Q: A contingency table shows the number of Americans in different age groups and whether they are more or less likely to vote. If $$275$$ Americans aged $$35-54$$ are more likely to vote out of $$400$$ total Americans, what is the probability that a randomly selected American is aged $$35-54$$ and more likely to vote?

$$P(35\text{-}54 \text{ and more likely}) = \frac{275}{400}$$

Q: A researcher fits a least squares regression line to predict selling price from square footage. The regression equation is $$\hat{y} = 50 + 0.2x$$, where $$x is $$square footage and $$y is $$price in thousands of dollars. What is the predicted price for a home with $$1400$$ square feet?

$$\hat{y} = 50 + 0.2 \times 1400 = 330$$ thousand dollars

Q: If $$X is a $$binomial random variable with $$n = 10$$ and $$p = 0.7$$, what is the expected value $$E(X)$$?

$$E(X) = n \times p = 10 \times 0.7 = 7$$

Question 1

A researcher wants to determine if there is a linear relationship between the number of hours studied and exam scores. She calculates the correlation coefficient $$r = 0.85$$ for her sample. Which of the following best describes the relationship?

Accepted Answer

There is a strong positive linear relationship between hours studied and exam scores.

Question 2

Suppose the birth weights of full-term babies are normally distributed with mean $$3500$$ grams and standard deviation $$510$$ grams. What is the probability that a randomly selected baby weighs more than $$4230$$ grams?

Accepted Answer

$$P(X \u003e 4230) = 0.0228$$

Question 3

A probability experiment is conducted with $$n = 6$$ independent trials and probability of success $$p = 0.4 on $$each trial. What is the probability of exactly $$2$$ successes?

Accepted Answer

$$P(X = 2) = \binom{6}{2} (0.4)^2$ (0.6)^4$

Question 4

A contingency table shows the number of Americans in different age groups and whether they are more or less likely to vote. If $$275$$ Americans aged $$35-54$$ are more likely to vote out of $$400$$ total Americans, what is the probability that a randomly selected American is aged $$35-54$$ and more likely to vote?

Accepted Answer

$$P(35	ext{-}54 	ext{ and more likely}) = \frac{275}{400}$$

Question 5

A random variable $$X$$ has the following probability distribution: $$P(X = x) = \frac{1}{6}$$ for $$x = 1, 2, 3, 4, 5, 6. $$What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = \frac{1+2+3+4+5+6}{6} = 3.5$$

Question 6

A sample of $$n = 10$$ has a correlation coefficient $$r = 0.65. $$Using the critical values table, is this correlation significant at the $$\alpha = 0.05$$ level?

Accepted Answer

No, because the critical value for $$n = 10 is$$ $$0.632$$ and $$|r| = 0.65 \u003e 0.632.$$

Question 7

A researcher fits a least squares regression line to predict selling price from square footage. The regression equation is $$\hat{y} = 50 + 0.2x$$, where $$x is $$square footage and $$y is $$price in thousands of dollars. What is the predicted price for a home with $$1400$$ square feet?

Accepted Answer

$$\hat{y} = 50 + 0.2 	imes 1400 = 330$$ thousand dollars

Question 8

A probability experiment consists of flipping a fair coin $$5$$ times. What is the probability of getting exactly $$3$$ heads?

Accepted Answer

$$P(X = 3) = \binom{5}{3} (0.5)^3$ (0.5)^2$

Question 9

Suppose the mean number of chocolate chips in a cookie is $$1000$$ with a standard deviation of $$30. $$What is the probability that a randomly selected cookie contains between $$970$$ and $$1030$$ chocolate chips?

Accepted Answer

$$P(970 \u003c X \u003c 1030) = P(-1 \u003c Z \u003c 1) = 0.6826$$

Question 10

A random sample of $$400$$ high school students shows that $$200$$ play organized sports. What is the probability that a randomly selected student from this sample plays organized sports?

Accepted Answer

$$P = \frac{200}{400} = 0.5$$

Question 11

If $$X is a $$binomial random variable with $$n = 10$$ and $$p = 0.7$$, what is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = n 	imes p = 10 	imes 0.7 = 7$$

Question 12

A normal distribution has mean $$\mu = 50$$ and standard deviation $$\sigma = 10. $$What is the $$z-$$score for $$x = 65$$?

Accepted Answer

$$z = \frac{65 - 50}{10} = 1.5$$

Question 13

Which of the following statements about the Empirical Rule is correct for a normal distribution?

Accepted Answer

About $$68\% of $$the data falls within $$1$$ standard deviation of the mean.

Question 14

A random variable $$X is $$defined as the number of heads in $$3$$ coin tosses. What are the possible values of $$X$$?

Accepted Answer

$$X = 0, 1, 2, 3$$

Question 15

A researcher wants to predict the number of calories consumed per day based on the number of snacks eaten. The regression equation is $$\hat{y} = 1200 + 150x$$, where $$x is $$the number of snacks. What is the predicted calorie intake for someone who eats $$3$$ snacks per day?

Accepted Answer

$$\hat{y} = 1200 + 150 	imes 3 = 1650$$ calories

Question 16

If the probability of an event $$A is$$ $$0.3$$ and the probability of event $$B is$$ $$0.5$$, and $$A$$ and $$B$$ are independent, what is the probability that both $$A$$ and $$B$$ occur?

Accepted Answer

$$P(A \cap B) = P(A) 	imes P(B) = 0.3 	imes 0.5 = 0.15$$

Question 17

A sample of $$n = 12$$ has a correlation coefficient $$r = 0.60. $$According to the critical values table, is this correlation significant at the $$\alpha = 0.05$$ level?

Accepted Answer

No, because the critical value for $$n = 12 is$$ $$0.576$$ and $$|r| = 0.60 \u003e 0.576.$$

Question 18

Suppose the mean time spent on social media by teenagers is $$2$$ hours per day with a standard deviation of $$0.5$$ hours. What proportion of teenagers spend between $$1.5$$ and $$2.5$$ hours per day on social media, assuming a normal distribution?

Accepted Answer

$$P(1.5 \u003c X \u003c 2.5) = P(-1 \u003c Z \u003c 1) = 0.6826$$

Comprehensive Study Notes: Core Concepts in College Statistics

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