Question

Consider a population consisting of the number of teachers per college at small 2-year colleges. Suppose that the number of teachers per college has an average \(\mu=175\) and a standard deviation \(\sigma=15 .\) a. Use Tchebysheff's Theorem to make a statement about the percentage of colleges that have between 145 and 205 teachers. b. Assume that the population is normally distributed. What fraction of colleges have more than 190 teachers?

Short answer

And assuming a normal distribution, what fraction of colleges has more than 190 teachers? Answer: Based on Tchebysheff's theorem, at least 75% of the colleges have between 145 and 205 teachers. Assuming a normal distribution, about 15.87% of the colleges have more than 190 teachers.

Step-by-step solution

Identify the given values and the range

The given average (mean) number of teachers per college is \(\mu=175\) and the standard deviation is \(\sigma=15\). We want to find the percentage of colleges that have between 145 and 205 teachers.

Calculate the number of standard deviations from the mean

To apply Tchebysheff's theorem, we first find how many standard deviations away from the mean the range of 145 to 205 teachers are. Lower boundary: \(k_l = \frac{175 - 145}{15} = 2\) Upper boundary: \(k_u = \frac{205 - 175}{15} = 2\) Both the lower and upper boundaries are 2 standard deviations away from the mean.

Apply Tchebysheff's Theorem

Tchebysheff's Theorem states that, for any non-negative real number \(k\), at least \((1-\frac{1}{k^2})\) of the distribution values are within \(k\) standard deviations from the mean. Since we have found that the range of 145 to 205 teachers is 2 standard deviations away from the mean, we can apply Tchebysheff's theorem: \(1-\frac{1}{k^2} = 1 - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4}\) This means that at least 75% of the distribution values (colleges) have between 145 and 205 teachers. #b. Assume normal distribution#

Calculate the z-score for the threshold of 190 teachers

To find the proportion of colleges with more than 190 teachers, we first calculate the z-score. The z-score is the number of standard deviations away from the mean: \(z = \frac{190 - 175}{15} = 1\)

Use the z-table to find the proportion of colleges with more than 190 teachers

Now, we will look for the area to the right of \(z=1\) in the standard normal table (z-table). From the table, we find that a z-score of 1 corresponds to an area of 0.8413. Since the z-table gives us the area to the left of the z-score, we subtract this area from 1 to get the area to the right of the z-score: \(P(z > 1) = 1 - 0.8413 = 0.1587\) So, about 15.87% of the colleges have more than 190 teachers, assuming the population of teachers per college is normally distributed.

Key concepts

Normal Distribution

The normal distribution, a fundamental concept in statistics, is often referred to as the bell curve due to its distinctive shape. This distribution is symmetrical around the mean, which implies that data is equally distributed on both sides of the average. In real world scenarios, this is an ideal distribution that many natural phenomena closely follow, such as heights, test scores, and, in this case, the number of teachers per college.
A normal distribution is characterized by two important parameters: the mean \(\mu\) (the central location) and the standard deviation \(\sigma\) (the spread). The mean is located at the center of the curve, and it represents the average value. The standard deviation determines how much the values in the dataset deviate from the mean.
Here are some key properties of the normal distribution:

• The mean, median, and mode of a normal distribution are all equal.
• About 68% of the data falls within one standard deviation of the mean.
• Approximately 95% of the data lies within two standard deviations.
• Roughly 99.7% is within three standard deviations, following the empirical rule.

Understanding these properties helps in estimating probabilities and makes it easier to apply statistical methods like z-scores and Tchebysheff's Theorem.

Z-Score Calculation

A z-score is a statistical measurement that describes the position of a raw score in terms of its distance from the mean when measured in standard deviations. Essentially, it tells us how far and in what direction the data point is from the mean.
The formula to calculate a z-score is:\[z = \frac{x - \mu}{\sigma} \\]where \(x\) is the raw score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Z-scores are incredibly useful because they allow comparison across different distributions or datasets. For example, here the z-score is used to determine the proportion of colleges that have more than 190 teachers by finding how many standard deviations 190 is from the average of 175.
After calculating the z-score, one can refer to a z-table to find the probability of a score occurring. In this problem, the z-score of 1 found that 190 teachers is one standard deviation above the mean, and using the z-table, approximately 84.13% of colleges fall below this threshold, leaving about 15.87% above.

Standard Deviation

Standard deviation is a crucial statistic that measures the amount of dispersion or variation in a set of values.
A low standard deviation indicates that the values tend to be close to the mean. Conversely, a high standard deviation means that the values are spread out over a wider range.
In the given exercise, the standard deviation is 15, showing the average deviation of the number of teachers from the mean (175 teachers) across colleges. This means that typically, the number of teachers is within 15 teachers above or below the average for the majority of colleges.
Standard deviation is often used in statistical analysis, particularly when assessing the risks in finance, variability in product quality, or any instance where understanding the variation is critical.
Additionally, understanding standard deviation helps in applying Tchebysheff's Theorem, which predicts the minimum percentage of observations within a certain number of standard deviations from the mean, regardless of the distribution's shape, although it is more informative in non-normal distributions.