Question

Although faculty salaries at colleges and universities in the United States continue to rise, they do not always keep pace with the cost of living nor with salaries in the private sector. In 2005 , the National Center for Educational Statistics indicated that the average salary for Assistant Professors at public four-year colleges was \(\$ 50,581 .^{11}\) Suppose that these salaries are normally distributed with a standard deviation of \(\$ 4000 .\) a. What proportion of assistant professors at public 4-year colleges will have salaries less than \(\$ 45,000 ?\) b. What proportion of these professors will have salaries between \(\$ 45,000\) and \(\$ 55,000 ?\)

Short answer

Answer: 8.16% of assistant professors have salaries less than $45,000, and 78.35% of assistant professors have salaries between $45,000 and $55,000.

Step-by-step solution

Calculate the z-scores for the salary ranges

To calculate the z-score for a given salary, we can use the following formula: \(z = \frac{x-\mu}{\sigma}\) where \(x\) is the salary value, \(\mu\) is the mean salary, and \(\sigma\) is the standard deviation. For the first salary range (less than \(\$ 45,000\)), we have: \(z_1 = \frac{45,000 - 50,581}{4,000} = -1.39525\) For the second salary range (between \(\$ 45,000\) and \(\$ 55,000\)), we need to calculate two z-scores: \(z_2 = \frac{45,000 - 50,581}{4,000} = -1.39525\) \(z_3 = \frac{55,000 - 50,581}{4,000}=1.10475\)

Find the proportions using the z-table

Now that we have the z-scores, we can use the z-table to find the proportions. a. To find the proportion of assistant professors with salaries less than \(\$ 45,000\), we will look up the z-score \(z_1 = -1.39525\) in a z-table. From the z-table, we get \(P(z < -1.39525) = 0.0816\). So, 8.16% of assistant professors have salaries less than \(\$ 45,000\). b. To find the proportion of assistant professors with salaries between \(\$ 45,000\) and \(\$ 55,000\), we will look up the z-scores \(z_2 = -1.39525\) and \(z_3 = 1.10475\) in a z-table and calculate the difference between them. First, we find the proportion of assistant professors with salaries less than \(\$ 55,000\) which corresponds to \(P(z < 1.10475) = 0.8651\). Then we subtract the proportion of professors with salaries less than \(\$ 45,000\) obtained earlier. Thus, \(P(-1.39525 < z < 1.10475)=0.8651-0.0816=0.7835\) So, 78.35% of assistant professors have salaries between \(\$ 45,000\) and \(\$ 55,000\).

Key concepts

z-score calculation

The z-score is a powerful statistical tool that lets us understand how far a data point is from the mean of a data set, expressed in terms of standard deviation. In simpler terms, it helps us know how unusual or typical a particular data point is compared to the average. It's defined by the formula:

• \( z = \frac{x - \mu}{\sigma} \)

Here, \( x \) is the individual data point or salary in this context, \( \mu \) is the mean salary, and \( \sigma \) is the standard deviation.

By calculating a z-score, we convert that specific salary value into "standardized" form. We then use a z-table to determine the probability or proportion associated with that z-score. In our exercise, the z-score calculation helps show the proportion of assistant professors earning below a certain salary level.

standard deviation

The standard deviation is a key concept in statistics that measures the extent of variation or dispersion of a set of data values. A smaller standard deviation signifies that the data points tend to be closer to the mean, while a larger standard deviation implies more spread out data.

In our problem, the standard deviation is \( \\(4000 \), which means that most of the salaries are within this range of the average salary of \( \\)50,581 \). The standard deviation is used in z-score calculation and helps us understand how dispersed the salaries are around the average. It's important to see if the data distribution is tight or scattered, which in turn affects our assessment of salaries in the universities.

• Helpful for risk assessment
• Indicates data spread
• Key for calculating z-scores

proportion of a normal distribution

When we talk about the proportion of a normal distribution, we're discussing how much of our data set falls within a certain range of values. Normal distribution, also known as the bell curve, is a probability distribution that's symmetric around the mean, showing that data near the mean are more frequent in occurrence.

In the given example, once we calculate the z-scores, we use them to find proportions in a z-table. This table provides probabilities or proportions for each z-score. For instance, we find that 8.16% of professors earn less than \( \$45,000 \), which means they are in the lower "tail" of the distribution.

• Visualize data probabilities easily
• Understand where most data points lie
• Helps in statistical analyses regarding probability

The concept of proportion of a normal distribution allows us to make sense of real-world data in terms of likelihood and frequency, particularly in assessing salaries within a given context.