0.1.1 - Order of Operations

The acronym PEMDAS, or the mnemonic "please excuse my dear aunt Sally," are sometimes used to help students remember the basic order of operations, where P = parentheses, E = exponents (and square roots), M = multiplication, D = division, A = addition, and S = subtraction. 

When performing a series of mathematical operations, begin inside the parentheses. Next, calculate any exponents or square roots. Then, multiplication and division. And finally, addition and subtraction. For a more in-depth review, we recommend the Khan Academy's Order of Operations lesson.

Order of Operations

In this course, we will be using fractions often. When working with fractions, you can imagine that the operations in the numerator are within parentheses and the operations in the denominator are in parentheses. Below are a few examples of mathematical operations that will be applied in this course. We'll learn about the applications of these operations later in the course, here, we're focusing only on the mathematical operations. 

Example: Confidence Interval for a Mean Section

A confidence interval for a mean can be computed using the equation \(\bar x \pm SE (t^*)\). Let's construct a confidence interval given \(\bar x = 5.770\), \(SE = 0.335\), and \(t^* = 2.080\). 

First, we'll plug in the given values.

\(5.770 \pm 0.355(2.080)\)

There are no operations within parentheses and no exponents or square roots, so our next step will be to multiply. 

\(5.770 \pm 0.697\)

The symbol ± tells use to both subtract and add.

\(5.770 - 0.697 = 5.073\)

\(5.770 + 0.697 = 6.467\)

The confidence interval is (5.073, 6.467).

Example: Pooled Proportion Section

Example: Test Statistic for a Proportion Section

A test statistic for a one sample proportion hypothesis test can be computed using the equation \(z = \frac {\hat p - p_0}{\sqrt{\frac{p_0 (1-p_0)}{n}}}\). Let's work through this formula using the following values: \(\hat p = 0.87\), \(p_0 = 0.8\), and \(n=100\).

First, we'll plug in the given values.

\(z = \frac {0.87 - 0.8}{\sqrt{\frac{0.8 (1- 0.8)}{100}}}\)

In the first few steps we'll work in the numerator and denominator separately. Let's start in the numerator, which contains only subtraction.

\(z = \frac {0.07}{\sqrt{\frac{0.8 (1- 0.8)}{100}}}\)

Next, let's focus on the denominator. The operation within the parentheses should be conducted first.

\(z = \frac {0.07}{\sqrt{\frac{0.8 (0.2)}{100}}}\)

Within the denominator, we can work within the top of the fraction first,

\(z = \frac {0.07}{\sqrt{\frac{0.16}{100}}}\)

Solve for fraction under the square root in the denominator.

\(z = \frac {0.07}{\sqrt{0.0016}}\)

Next, take the square root in the denominator.

\(z = \frac {0.07}{0.04}\)

And finally, divide the numerator by the denominator.

\(z = 1.75\)