Question

College graduates with STEM majors have starting salaries that appear to be much better than those in non-STEM majors. \({ }^{14}\) Starting salaries for 50 randomly selected graduates in electrical engineering and 50 randomly selected graduates in computer science were compiled with the information that follows. $$\begin{array}{lccc}\hline & & \text { Standard } & \text { Sample } \\\\\text { Field } & \text { Mean } & \text { Deviation } & \text { Size } \\\\\hline \text { Electrical engineering } & \$ 62,428 & \$ 12,500 & 50 \\\\\text { Computer science } & \$ 57,762 & \$ 13,330 & 50 \\\\\hline\end{array}$$ a. What is the point estimate of the mean difference between starting salaries for electrical engineers and computer scientists? b. Find a \(95 \%\) lower confidence bound for the mean difference between starting salaries for electrical engineers and computer scientists. Does it appear that electrical engineers have a higher starting salary than computer scientists?

Short answer

Answer: Yes, it appears that electrical engineers have a higher starting salary than computer scientists, as the 95% lower confidence bound is greater than 0.

Step-by-step solution

Calculate the mean difference

To compute the point estimate of the mean difference between starting salaries for electrical engineers and computer scientists, subtract the mean salary of computer science graduates from the mean salary of electrical engineering graduates: Point Estimate = Mean(EE) - Mean(CS) = \(62,428 - \)57,762 = $4,666 #Step 2: Calculate the standard error of the mean difference#

Standard error formula

The standard error of the mean difference can be calculated using the formula: SE = \(\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\) Where \(s_1\) and \(s_2\) are the standard deviations of the two groups, and \(n_1\) and \(n_2\) are their sample sizes. #Step 3: Plug in the values and calculate the standard error#

Compute the standard error

SE = \(\sqrt{\frac{(\$12,500)^2}{50}+\frac{(\$13,330)^2}{50}}\) = \(\sqrt{\frac{156,250,000}{50}+\frac{177,688,900}{50}}\) = \(\sqrt{3,125,000+3,553,778}\) = \(\sqrt{6,678,778}\) ≈ \(2,585\) #Step 4: Calculate the degrees of freedom#

Degrees of freedom calculation

The degrees of freedom (df) for the t-distribution can be approximated using the following formula: df = \(\frac{((\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2})^2)}{\frac{(\frac{s_1^2}{n_1})^2}{n_1-1}+\frac{(\frac{s_2^2}{n_2})^2}{n_2-1}}\) #Step 5: Plug in the values and calculate the degrees of freedom#

Compute the degrees of freedom

df = \(\frac{((\frac{156,250,000}{50}+\frac{177,688,900}{50})^2)}{\frac{(\frac{156,250,000}{50})^2}{49}+\frac{(\frac{177,688,900}{50})^2}{49}}\) ≈ 98 #Step 6: Determine the t-score for a 95% lower confidence bound#

T-score computation

With 98 degrees of freedom and a 95% confidence level, we can look up the t-score in a t-table or use a t-distribution calculator. The t-score is approximately 1.66. #Step 7: Calculate the 95% lower confidence bound#

Confidence bound computation

To calculate the 95% lower confidence bound for the mean difference, we use this formula: Lower Bound = Point Estimate - (t-score * SE) = \(4,666 - (1.66 * \)2,585) ≈ $896 #Step 8: Interpret the results#

Analyze the result

Since the 95% lower confidence bound (\(896) is greater than \)0, it appears that electrical engineers have a higher starting salary than computer scientists.

Key concepts

Confidence Interval

The concept of a confidence interval is pivotal in inferential statistics. It provides a range of values within which we can be 'confident' the true parameter, like a population mean, lies. A confidence interval is not a single number but an interval estimate, coupled with a probability statement.

For example, a 95% confidence interval around a sample mean indicates that if we were to take many, many samples and calculate the confidence interval for each, we would expect about 95% of those intervals to contain the true population mean. It's crucial to note that the true mean is fixed, and it's the intervals that vary from sample to sample.

In the exercise concerning starting salaries, the 95% lower confidence bound suggests that we are 95% confident that the true mean difference between salaries for electrical engineers and computer scientists is greater than the calculated bound. If this bound is notably higher than zero, it would suggest electrical engineers do indeed have a higher starting salary on average compared to computer scientists.

Point Estimate Mean Difference

A point estimate is a single value given as the estimate of a population parameter that is of interest, for example, the difference in average starting salaries between two groups. It is obtained by taking a statistic from sample data as an approximation of the population parameter.

In the context of the exercise, the point estimate is the sample mean difference between the starting salaries of electrical engineers and computer scientists. It is calculated simply by subtracting the sample mean of the computer scientists' salaries from the electrical engineers' salaries. This figure gives a snapshot estimate of the difference but doesn't reflect any measure of precision or uncertainty which is where confidence intervals play their part.

Standard Error

The term standard error (SE) refers to the standard deviation of the sampling distribution of a statistic, commonly the mean. Standard error is a critical concept as it quantifies the variability of the estimate across different hypothetical samples from the same population, giving us an idea of how precise our point estimate is.

To illustrate with our example, the standard error of the mean difference was calculated using given standard deviations and sample sizes of two distinct groups of graduates. It quantifies the expected variation in the mean differences if new samples were taken under the same conditions. A smaller standard error suggests our estimate is closer to the true mean difference for the entire population. Thus, when calculating confidence intervals or conducting hypothesis tests, the standard error is essential in determining the reliability of the estimates.