Q: A medical test for a disease has a sensitivity of 90% and a specificity of 80%. If the disease prevalence is 5%, what is the probability that a randomly selected person tests positive?

$$P(\text{Positive}) = P(\text{Positive} | \text{Disease}) \times P(\text{Disease}) + P(\text{Positive} | \text{No Disease}) \times P(\text{No Disease}) = 0.9 \times 0.05 + 0.2 \times 0.95 = 0.045 + 0.19 = 0.235$$

Q: In a binomial experiment with $$n = 10$$ trials and probability of success $$p = 0.4$$, what is the probability of exactly 4 successes?

$$P(X = 4) = \binom{10}{4} \times 0.4^4$ \times 0.6^6$

Question 1

Which of the following best describes a nominal measurement scale?

Accepted Answer

It categorizes data without any order or ranking.

Question 2

A researcher wants to visualize the relationship between two categorical variables, such as gender and preferred beverage. Which method is most appropriate?

Accepted Answer

A contingency table

Question 3

Given a dataset with a mean of 50 and a standard deviation of 10, what percentage of data falls within two standard deviations of the mean according to the Empirical Rule?

Accepted Answer

Approximately 95%

Question 4

Chebyshev’s theorem states that for any dataset, at least what percentage of values lie within $$k$$ standard deviations of the mean, where $$k \u003e 1$$?

Accepted Answer

$$1 - \frac{1}{k^2$}$$

Question 5

A boxplot is used to visualize which of the following?

Accepted Answer

The distribution, median, quartiles, and potential outliers of a numerical variable

Question 6

If the probability of event A is 0.3 and the probability of event B is 0.5, and A and B are mutually exclusive, what is the probability that either A or B occurs?

Accepted Answer

$$P(A \cup B) = P(A) + P(B) = 0.8$$

Question 7

A medical test for a disease has a sensitivity of 90% and a specificity of 80%. If the disease prevalence is 5%, what is the probability that a randomly selected person tests positive?

Accepted Answer

$$P(	ext{Positive}) = P(	ext{Positive} | 	ext{Disease}) 	imes P(	ext{Disease}) + P(	ext{Positive} | 	ext{No Disease}) 	imes P(	ext{No Disease}) = 0.9 	imes 0.05 + 0.2 	imes 0.95 = 0.045 + 0.19 = 0.235$$

Question 8

In a binomial experiment with $$n = 10$$ trials and probability of success $$p = 0.4$$, what is the probability of exactly 4 successes?

Accepted Answer

$$P(X = 4) = \binom{10}{4} 	imes 0.4^4$ 	imes 0.6^6$

Question 9

A random variable $$X$$ follows a normal distribution with mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the $$Z-$$score for $$X = 130$$?

Accepted Answer

$$Z = \frac{130 - 100}{15} = 2$$

Question 10

A sample of 50 students is randomly selected from a university population. Which sampling method is being used?

Accepted Answer

Simple random sampling

Question 11

A confidence interval for the mean is calculated as $$\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}. $$Which condition must be met for this formula to be valid?

Accepted Answer

The population standard deviation $$\sigma$$ must be known.

Question 12

A pharmaceutical company wants to estimate the proportion of patients who experience side effects from a new drug. Which formula should they use to calculate the required sample size for a desired margin of error $$E$$?

Accepted Answer

$$n = \frac{z_{\alpha/2}^2 p (1-p)}{E^2$}$$

Question 13

Which of the following best describes a Type I error in hypothesis testing?

Accepted Answer

Rejecting the null hypothesis when it is actually true

Question 14

A researcher tests whether the mean cholesterol level in a population is different from 200 mg/dL. The population standard deviation is unknown. Which test should be used?

Accepted Answer

A $$t-$$test for the mean

Question 15

In a matched-pairs study, the confidence interval for the mean difference is calculated as $$\bar{d} \pm t_{\alpha/2} \frac{s_d}{\sqrt{n}}. $$What does $$\bar{d}$$ represent?

Accepted Answer

The average difference between paired observations

Question 16

In multiple linear regression, what does the adjusted $$R^2$$ statistic account for?

Accepted Answer

The number of predictors in the model, penalizing for unnecessary variables

Question 17

A dummy variable in regression analysis is used to:

Accepted Answer

Represent categorical data as numerical values

Question 18

Which formula is used to test the significance of an individual regression coefficient $$\beta_j$$?

Accepted Answer

$$t = \frac{\hat{\beta}_j}{SE(\hat{\beta}_j)}$$

Question 19

A general test of linear restrictions in regression is conducted using which statistic?

Accepted Answer

$$F = \frac{(SSR_{restricted} - SSR_{unrestricted}) / q}{SSR_{unrestricted} / (n - k - 1)}$$

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