Step-by-Step Guidance for College Statistics Practice Questions

Q1. Approximately what percentage of people aged 25-40 had a systolic blood pressure reading between 110 and 139 inclusive, based on a relative-frequency histogram?

Background

Topic: Interpreting Histograms and Relative Frequency

This question tests your ability to read and interpret a relative-frequency histogram, specifically to estimate the proportion of data within a given interval.

Key Terms and Concepts:

• Relative Frequency: The proportion or percentage of data values that fall within a specified class interval.

• Histogram: A graphical representation of data using bars to show the frequency of data in intervals.

Step-by-Step Guidance

1. Identify the class intervals on the histogram that include readings from 110 to 139 (inclusive).

2. For each relevant bar, note the relative frequency (height of the bar) corresponding to that interval.

3. Add the relative frequencies for all intervals that fall within 110 to 139.

4. Convert the sum to a percentage if necessary (multiply by 100 if the frequencies are in decimal form).

Try solving on your own before revealing the answer!

Q2. Find the mean study time for the following data: 1.5, 8.3, 6.9, 1.8, 5.3 (in hours). Round your answer to one more decimal place than the data values.

Background

Topic: Measures of Central Tendency (Mean)

This question tests your ability to calculate the arithmetic mean (average) for a set of sample data.

Key Formula:

$\bar{x} = \frac{\sum x_i}{n}$

• $\bar{x}$ = sample mean

• $x_i$ = each individual data value

• $n$ = number of data values

Step-by-Step Guidance

1. Add up all the study times: $1.5 + 8.3 + 6.9 + 1.8 + 5.3$.

2. Count the number of data values ($n$), which is 5 in this case.

3. Divide the total sum by the number of values: $\bar{x} = \frac{\text{sum}}{5}$.

4. Round your answer to one more decimal place than the original data (which is to the nearest tenth, so round to the nearest hundredth).

Try solving on your own before revealing the answer!

Q3. Use the data to create a stemplot for the following weights (in lbs): 144, 152, 142, 151, 160, 152, 131, 164, 141, 153, 140, 144, 175, 156, 147, 133, 172, 159, 135, 159, 148, 171.

Background

Topic: Data Visualization (Stemplots)

This question tests your ability to organize quantitative data into a stemplot (stem-and-leaf plot), which helps visualize the distribution.

Key Terms:

• Stemplot: A graphical method of displaying quantitative data where each data value is split into a "stem" (all but the final digit) and a "leaf" (the final digit).

Step-by-Step Guidance

1. Order the data from smallest to largest.

2. Determine the stems (e.g., for 140-149, the stem is 14).

3. List each leaf (final digit) next to its corresponding stem.

4. Ensure all data values are included and the plot is neatly organized.

Try constructing the stemplot before checking the answer!

Q4. Find the standard deviation for the sample data: 47, 55, 71, 41, 82, 57, 25, 66, 81. Round your answer to one more decimal place than the data.

Background

Topic: Measures of Spread (Standard Deviation)

This question tests your ability to calculate the sample standard deviation, which measures the spread of data around the mean.

Key Formula:

$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

• $s$ = sample standard deviation

• $x_i$ = each data value

• $\bar{x}$ = sample mean

• $n$ = number of data values

Step-by-Step Guidance

1. Calculate the sample mean $\bar{x}$ by summing all values and dividing by $n$.

2. Subtract the mean from each data value to find $x_i - \bar{x}$ for each $x_i$.

3. Square each difference: $(x_i - \bar{x})^2$.

4. Sum all squared differences.

5. Divide the sum by $n-1$ (since this is a sample), then take the square root.

Try calculating the standard deviation before checking the answer!

Q5. A manufacturing process has a 70% yield. If three products are randomly selected, what is the probability that all are acceptable?

Background

Topic: Basic Probability (Multiplication Rule for Independent Events)

This question tests your understanding of independent events and how to calculate the probability that all selected items meet a criterion.

