PCAT (Pharmacy College Admission Test) Quantitative Reasoning Flashcards

24 sample cards from the Quantitative Reasoning deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. State the product, quotient, and power rules for logarithms.

   $\log_b(MN) = \log_b M + \log_b N$; $\log_b\!\left(\frac{M}{N}\right) = \log_b M - \log_b N$; $\log_b(M^{p}) = p\log_b M$.

2. What is the change-of-base formula for logarithms?

   $$\log_b x = \frac{\log_c x}{\log_c b},$$ commonly written as $\frac{\ln x}{\ln b}$ or $\frac{\log x}{\log b}$.

3. What is the formula for continuous exponential growth/decay?

   $A = A_0 e^{kt}$, where $A_0$ is the initial amount, $k$ is the rate constant ($k>0$ growth, $k<0$ decay), and $t$ is time.

4. How is the half-life of an exponentially decaying quantity related to its decay constant $k$?

   $t_{1/2} = \dfrac{\ln 2}{|k|}$, the time for the quantity to drop to half its value (using $A = A_0 e^{-kt}$).

5. How do you compute the mean of a data set?

   $$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i,$$ the sum of all values divided by the number of values $n$.

6. Define the median and the mode of a data set.

   The median is the middle value when data are ordered (the average of the two middle values if $n$ is even). The mode is the value that occurs most frequently.

7. What is the formula for the sample standard deviation?

   $$s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^{2}}.$$ The variance $s^{2}$ is this quantity without the square root.

8. How is the range and the interquartile range (IQR) of a data set defined?

   Range $= \text{max} - \text{min}$. The IQR $= Q_3 - Q_1$, the difference between the third and first quartiles, measuring the spread of the middle 50% of data.

9. For right-skewed and left-skewed distributions, how do the mean and median compare?

   Right-skewed (positive skew): mean > median. Left-skewed (negative skew): mean < median. The mean is pulled toward the longer tail.

10. What is the classical (theoretical) definition of the probability of an event?

    $$P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of equally likely outcomes}},$$ a value between 0 and 1 inclusive.

11. What is the probability of the complement of an event $E$?

    $P(E^{c}) = 1 - P(E)$. The probability that $E$ does not occur.

12. State the addition rule for the probability of $A$ or $B$.

    $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. For mutually exclusive events, $P(A \cap B) = 0$, so $P(A \cup B) = P(A) + P(B)$.

13. State the multiplication rule for independent events, and the general (conditional) form.

    Independent: $P(A \cap B) = P(A)\,P(B)$. General: $P(A \cap B) = P(A)\,P(B \mid A)$, where $P(B \mid A)$ is the conditional probability.

14. What are the formulas for permutations and combinations of $n$ items taken $r$ at a time?

    Permutations (order matters): $P(n,r) = \dfrac{n!}{(n-r)!}$. Combinations (order doesn't matter): $C(n,r) = \dbinom{n}{r} = \dfrac{n!}{r!(n-r)!}$.

15. What are the key properties of a normal distribution?

    It is symmetric and bell-shaped about its mean $\mu$; mean = median = mode; total area under the curve is 1; its spread is set by the standard deviation $\sigma$.

16. State the empirical (68-95-99.7) rule for a normal distribution.

    About 68% of data lie within $\mu \pm 1\sigma$, about 95% within $\mu \pm 2\sigma$, and about 99.7% within $\mu \pm 3\sigma$.

17. What is a z-score and how is it computed?

    A z-score gives how many standard deviations a value is from the mean: $$z = \frac{x - \mu}{\sigma}.$$ It standardizes values to compare across distributions.

18. What does the Central Limit Theorem state about sample means?

    For sufficiently large $n$, the sampling distribution of the sample mean is approximately normal regardless of the population's shape, with mean $\mu$ and standard error $\dfrac{\sigma}{\sqrt{n}}$.

19. What does a p-value represent in hypothesis testing?

    The probability of obtaining results at least as extreme as those observed, assuming the null hypothesis is true. If $p \leq \alpha$ (e.g. 0.05), the null hypothesis is rejected.

20. State the three primary trigonometric ratios in a right triangle (SOH-CAH-TOA).

    $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$.

21. State the Pythagorean trigonometric identity.

    $\sin^{2}\theta + \cos^{2}\theta = 1$. Dividing by $\cos^{2}\theta$ gives $1 + \tan^{2}\theta = \sec^{2}\theta$.

22. What is the limit definition of the derivative of $f(x)$?

    $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.$$ It gives the instantaneous rate of change (slope of the tangent line).

23. State the power rule for differentiation and the basic derivatives of $e^{x}$ and $\ln x$.

    Power rule: $\dfrac{d}{dx}x^{n} = nx^{n-1}$. Also $\dfrac{d}{dx}e^{x} = e^{x}$ and $\dfrac{d}{dx}\ln x = \dfrac{1}{x}$.

24. State the Fundamental Theorem of Calculus (the evaluation form) and the power rule for integration.

    $$\int_a^b f(x)\,dx = F(b) - F(a),\quad \text{where } F'=f.$$ Power rule: $\displaystyle\int x^{n}\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.

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