Q: Suppose you randomly select a card from a standard deck, replace it, and select another. What is the probability that both cards are red?

$$P(\text{red}) = \frac{26}{52} = 0.5$$, so $$0.5 \times 0.5 = 0.25$$

Q: If a random variable $$X$$ follows a binomial distribution with $$n=53$$ and $$p=0.25$$, what is the standard deviation of $$X$$?

$$\sigma = \sqrt{np(1-p)} = \sqrt{53 \times 0.25 \times 0.75} \approx 3.15$$

Q: Which formula would you use to calculate the probability of at least one event occurring in $$n$$ independent trials, each with probability $$p$$?

$$P(\text{at least one}) = 1 - (1-p)^n$$

Q: A random variable $$X$$ has the following probability distribution: $$P(X=0)=0.659$$, $$P(X=1)=0.286$$, $$P(X=2)=0.050$$, $$P(X=3)=0.004$$, $$P(X=4)=0.001. $$What is $$E[X]$$?

$$E[X] = 0 \times 0.659 + 1 \times 0.286 + 2 \times 0.050 + 3 \times 0.004 + 4 \times 0.001 = 0.401$$

Question 1

Which of the following best describes the variable 'calcium concentration in water' as measured in the UK city study?

Accepted Answer

Quantitative continuous, because it can take any value within a range.

Question 2

Given the histogram of calcium concentrations in UK cities (see $$image_6$$), which of the following best describes the shape of the distribution?

Accepted Answer

Right-skewed and unimodal

Question 3

Suppose you randomly select a card from a standard deck, replace it, and select another. What is the probability that both cards are red?

Accepted Answer

$$P(	ext{red}) = \frac{26}{52} = 0.5$$, so $$0.5 	imes 0.5 = 0.25$$

Question 4

A hospital had 1624 C-section births in 2023. Assuming all days are the same, what is the mean number of C-section births per day?

Accepted Answer

$$\mu = \frac{1624}{365} \approx 4.45$$

Question 5

Which type of graph is most appropriate for displaying the distribution of a categorical variable such as 'recipe type'?

Accepted Answer

Bar graph

Question 6

Given the stemplot in $$image_4$$, what is the median value of the dataset?

Accepted Answer

35

Question 7

If a random variable $$X$$ follows a binomial distribution with $$n=53$$ and $$p=0.25$$, what is the standard deviation of $$X$$?

Accepted Answer

$$\sigma = \sqrt{np(1-p)} = \sqrt{53 	imes 0.25 	imes 0.75} \approx 3.15$$

Question 8

A contingency table summarizes the relationship between two variables. What are the sums of the rows and columns called?

Accepted Answer

Marginal totals

Question 9

Which of the following is NOT a property of the normal distribution?

Accepted Answer

It is always skewed to the right.

Question 10

Suppose the mean head circumference for adults is 57 cm with a standard deviation of 1.83 cm. According to the empirical rule, what percentage of adults have head circumferences between 55.17 cm and 58.83 cm?

Accepted Answer

About 68%

Question 11

Which formula would you use to calculate the probability of at least one event occurring in $$n$$ independent trials, each with probability $$p$$?

Accepted Answer

$$P(	ext{at least one}) = 1 - (1-p)^n$$

Question 12

If you want to find the probability of drawing a 2 from a deck, given that you drew a spade, which probability rule applies?

Accepted Answer

Conditional probability

Question 13

A frequency distribution is most useful for which type of variable?

Accepted Answer

Both categorical and quantitative variables

Question 14

Which of the following is the correct formula for the interquartile range (IQR)?

Accepted Answer

$$IQR = Q_3$ - Q_1$

Question 15

If you have 8 questions and want to select 5 for a test, how many different tests can you make (order matters)?

Accepted Answer

$$P(8,5) = \frac{8!}{(8-5)!} = 6720$$

Question 16

Which summary statistic is most appropriate for describing the center of a right-skewed distribution?

Accepted Answer

Median

Question 17

A random variable $$X$$ has the following probability distribution: $$P(X=0)=0.659$$, $$P(X=1)=0.286$$, $$P(X=2)=0.050$$, $$P(X=3)=0.004$$, $$P(X=4)=0.001. $$What is $$E[X]$$?

Accepted Answer

$$E[X] = 0 	imes 0.659 + 1 	imes 0.286 + 2 	imes 0.050 + 3 	imes 0.004 + 4 	imes 0.001 = 0.401$$

Question 18

Which of the following is a property of probability density functions for continuous random variables?

Accepted Answer

The total area under the curve is 1.

Statistics Test 1 Review: Descriptive Statistics, Probability, and Distributions

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