Key Formula:

$P(\text{all acceptable}) = P(\text{acceptable})^n$

• $P(\text{acceptable})$ = probability a single product is acceptable (0.70)

• $n$ = number of products selected (3)

Step-by-Step Guidance

1. Identify the probability that one product is acceptable: $P = 0.70$.

2. Since the selections are independent, multiply the probability for each product: $0.70 \times 0.70 \times 0.70$.

3. Alternatively, use the formula: $P = (0.70)^3$.

Try solving on your own before revealing the answer!

Q6. You are dealt two cards successively (without replacement) from a shuffled deck of 52 cards. What is the probability that the first card is a King and the second card is a Queen? Express as a simplified fraction.

Background

Topic: Probability with Dependent Events (Without Replacement)

This question tests your ability to calculate probabilities for dependent events, where the outcome of the first event affects the second.

Key Formula:

$P(A \text{ and } B) = P(A) \times P(B|A)$

• $P(A)$ = probability first card is a King

• $P(B|A)$ = probability second card is a Queen, given first was a King

Step-by-Step Guidance

1. Find $P($first card is a King$) = \frac{4}{52}$.

2. After removing a King, there are 51 cards left. Find $P($second card is a Queen$|$first was King$) = \frac{4}{51}$.

3. Multiply the two probabilities: $\frac{4}{52} \times \frac{4}{51}$.

4. Simplify the resulting fraction if possible.

Try solving on your own before revealing the answer!

Q7. A sample of 4 calculators is randomly selected from a group of 18 defective and 40 non-defective calculators. What is the probability that at least one is defective? (Round to the nearest thousandth.)

Background

Topic: Probability (At Least One Event)

This question tests your ability to use the complement rule to find the probability that at least one event occurs.

Key Formula:

$P(\text{at least one defective}) = 1 - P(\text{none defective})$

• $P(\text{none defective})$ = probability all selected calculators are non-defective

Step-by-Step Guidance

1. Calculate the total number of calculators: $18 + 40 = 58$.

2. Find the probability that all 4 selected are non-defective (use combinations or multiplication rule without replacement).

3. Subtract this probability from 1 to get the probability that at least one is defective.

4. Round your answer to the nearest thousandth.

Try solving on your own before revealing the answer!

Q8. Is the height of a randomly selected student a discrete or continuous random variable?

Background

Topic: Types of Random Variables

This question tests your understanding of the difference between discrete and continuous random variables.

Key Terms:

• Discrete Random Variable: Takes on countable values (e.g., number of students).

• Continuous Random Variable: Can take on any value within a range (e.g., height, weight).

Step-by-Step Guidance

1. Consider whether height can be measured in infinitely many possible values within a range, or only specific, countable values.

2. Recall the definitions of discrete and continuous variables to classify height appropriately.

Try classifying before checking the answer!

Q9. In a town where 70% of adults have a college degree, and among 4 randomly selected adults, the probability distribution for the number with a degree is given. Find the standard deviation for this probability distribution (round to the nearest hundredth).

Background

Topic: Binomial Probability Distribution (Standard Deviation)

This question tests your ability to calculate the standard deviation for a binomial distribution.

Key Formula:

$\sigma = \sqrt{n p (1-p)}$

• $n$ = number of trials (4)

• $p$ = probability of success (0.70)

Step-by-Step Guidance

1. Identify $n = 4$ and $p = 0.70$.

2. Calculate $1 - p = 0.30$.

3. Plug values into the formula: $\sigma = \sqrt{4 \times 0.70 \times 0.30}$.

4. Compute the value under the square root, then take the square root (but stop before the final calculation).

Try solving on your own before revealing the answer!

Q10. Suppose you buy 1 ticket for $1 out of a lottery of 1,000 tickets where the prize for the one winning ticket is $500. What is your expected value?

Background

Topic: Expected Value (Probability)

This question tests your ability to calculate the expected value of a random variable in a lottery scenario.

