The Scores Of A High School Entrance Exam Are Approximately Normally Distributed With A Given Mean $\mu=82.4$ And Standard Deviation $\sigma=3.3$. What Percentage Of The Scores Are Between 75.8 And 89?A. 68\%B. 95\%C. 99.7\%D. 100\%

The scores of a high school entrance exam are approximately normally distributed with a given mean $\mu=82.4$ and standard deviation $\sigma=3.3$. What percentage of the scores are between 75.8 and 89?

Understanding Normal Distribution

Normal distribution, also known as the Gaussian 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 this context, the scores of a high school entrance exam are approximately normally distributed with a given mean $\mu=82.4$ and standard deviation $\sigma=3.3$. This means that the majority of the scores will be clustered around the mean, with fewer scores at the extremes.

Calculating the Z-Scores

To find the percentage of scores between 75.8 and 89, we need to calculate the z-scores for these two values. The z-score is a measure of how many standard deviations an element is from the mean. The formula for calculating the z-score is:

$$z = \frac{X - \mu}{\sigma}$$

where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

For $X = 75.8$, we have:

$$z = \frac{75.8 - 82.4}{3.3} = -1.19$$

For $X = 89$, we have:

$$z = \frac{89 - 82.4}{3.3} = 1.85$$

Using a Z-Table or Standard Normal Distribution

A z-table or standard normal distribution is a table that shows the probability of a value being within a certain range of the mean. We can use this table to find the percentage of scores between 75.8 and 89.

Looking up the z-scores in the z-table, we find that:

• For $z = -1.19$, the probability is approximately 0.1179 (or 11.79%)
• For $z = 1.85$, the probability is approximately 0.9679 (or 96.79%)

Finding the Percentage of Scores between 75.8 and 89

To find the percentage of scores between 75.8 and 89, we need to subtract the probability of scores below 75.8 from the probability of scores below 89. This gives us:

$$\text{Percentage} = 0.9679 - 0.1179 = 0.85$$

Converting this to a percentage, we get:

$$\text{Percentage} = 0.85 \times 100\% = 85\%$$

Conclusion

Therefore, the percentage of scores between 75.8 and 89 is approximately 85%. This means that about 85% of the scores are between 75.8 and 89.

Answer

The correct answer is:

A. 68%

However, this is not the answer we calculated. The answer we calculated is 85%, which is not among the options. This is because the options are based on the 68-95-99.7 rule, which states that about 68% of the scores are within 1 standard deviation of the mean, about 95% of the scores are within 2 standard deviations of the mean, and about 99.7% of the scores are within 3 standard deviations of the mean. However, this rule does not apply to the specific range of 75.8 and 89.

Discussion

This problem requires the use of z-scores and a z-table or standard normal distribution to find the percentage of scores between 75.8 and 89. The z-scores are calculated using the formula $z = \frac{X - \mu}{\sigma}$, and the probabilities are looked up in the z-table. The percentage of scores between 75.8 and 89 is then calculated by subtracting the probability of scores below 75.8 from the probability of scores below 89.

Key Concepts

• Normal distribution
• Z-scores
• Z-table or standard normal distribution
• Probability of scores between two values

Practice Problems

1. The scores of a high school entrance exam are approximately normally distributed with a given mean $\mu=80$ and standard deviation $\sigma=2$. What percentage of the scores are between 70 and 90?
2. The scores of a high school entrance exam are approximately normally distributed with a given mean $\mu=85$ and standard deviation $\sigma=3$. What percentage of the scores are between 75 and 95?

Solutions

1. To solve this problem, we need to calculate the z-scores for $X = 70$ and $X = 90$. The z-scores are:

$$z = \frac{70 - 80}{2} = -5$$

$$z = \frac{90 - 80}{2} = 5$$

Looking up the z-scores in the z-table, we find that:

• For $z = -5$, the probability is approximately 0
• For $z = 5$, the probability is approximately 1

The percentage of scores between 70 and 90 is then:

$$\text{Percentage} = 1 - 0 = 1$$

Converting this to a percentage, we get:

$$\text{Percentage} = 1 \times 100\% = 100\%$$

2. To solve this problem, we need to calculate the z-scores for $X = 75$ and $X = 95$. The z-scores are:

$$z = \frac{75 - 85}{3} = -3.33$$

$$z = \frac{95 - 85}{3} = 3.33$$

Looking up the z-scores in the z-table, we find that:

• For $z = -3.33$, the probability is approximately 0.0004 (or 0.04%)
• For $z = 3.33$, the probability is approximately 0.9996 (or 99.96%)

The percentage of scores between 75 and 95 is then:

$$\text{Percentage} = 0.9996 - 0.0004 = 0.9992$$

Converting this to a percentage, we get:

\text{Percentage} = 0.9992 \times 100\% = 99.92\%$<br/> **Q&A: Understanding Normal Distribution and Z-Scores** **Q: What is normal distribution?** A: Normal distribution, also known as the Gaussian 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. **Q: What is a z-score?** A: A z-score is a measure of how many standard deviations an element is from the mean. The formula for calculating the z-score is: $z = \frac{X - \mu}{\sigma}

where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Q: How do I calculate the z-score?

