Q: A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads? Use the binomial formula $$ P(X=k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$ $$ with $$ n=5, k=3, p=0.5 .$$

$$ P(X=3) = \binom{5}{3} (0.5)^3$ (0.5)^2$ = 10 \times 0.125 \times 0.25 = 0.3125 $$

Q: Which formula is used to calculate the theoretical probability of an event A?

$$ P(A) = \frac{\text{Number of ways event A can occur}}{\text{Total number of possible outcomes}} $$

Question 1

Which of the following best describes ordinal data?

Accepted Answer

Categories with a natural order, such as movie ratings (bad, fair, good).

Question 2

A researcher records the number of patients admitted to a hospital each day for a month. What type of data is this?

Accepted Answer

Discrete quantitative data

Question 3

Given the dataset: 12, 15, 18, 21, 24, what is the sample mean? Use the formula $$ x\bar = \frac{\sum x}{n} .$$

Accepted Answer

$$ x\bar = \frac{12+15+18+21+24}{5} = \frac{90}{5} = 18 $$

Question 4

For the dataset: 3, 7, 8, 12, 15, what is the median? Use the location formula $$ \frac{2n+1}{2} .$$

Accepted Answer

Median is the 3rd value, which is 8.

Question 5

A dataset contains the values: 5, 7, 7, 8, 10. What is the mode?

Accepted Answer

7

Question 6

A student measures the temperature in a laboratory every hour for a day. Which type of graph is most appropriate to visualize this continuous quantitative data?

Accepted Answer

Histogram

Question 7

Given the dataset: 2, 4, 6, 8, 10, what is the range?

Accepted Answer

$$ 10 - 2 = 8 $$

Question 8

For the dataset: 1, 2, 4, 5, 5, 6, 7, 10, what is the interquartile range (IQR)?

Accepted Answer

$$ IQR = Q3 - Q1 = 6.5 - 3 = 3.5 $$

Question 9

A sample of 8 values has a mean of 5. The squared deviations from the mean are: 16, 9, 1, 0, 0, 1, 4, 25. What is the sample standard deviation? Use $$ s = \sqrt{\frac{\sum (x - x\$$bar)^2$$}{n-1}} .$$

Accepted Answer

$$ s = \sqrt{\frac{56}{7}} = \sqrt{8} \approx 2.8 $$

Question 10

Which of the following statements about the Law of Large Numbers is correct?

Accepted Answer

As the number of trials increases, experimental probability tends to converge towards theoretical probability.

Question 11

A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads? Use the binomial formula $$ P(X=k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$ $$ with $$ n=5, k=3, p=0.5 .$$

Accepted Answer

$$ P(X=3) = \binom{5}{3} (0.5)^3$ (0.5)^2$ = 10 	imes 0.125 	imes 0.25 = 0.3125 $$

Question 12

Which of the following is NOT a characteristic of a binomial experiment?

Accepted Answer

The probability of success changes from trial to trial.

Question 13

A pie chart is most appropriate for which type of data?

Accepted Answer

Qualitative data showing proportions of categories.

Question 14

Which formula is used to calculate the theoretical probability of an event A?

Accepted Answer

$$ P(A) = \frac{	ext{Number of ways event A can occur}}{	ext{Total number of possible outcomes}} $$

Question 15

A box plot is constructed for a dataset of student heights. Which values are represented in the five-number summary?

Accepted Answer

Minimum, Q1, Median (Q2), Q3, Maximum

Question 16

A teacher records students' favorite school subjects: Math, Science, English, History. Which graph should be used to display the frequency of each subject?

Accepted Answer

Bar chart

Question 17

If a die is rolled twice, what is the probability of rolling a 3 both times? Use $$ P(3 	ext{ and } 3) = P(3) 	imes P(3) .$$

Accepted Answer

$$ P(3 	ext{ and } 3) = \frac{1}{6} 	imes \frac{1}{6} = \frac{1}{36} $$

Question 18

Which of the following best describes a discrete probability distribution?

Accepted Answer

A list of all possible values of a discrete random variable and their associated probabilities.

Question 19

A student rolls a die three times and counts the number of sixes. Which formula gives the probability of rolling exactly k sixes?

Accepted Answer

$$ P(X=k) = \binom{3}{k} \left(\frac{1}{6}\right)^k \left(\frac{5}{6}\$$right)^{3-k}$$ $$

Core Concepts in Statistics: Data Types, Descriptive Statistics, Probability, and Discrete Distributions

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