Key Formula:

$E(X) = \sum [x_i \cdot P(x_i)]$

• $x_i$ = possible outcomes (net gain/loss)

• $P(x_i)$ = probability of each outcome

Step-by-Step Guidance

1. Determine the possible outcomes: win ($500 - $1 = $499) or lose ($-1).

2. Find the probability of winning: $\frac{1}{1000}$; probability of losing: $\frac{999}{1000}$.

3. Multiply each outcome by its probability: $499 \times \frac{1}{1000}$ and $(-1) \times \frac{999}{1000}$.

4. Add the two products to get the expected value.

Try calculating the expected value before checking the answer!

Q11. Does choosing 5 people (without replacement) from a group of 40 (15 women), keeping track of the number of men chosen, result in a binomial distribution? If not, why?

Background

Topic: Binomial Distributions (Conditions)

This question tests your understanding of the requirements for a binomial distribution, especially the independence of trials.

Key Concepts:

• Binomial distribution requires: fixed number of trials, only two outcomes per trial, constant probability, and independent trials.

Step-by-Step Guidance

1. Check if the number of trials is fixed (yes, 5).

2. Check if each trial has only two outcomes (man or not man).

3. Check if the probability of choosing a man remains constant for each selection (without replacement, so probability changes).

4. Determine if the trials are independent (they are not, due to no replacement).

Try reasoning through the binomial conditions before checking the answer!

Q12. For a binomial distribution with n = 30, x = 12, p = 0.20, use the binomial probability formula to find the probability of 12 successes. Round to three decimal places.

Background

Topic: Binomial Probability Formula

This question tests your ability to use the binomial formula to calculate the probability of a specific number of successes.

Key Formula:

$P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$

• $n$ = number of trials (30)

• $x$ = number of successes (12)

• $p$ = probability of success (0.20)

Step-by-Step Guidance

1. Calculate the binomial coefficient: $\binom{30}{12}$.

2. Compute $p^x = (0.20)^{12}$.

3. Compute $(1-p)^{n-x} = (0.80)^{18}$.

4. Multiply all three components together to get the probability.

Try calculating the probability before checking the answer!

Q13. Find the mean, μ, for the binomial distribution with n = 676 and p = 0.7. Round to the nearest tenth.

Background

Topic: Binomial Distribution (Mean)

This question tests your ability to find the mean (expected value) of a binomial distribution.

Key Formula:

$\mu = n p$

• $n$ = number of trials (676)

• $p$ = probability of success (0.7)

Step-by-Step Guidance

1. Multiply $n$ by $p$: $\mu = 676 \times 0.7$.

2. Round the result to the nearest tenth.

Try calculating the mean before checking the answer!

Q14. At one college, GPAs are normally distributed with a mean of 3 and a standard deviation of 0.6. What percentage of students have a GPA between 2.4 and 3.6?

Background

Topic: Normal Distribution (Empirical Rule)

This question tests your ability to apply the empirical rule (68-95-99.7 rule) to a normal distribution.

Key Concepts:

• Empirical Rule: About 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

Step-by-Step Guidance

1. Calculate 1 standard deviation above and below the mean: $3 \pm 0.6$ gives $2.4$ and $3.6$.

2. Recall the empirical rule: the percentage of data within 1 standard deviation of the mean.

3. Match this interval to the empirical rule's percentages.

Try applying the empirical rule before checking the answer!

Q15. The mean of a set of data is 108.06 and its standard deviation is 115.45. Find the z score for a value of 489.67. Round to the nearest hundredth.

Background

Topic: Z-Scores (Standardization)

This question tests your ability to calculate a z-score, which measures how many standard deviations a value is from the mean.