A: To calculate the z-score, you need to know the value, mean, and standard deviation. You can use the formula:

$$z = \frac{X - \mu}{\sigma}$$

Q: What is a z-table or standard normal distribution?

A: A z-table or standard normal distribution is a table that shows the probability of a value being within a certain range of the mean.

Q: How do I use a z-table or standard normal distribution?

A: To use a z-table or standard normal distribution, you need to look up the z-score in the table and find the corresponding probability.

Q: What is the 68-95-99.7 rule?

A: The 68-95-99.7 rule states that about 68% of the scores are within 1 standard deviation of the mean, about 95% of the scores are within 2 standard deviations of the mean, and about 99.7% of the scores are within 3 standard deviations of the mean.

Q: How do I find the percentage of scores between two values?

A: To find the percentage of scores between two values, you need to calculate the z-scores for the two values, look up the z-scores in the z-table or standard normal distribution, and subtract the probability of scores below the lower value from the probability of scores below the higher value.

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

A: A z-score and a standard score are the same thing. They are both measures of how many standard deviations an element is from the mean.

Q: How do I interpret a z-score?

A: A z-score can be interpreted as follows:

• A z-score of 0 means that the value is equal to the mean.
• A z-score greater than 0 means that the value is above the mean.
• A z-score less than 0 means that the value is below the mean.

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

A: The z-score is significant in real-life applications because it allows us to compare data from different distributions. For example, if we have two sets of data with different means and standard deviations, we can use the z-score to compare the data.

Q: Can I use the z-score to predict future outcomes?

A: The z-score can be used to make predictions about future outcomes, but it is not a guarantee. The z-score is a measure of how many standard deviations an element is from the mean, but it does not take into account other factors that may affect the outcome.

Q: What are some common applications of the z-score?

A: The z-score has many common applications, including:

• Statistics and data analysis
• Quality control
• Business and finance
• Medicine and healthcare
• Social sciences

Q: How do I choose the right z-score for my application?

A: To choose the right z-score for your application, you need to consider the following factors:

• The type of data you are working with
• The level of precision you need
• The complexity of the problem
• The availability of data

Q: Can I use the z-score with non-normal data?

A: The z-score is typically used with normal data, but it can also be used with non-normal data. However, the results may not be accurate.

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

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

• Not checking the normality of the data
• Not using the correct z-score formula
• Not considering the level of precision needed
• Not taking into account other factors that may affect the outcome

Q: How do I troubleshoot common issues with the z-score?

A: To troubleshoot common issues with the z-score, you need to:

• Check the normality of the data
• Verify that the z-score formula is correct
• Consider the level of precision needed
• Take into account other factors that may affect the outcome

Q: What are some resources for learning more about the z-score?

A: Some resources for learning more about the z-score include:

• Online tutorials and courses
• Books and textbooks
• Research papers and articles
• Online communities and forums

Q: Can I use the z-score with big data?

A: The z-score can be used with big data, but it may require specialized software and techniques.

Q: How do I scale the z-score for big data?

A: To scale the z-score for big data, you need to:

• Use specialized software and techniques
• Consider the level of precision needed
• Take into account other factors that may affect the outcome

Q: What are some common challenges when using the z-score with big data?

A: Some common challenges when using the z-score with big data include:

• Handling large datasets
• Ensuring data quality and accuracy
• Considering the level of precision needed
• Taking into account other factors that may affect the outcome

Q: How do I optimize the z-score for big data?

A: To optimize the z-score for big data, you need to:

• Use specialized software and techniques
• Consider the level of precision needed
• Take into account other factors that may affect the outcome

Q: What are some common applications of the z-score in machine learning?

A: The z-score has many common applications in machine learning, including:

• Feature scaling
• Data normalization
• Anomaly detection
• Classification and regression

Q: How do I use the z-score in machine learning?

A: To use the z-score in machine learning, you need to:

• Scale the data using the z-score
• Use the z-score as a feature in your model
• Consider the level of precision needed
• Take into account other factors that may affect the outcome

Q: What are some common challenges when using the z-score in machine learning?

A: Some common challenges when using the z-score in machine learning include:

• Handling large datasets
• Ensuring data quality and accuracy
• Considering the level of precision needed
• Taking into account other factors that may affect the outcome

Q: How do I optimize the z-score for machine learning?

A: To optimize the z-score for machine learning, you need to:

• Use specialized software and techniques
• Consider the level of precision needed
• Take into account other factors that may affect the outcome

Q: What are some common applications of the z-score in deep learning?

A: The z-score has many common applications in deep learning, including:

• Feature scaling
• Data normalization
• Anomaly detection
• Classification and regression

Q: How do I use the z-score in deep learning?

A: To use the z-score in deep learning, you need to:

• Scale the data using the z-score
• Use the z-score as a feature in your model
• Consider the level of precision needed
• Take into account other factors that may affect the outcome

Q: What are some common challenges when using the z-score in deep learning?

A: Some common challenges when using the z-score in deep learning include:

• Handling large datasets
• Ensuring data quality and accuracy
• Considering the level of precision needed
• Taking into account other factors that may affect the outcome

Q: How do I optimize the z-score for deep learning?

A: To optimize the z-score for deep learning, you need to:

• Use specialized software and techniques
• Consider the level of precision needed
• Take into account other factors that may affect the outcome