount of variation or dispersion of the number of girls among the 8 children.

To find the standard deviation, we need to calculate the variance first. The variance is the average of the squared differences from the mean.

Let's calculate the squared differences from the mean:

(4 - 3.5)^2 = 0.25 (3 - 3.5)^2 = 0.25 (5 - 3.5)^2 = 2.25 (2 - 3.5)^2 = 2.25 (6 - 3.5)^2 = 6.25 (1 - 3.5)^2 = 12.25 (7 - 3.5)^2 = 14.25 (0 - 3.5)^2 = 12.25

The sum of the squared differences is:

0.25 + 0.25 + 2.25 + 2.25 + 6.25 + 12.25 + 14.25 + 12.25 = 50

There are 8 sets of parents, so we divide the sum by 8:

50 ÷ 8 = 6.25

The variance is 6.25. The standard deviation is the square root of the variance:

√6.25 = 2.5

Therefore, the standard deviation of the random variable XX is 2.5.

Conclusion

Q: What is the purpose of the accompanying table?

A: The accompanying table describes results from groups of 8 births from 8 different sets of parents. The table is used to analyze the number of girls among the 8 children.

Q: What is the random variable X?

A: The random variable X represents the number of girls among 8 children.

Q: What is the mean of the random variable X?

A: The mean of the random variable X is 3.5, which is the average number of girls among the 8 children.

Q: What is the median of the random variable X?

A: The median of the random variable X is 3.5, which is the middle value of the number of girls among the 8 children when they are arranged in order from smallest to largest.

Q: Does the mode of the random variable X exist?

A: No, the mode of the random variable X does not exist because there is no value that appears more than once.

Q: What is the standard deviation of the random variable X?

A: The standard deviation of the random variable X is 2.5, which is a measure of the amount of variation or dispersion of the number of girls among the 8 children.

Q: How is the standard deviation calculated?

A: The standard deviation is calculated by first finding the variance, which is the average of the squared differences from the mean. The variance is then used to find the standard deviation, which is the square root of the variance.

Q: What is the significance of the standard deviation?

A: The standard deviation is a measure of the amount of variation or dispersion of the number of girls among the 8 children. A small standard deviation indicates that the number of girls is consistent across the different sets of parents, while a large standard deviation indicates that the number of girls varies significantly.

Q: Can the results of the accompanying table be generalized to a larger population?

A: The results of the accompanying table can be generalized to a larger population if the sample is representative of the population. However, if the sample is not representative, the results may not be generalizable to the larger population.

Q: What are some potential limitations of the accompanying table?

A: Some potential limitations of the accompanying table include:

  • The sample size is small (8 sets of parents).
  • The sample may not be representative of the larger population.
  • The results may be influenced by random chance.

Q: How can the results of the accompanying table be used in real-world applications?

A: The results of the accompanying table can be used in real-world applications such as:

  • Predicting the number of girls in a future birth.
  • Understanding the factors that influence the number of girls in a birth.
  • Making informed decisions about family planning.

Conclusion

In this article, we have answered some frequently asked questions about the accompanying table that describes results from groups of 8 births. We have discussed the purpose of the table, the random variable X, the mean, median, mode, and standard deviation of the random variable X, and some potential limitations of the table. We have also discussed how the results of the table can be used in real-world applications.