Key Formula:

$z = \frac{x - \mu}{\sigma}$

• $x$ = value (489.67)

• $\mu$ = mean (108.06)

• $\sigma$ = standard deviation (115.45)

Step-by-Step Guidance

1. Subtract the mean from the value: $489.67 - 108.06$.

2. Divide the result by the standard deviation: $\frac{\text{result}}{115.45}$.

3. Round the z-score to the nearest hundredth.

Try calculating the z-score before checking the answer!

Q16. A class consists of 27 women and 32 men. If a student is randomly selected, what is the probability that the student is a woman?

Background

Topic: Basic Probability (Classical Definition)

This question tests your ability to calculate the probability of a single event from a finite sample space.

Key Formula:

$P(\text{woman}) = \frac{\text{number of women}}{\text{total students}}$

• Number of women = 27

• Total students = 27 + 32

Step-by-Step Guidance

1. Add the number of women and men to get the total number of students.

2. Divide the number of women by the total number of students.

3. Simplify the fraction if possible.

Try calculating the probability before checking the answer!

Q17. Flip a coin twice. What is the sample space of possible outcomes?

Background

Topic: Sample Space (Probability)

This question tests your ability to list all possible outcomes of a simple random experiment.

Key Concepts:

• Each flip has two outcomes: Heads (H) or Tails (T).

• Sample space: all possible ordered pairs of outcomes.

Step-by-Step Guidance

1. List the possible outcomes for the first flip (H or T).

2. For each outcome of the first flip, list the possible outcomes for the second flip.

3. Combine to form all possible pairs (e.g., HH, HT, TH, TT).

Try listing the sample space before checking the answer!

Q18. A sample of 100 wood and 100 graphite tennis rackets is taken. If 7 wood and 14 graphite are defective, what is the probability that a randomly selected racket is wood or defective?

Background

Topic: Probability (Addition Rule, Overlap)

This question tests your ability to use the addition rule for probabilities, accounting for overlap between events.

Key Formula:

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• $P(A)$ = probability racket is wood

• $P(B)$ = probability racket is defective

• $P(A \text{ and } B)$ = probability racket is both wood and defective

Step-by-Step Guidance

1. Calculate $P($wood$) = \frac{100}{200}$.

2. Calculate $P($defective$) = \frac{7 + 14}{200}$.

3. Calculate $P($wood and defective$) = \frac{7}{200}$.

4. Apply the addition rule: $P($wood or defective$) = P($wood$) + P($defective$) - P($wood$\text{ and }$defective$)$.

Try applying the addition rule before checking the answer!

Q19. A card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing a face card or a 4?

Background

Topic: Probability (Addition Rule, Mutually Exclusive Events)

This question tests your ability to calculate the probability of one or another event occurring, considering possible overlap.

Key Concepts:

• Face cards: J, Q, K (3 per suit, 12 total).

• There are four 4s in the deck.

• Face cards and 4s are mutually exclusive (no overlap).

Step-by-Step Guidance

1. Count the number of face cards (12) and number of 4s (4).

2. Add the two counts: $12 + 4$.

3. Divide by the total number of cards (52) to get the probability.

4. Simplify the fraction if possible.

Try calculating the probability before checking the answer!

Q20. The table below describes the smoking habits of a group of asthma sufferers. Find the probability of selecting a man or a heavy smoker.

Background

Topic: Probability (Addition Rule, Overlapping Events)

This question tests your ability to use the addition rule for probabilities, especially when events may overlap.

Key Formula:

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• $P(A)$ = probability of selecting a man

• $P(B)$ = probability of selecting a heavy smoker

• $P(A \text{ and } B)$ = probability of selecting a man who is also a heavy smoker

Step-by-Step Guidance

1. From the table, find the total number of men, total number of heavy smokers, and number of men who are heavy smokers.

2. Calculate each probability by dividing the relevant count by the total number of people.

3. Apply the addition rule: $P($man or heavy smoker$) = P($man$) + P($heavy smoker$) - P($man$\text{ and }$heavy smoker$)$.

Try applying the addition rule before checking the answer!