Question

A group of students at a school takes a history test. The distribution is normal with a mean of \(25,\) and a standard deviation of \(4 .\) (a) Everyone who scores in the top \(30 \%\) of the distribution gets a certificate. What is the lowest score someone can get and still earn a certificate? (b) The top \(5 \%\) of the scores get to compete in a statewide history contest. What is the lowest score someone can get and still go onto compete with the rest of the state?

Short answer

Minimum score for a certificate: 27.1. Minimum score for contest: 31.6.

Step-by-step solution

Understanding the Problem - Part (a)

First, we need to understand that the problem involves finding specific scores on a normal distribution curve. We need to find the score that separates the top 30% from the rest, which means finding the 70th percentile since 100% - 30% = 70%.

Finding the 70th Percentile

We use the Z-score formula for a normal distribution \[ Z = \frac{X - \, \mu}{\sigma} \] where \( X \) is the score, \( \mu \) is the mean (25), and \( \sigma \) is the standard deviation (4).Using a Z-table (or calculator), find the Z-score corresponding to the 70th percentile, which is approximately 0.524.

Calculating the Corresponding Score - Part (a)

Rearrange the Z-score formula to solve for \( X \):\[ X = Z \cdot \sigma + \mu \]Substituting the values we get:\[ X = 0.524 \cdot 4 + 25 \]Calculate to find \( X \approx 27.096. \) Thus, the minimum score required to earn a certificate is approximately 27.1.

Understanding the Problem - Part (b)

Now, we need to find the score that separates the top 5% from the rest. This corresponds to the 95th percentile (100% - 5% = 95%).

Finding the 95th Percentile

Using a Z-table (or calculator), find the Z-score corresponding to the 95th percentile, which is approximately 1.645.

Calculating the Corresponding Score - Part (b)

Rearrange the Z-score formula to solve for \( X \):\[ X = Z \cdot \sigma + \mu \]Substituting the values we get:\[ X = 1.645 \cdot 4 + 25 \]Calculate to find \( X \approx 31.58. \) Thus, the minimum score required to participate in the statewide contest is approximately 31.6.

Key concepts

Normal Distribution

A normal distribution is a way to represent data that clusters around a mean, or average, value. It's often depicted as a bell curve. The mean, median, and mode of a normal distribution are all the same. It is symmetrical around the mean.
Imagine plotting the scores of students on a history test. If these scores are normally distributed, most students will score around the average, and fewer students will have much higher or lower scores.

• The highest point on the curve represents the mean score.
• The bell shape shows the spread of scores, with most being close to the average and fewer as you move away from the mean.

Understanding normal distribution helps in predicting the way scores will spread out. This concept is crucial when working with data that involves probabilities and averages.

Percentile Calculation

Percentiles are a way of ranking data points within a dataset. When you are told a score is at the 70th percentile, it means that score is higher than 70% of the values in the dataset.
In our exercise, we need scores from the top 30% for certificates. This means we look at the 70th percentile because 100% - 30% = 70%.

• To find percentiles, we use a Z-score, which is a measure of how many standard deviations an element is from the mean.
• Percentiles help in understanding average and extreme scores in a dataset.

To find percentiles, statistical tables or software can provide the corresponding Z-scores. Choosing the correct percentile is key to solving related exercises.

Z-score

A Z-score tells you how many standard deviations a data point is from the mean. It's a way to measure the position of a specific score within the distribution. The formula for Z-score is given by: \[ Z = \frac{X - \mu}{\sigma} \]where \( X \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
This score helps in standardizing different data points across a common distribution. In our exercise, the Z-score helps to determine which scores fall within the top percentage for awards and contests.

• A high positive Z-score means the score is notably above average.
• A Z-score of zero indicates the score is exactly at the average.
• A negative Z-score signifies a score below the average.

Understanding Z-scores allows us to calculate where a score lies in relation to the rest of the data.

Standard Deviation

Standard deviation is a statistical measure of the spread or dispersion of a set of data. It tells us how much the data deviates from the mean on average. A small standard deviation indicates that the values tend to be close to the mean, whereas a large standard deviation indicates the values are spread out over a wider range.
In the case of our history test, a standard deviation of 4 means most test scores are within 4 points of the mean of 25.

• The standard deviation is crucial when calculating Z-scores and understanding data spread.
• An understanding of standard deviation helps to gauge the variability in data.

Calculating standard deviation involves determining the variance first and then taking the square root. It’s an essential element in normal distribution and truly shapes how we interpret data variability.