The Scores Of The Students On A Standardized Test Are Normally Distributed, With A Mean Of 500 And A Standard Deviation Of 110. What Is The Probability That A Randomly Selected Student Has A Score Between 350 And 550? Use The Portion Of The Standard

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Understanding the Problem

The problem involves finding the probability of a randomly selected student having a score between 350 and 550 on a standardized test. The scores are normally distributed with a mean of 500 and a standard deviation of 110. To solve this problem, we will use the standard normal distribution table, also known as the z-table.

Standard Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Any normal distribution can be converted to a standard normal distribution by subtracting the mean and dividing by the standard deviation. This is known as standardization.

Standardizing the Scores

To standardize the scores, we will subtract the mean (500) and divide by the standard deviation (110).

  • For a score of 350: z = (350 - 500) / 110 = -150 / 110 = -1.36
  • For a score of 550: z = (550 - 500) / 110 = 50 / 110 = 0.45

Using the Standard Normal Distribution Table

The standard normal distribution table shows the probability of a z-score being less than or equal to a given value. We will use this table to find the probability of a z-score being less than or equal to -1.36 and 0.45.

Finding the Probability

Using the standard normal distribution table, we find that:

  • P(z ≤ -1.36) = 0.0864
  • P(z ≤ 0.45) = 0.6764

Finding the Probability of a Score Between 350 and 550

To find the probability of a score between 350 and 550, we need to find the probability of a z-score being between -1.36 and 0.45. This can be done by subtracting the probability of a z-score being less than or equal to -1.36 from the probability of a z-score being less than or equal to 0.45.

P(-1.36 < z < 0.45) = P(z ≤ 0.45) - P(z ≤ -1.36) = 0.6764 - 0.0864 = 0.5900

Conclusion

The probability of a randomly selected student having a score between 350 and 550 on a standardized test is 0.5900 or 59.00%.

Discussion

The standard normal distribution table is a powerful tool for finding probabilities in a normal distribution. By standardizing the scores and using the standard normal distribution table, we can find the probability of a score being between any two values. This can be useful in a variety of applications, such as finding the probability of a student scoring above a certain threshold or finding the probability of a group of students scoring within a certain range.

Example

Suppose we want to find the probability of a student scoring above 500 on the test. We can standardize the score by subtracting the mean and dividing by the standard deviation:

z = (500 - 500) / 110 = 0 / 110 = 0

Using the standard normal distribution table, we find that P(z ≤ 0) = 0.5. Therefore, the probability of a student scoring above 500 is 1 - 0.5 = 0.5 or 50%.

Applications

The standard normal distribution table has a variety of applications in statistics and data analysis. Some examples include:

  • Finding the probability of a student scoring above a certain threshold
  • Finding the probability of a group of students scoring within a certain range
  • Finding the mean and standard deviation of a normal distribution
  • Finding the probability of a z-score being less than or equal to a given value

Conclusion

In conclusion, the standard normal distribution table is a powerful tool for finding probabilities in a normal distribution. By standardizing the scores and using the standard normal distribution table, we can find the probability of a score being between any two values. This can be useful in a variety of applications, such as finding the probability of a student scoring above a certain threshold or finding the probability of a group of students scoring within a certain range.

References

  • Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
  • Ross, S. M. (2014). Introduction to probability models. Academic Press.
  • Johnson, R. A., & Wichern, D. W. (2014). Applied multivariate statistical analysis. Pearson Education.

Further Reading

  • For more information on the standard normal distribution table, see the following resources:
  • "Standard Normal Distribution Table" by Math Is Fun
  • "Z-Table" by Stat Trek
  • "Standard Normal Distribution" by Wolfram MathWorld
  • For more information on statistics and data analysis, see the following resources:
  • "Statistics" by Khan Academy
  • "Data Analysis" by Coursera
  • "Statistics and Data Analysis" by edX

Q: What is the standard normal distribution?

A: The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is also known as the z-distribution.

Q: How is the standard normal distribution used?

A: The standard normal distribution is used to find probabilities in a normal distribution. By standardizing the scores and using the standard normal distribution table, we can find the probability of a score being between any two values.

Q: What is the z-score?

A: The z-score is a measure of how many standard deviations an observation is away from the mean. It is calculated by subtracting the mean and dividing by the standard deviation.

Q: How do I calculate the z-score?

A: To calculate the z-score, you need to know the mean and standard deviation of the distribution. The formula for the z-score is:

z = (X - μ) / σ

where X is the value of the observation, μ is the mean, and σ is the standard deviation.

Q: What is the standard normal distribution table?

A: The standard normal distribution table is a table that shows the probability of a z-score being less than or equal to a given value. It is also known as the z-table.

Q: How do I use the standard normal distribution table?

A: To use the standard normal distribution table, you need to know the z-score of the value you are interested in. Then, you can look up the probability in the table.

Q: What is the difference between the standard normal distribution and the normal distribution?

A: The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The normal distribution is a normal distribution with any mean and standard deviation.

Q: Can I use the standard normal distribution table for any normal distribution?

A: Yes, you can use the standard normal distribution table for any normal distribution. By standardizing the scores, you can convert any normal distribution to a standard normal distribution.

Q: What are some common applications of the standard normal distribution?

A: Some common applications of the standard normal distribution include:

  • Finding the probability of a student scoring above a certain threshold
  • Finding the probability of a group of students scoring within a certain range
  • Finding the mean and standard deviation of a normal distribution
  • Finding the probability of a z-score being less than or equal to a given value

Q: What are some common mistakes to avoid when using the standard normal distribution?

A: Some common mistakes to avoid when using the standard normal distribution include:

  • Not standardizing the scores correctly
  • Not using the correct z-table
  • Not understanding the concept of z-scores
  • Not checking the assumptions of the normal distribution

Q: Where can I find more information about the standard normal distribution?

A: You can find more information about the standard normal distribution in the following resources:

  • "Standard Normal Distribution Table" by Math Is Fun
  • "Z-Table" by Stat Trek
  • "Standard Normal Distribution" by Wolfram MathWorld
  • "Statistics" by Khan Academy
  • "Data Analysis" by Coursera
  • "Statistics and Data Analysis" by edX

Q: Can I use the standard normal distribution for non-normal data?

A: No, you cannot use the standard normal distribution for non-normal data. The standard normal distribution is only applicable to normal data.

Q: What are some common non-normal distributions?

A: Some common non-normal distributions include:

  • Binomial distribution
  • Poisson distribution
  • Exponential distribution
  • Gamma distribution

Q: How do I determine if my data is normal or non-normal?

A: You can use statistical tests such as the Shapiro-Wilk test or the Kolmogorov-Smirnov test to determine if your data is normal or non-normal.

Q: What are some common applications of non-normal distributions?

A: Some common applications of non-normal distributions include:

  • Modeling the number of defects in a manufacturing process
  • Modeling the time between events in a Poisson process
  • Modeling the amount of money spent on a project
  • Modeling the number of people in a population

Q: Where can I find more information about non-normal distributions?

A: You can find more information about non-normal distributions in the following resources:

  • "Binomial Distribution" by Math Is Fun
  • "Poisson Distribution" by Stat Trek
  • "Exponential Distribution" by Wolfram MathWorld
  • "Gamma Distribution" by Khan Academy
  • "Statistics" by Coursera
  • "Data Analysis" by edX