On A Standardized Exam, The Scores Are Normally Distributed With A Mean Of Μ = 208 \mu = 208 Μ = 208 And A Standard Deviation Of 7.59. The Z Z Z -score Is 0.88.Find The Raw Score For The Z Z Z -score. Round To The Nearest Whole Number.Formula:

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In statistics, a normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In a normal distribution, the mean, median, and mode are all equal. The standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

The Formula for Finding Raw Scores

The formula for finding the raw score from a given z-score is:

x = μ + (z * σ)

Where:

  • x is the raw score
  • μ is the mean of the distribution
  • z is the z-score
  • σ is the standard deviation of the distribution

Given Information

In this problem, we are given the following information:

  • The mean (μ) is 208
  • The standard deviation (σ) is 7.59
  • The z-score is 0.88

Finding the Raw Score

Using the formula for finding the raw score, we can plug in the given values:

x = 208 + (0.88 * 7.59)

First, we multiply the z-score by the standard deviation:

0.88 * 7.59 = 6.67

Now, we add the result to the mean:

x = 208 + 6.67 x = 214.67

Rounding to the Nearest Whole Number

Since we are asked to round the raw score to the nearest whole number, we round 214.67 to 215.

Conclusion

In this problem, we used the formula for finding the raw score from a given z-score to find the raw score for a z-score of 0.88. We plugged in the given values, performed the calculation, and rounded the result to the nearest whole number. The final answer is 215.

Example Use Case

This formula can be used in a variety of situations, such as:

  • Finding the raw score for a student who scored a certain number of standard deviations above or below the mean on a standardized test
  • Determining the raw score for a patient who scored a certain number of standard deviations above or below the mean on a medical test
  • Calculating the raw score for a product that scored a certain number of standard deviations above or below the mean in a quality control test

Common Mistakes to Avoid

When using this formula, it's essential to:

  • Make sure to use the correct values for the mean, standard deviation, and z-score
  • Perform the calculation carefully to avoid errors
  • Round the result to the nearest whole number as required

Real-World Applications

This formula has numerous real-world applications, including:

  • Standardized testing: This formula can be used to find the raw score for a student who scored a certain number of standard deviations above or below the mean on a standardized test.
  • Medical testing: This formula can be used to determine the raw score for a patient who scored a certain number of standard deviations above or below the mean on a medical test.
  • Quality control: This formula can be used to calculate the raw score for a product that scored a certain number of standard deviations above or below the mean in a quality control test.

Limitations of the Formula

While this formula is useful for finding the raw score from a given z-score, it has some limitations. For example:

  • The formula assumes that the data is normally distributed, which may not always be the case.
  • The formula requires the mean and standard deviation of the distribution, which may not always be known.
  • The formula is sensitive to errors in the input values, which can lead to incorrect results.
    Frequently Asked Questions (FAQs) =====================================

Q: What is a z-score, and why is it important?

A: A z-score is a measure of how many standard deviations an element is from the mean. It's a way to compare the performance of different elements in a dataset. The z-score is important because it allows us to understand how an element's performance compares to the rest of the dataset.

Q: How do I calculate the z-score?

A: To calculate the z-score, you need to know the mean (μ) and standard deviation (σ) of the dataset. The formula for calculating the z-score is:

z = (x - μ) / σ

Where:

  • x is the value you want to calculate the z-score for
  • μ is the mean of the dataset
  • σ is the standard deviation of the dataset

Q: What is the difference between a z-score and a raw score?

A: A z-score is a measure of how many standard deviations an element is from the mean, while a raw score is the actual value of the element. For example, if a student scores 85 on a test, their raw score is 85. If the mean of the test is 70 and the standard deviation is 10, the student's z-score would be (85 - 70) / 10 = 1.5.

Q: Can I use the z-score formula to find the raw score?

A: Yes, you can use the z-score formula to find the raw score. The formula is:

x = μ + (z * σ)

Where:

  • x is the raw score
  • μ is the mean of the dataset
  • z is the z-score
  • σ is the standard deviation of the dataset

Q: What is the significance of the z-score in real-world applications?

A: The z-score is significant in real-world applications because it allows us to compare the performance of different elements in a dataset. For example, in standardized testing, the z-score can be used to compare the performance of students from different schools or regions. In medical testing, the z-score can be used to compare the performance of patients with different medical conditions.

Q: Can I use the z-score formula to find the z-score for a non-normal distribution?

A: No, the z-score formula is only applicable to normal distributions. If the distribution is not normal, you may need to use a different formula or method to calculate the z-score.

Q: What are some common mistakes to avoid when using the z-score formula?

A: Some common mistakes to avoid when using the z-score formula include:

  • Using the wrong values for the mean and standard deviation
  • Performing the calculation incorrectly
  • Not rounding the result to the nearest whole number
  • Not considering the limitations of the formula (e.g. assuming a normal distribution)

Q: Can I use the z-score formula to find the z-score for a large dataset?

A: Yes, you can use the z-score formula to find the z-score for a large dataset. However, you may need to use a computer program or calculator to perform the calculation, as the formula can be complex and time-consuming to calculate by hand.

Q: What are some real-world applications of the z-score formula?

A: Some real-world applications of the z-score formula include:

  • Standardized testing: The z-score formula can be used to compare the performance of students from different schools or regions.
  • Medical testing: The z-score formula can be used to compare the performance of patients with different medical conditions.
  • Quality control: The z-score formula can be used to compare the performance of products or processes.
  • Finance: The z-score formula can be used to compare the performance of stocks or other financial instruments.