5.1 MCQs-Statistics

Statistics MCQs

Descriptive Statistics

1. Which of the following is NOT a measure of central tendency?

1. Mean

2. Median

3. Standard deviation

4. Mode

Show me the answer

Answer: 3. Standard deviation

Explanation:

• Measures of central tendency describe the center or typical value of a dataset. The mean, median, and mode are all measures of central tendency.

• Standard deviation is a measure of dispersion (spread or variability) of the data around the mean.

• Other measures of dispersion include variance and range.

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2. The mean of the dataset $\{2, 4, 6, 8, 10\}$ is:

1. 5

2. 6

3. 7

4. 8

Show me the answer

Answer: 2. 6

Explanation:

• The mean (arithmetic average) is calculated as the sum of all values divided by the number of values.

• $\text{Mean} = \frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6$

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3. For a skewed distribution, which measure of central tendency is most appropriate?

1. Mean

2. Median

3. Mode

4. All are equally appropriate

Show me the answer

Answer: 2. Median

Explanation:

• The mean is sensitive to extreme values (outliers). In a skewed distribution, the mean gets pulled toward the tail.

• The median is the middle value when data is ordered and is not affected by extreme values, making it a better measure of the "center" for skewed data.

• The mode is the most frequent value but may not represent the center well.

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4. The range of the dataset $\{12, 5, 8, 19, 3\}$ is:

1. 8

2. 12

3. 16

4. 47

Show me the answer

Answer: 3. 16

Explanation:

• The range is the difference between the maximum and minimum values in the dataset.

• Maximum value = 19, Minimum value = 3.

• Range = 19 - 3 = 16.

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5. If the variance of a dataset is 9, what is its standard deviation?

1. 3

2. 9

3. 81

4. Cannot be determined

Show me the answer

Answer: 1. 3

Explanation:

• Standard deviation ($\sigma$ or s) is the square root of the variance ($\sigma^2$ or $s^2$).

• $\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{9} = 3$

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Probability

6. The probability of an event A, denoted $P(A)$, always satisfies:

1. $0 \le P(A) \le 1$

2. $-1 \le P(A) \le 1$

3. $P(A) > 0$

4. $P(A) < 1$

Show me the answer

Answer: 1. $0 \le P(A) \le 1$

Explanation:

• This is a fundamental axiom of probability. Probability values range from 0 to 1 inclusive.

• $P(A) = 0$ means the event is impossible.

• $P(A) = 1$ means the event is certain.

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7. If two events A and B are mutually exclusive, then $P(A \cap B)$ equals:

1. $P(A) \times P(B)$

2. $P(A) + P(B)$

3. 0

4. 1

Show me the answer

Answer: 3. 0

Explanation:

• Mutually exclusive (disjoint) events cannot occur at the same time. They have no outcomes in common.

• Therefore, the probability of their intersection is zero: $P(A \cap B) = 0$.

• For mutually exclusive events, the addition rule simplifies to $P(A \cup B) = P(A) + P(B)$.

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8. If two events A and B are independent, then $P(A \cap B)$ equals:

1. $P(A) \times P(B)$

2. $P(A) + P(B)$

3. 0

4. $P(A \mid B)$

Show me the answer

Answer: 1. $P(A) \times P(B)$

Explanation:

• This is the multiplication rule for independent events. The occurrence of one event does not affect the probability of the other.

• For independent events, $P(A \mid B) = P(A)$ and $P(B \mid A) = P(B)$.

• Therefore, $P(A \cap B) = P(A) \cdot P(B)$.

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9. The conditional probability of A given B is defined as:

1. $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$, provided $P(B) > 0$

2. $P(A \mid B) = \frac{P(B \mid A)}{P(A)}$

3. $P(A \mid B) = P(A)$

4. $P(A \mid B) = P(A) + P(B)$

Show me the answer

Answer: 1. $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$, provided $P(B) > 0$

Explanation:

• Conditional probability is the probability of event A occurring given that event B has already occurred.

• The formula is derived from the concept of restricting the sample space to the event B.

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10. According to Bayes' Theorem, $P(A \mid B)$ equals:

1. $\frac{P(B \mid A) P(A)}{P(B)}$

2. $\frac{P(A \cap B)}{P(B)}$

3. $P(B \mid A)$

4. $P(A) + P(B)$

Show me the answer

Answer: 1. $\frac{P(B \mid A) P(A)}{P(B)}$

Explanation:

• Bayes' Theorem relates a conditional probability to its inverse. It is a direct application of the definition of conditional probability.

• $P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)}$.

• It is used to update the probability of a hypothesis (A) based on new evidence (B).

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Distributions

11. The normal distribution is characterized by:

1. Its mean ($\mu$) and variance ($\sigma^2$)

2. Its median and mode

3. Its range

4. Its skewness

Show me the answer

Answer: 1. Its mean ($\mu$) and variance ($\sigma^2$)

Explanation:

• A normal distribution is a continuous probability distribution that is completely defined by two parameters: its mean ($\mu$), which determines the center, and its variance ($\sigma^2$), which determines the spread.

• It is symmetric, so mean = median = mode.

• It has the familiar bell-shaped curve.

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12. In a binomial experiment with n trials and probability of success p, the mean (expected value) is:

1. $n$

2. $p$

3. $np$

4. $\sqrt{np(1-p)}$

Show me the answer

Answer: 3. $np$

Explanation:

• For a binomial random variable $X \sim Binomial(n, p)$:

  • Mean (Expected Value): $E(X) = np$

  • Variance: $Var(X) = np(1-p)$

  • Standard Deviation: $\sqrt{np(1-p)}$

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13. The standard normal distribution has:

1. Mean = 0, Variance = 1

2. Mean = 1, Variance = 0

3. Mean = 0, Variance = 0

4. Mean = 1, Variance = 1

Show me the answer

Answer: 1. Mean = 0, Variance = 1

Explanation:

• A standard normal distribution is a special case of the normal distribution with mean ($\mu$) equal to 0 and variance ($\sigma^2$) equal to 1.

• Any normal random variable $X$ can be standardized to a Z-score: $Z = \frac{X - \mu}{\sigma}$, where $Z$ follows the standard normal distribution.

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Hypothesis Testing

14. In hypothesis testing, the null hypothesis ($H_0$) is:

1. The hypothesis the researcher wants to prove

2. Always a statement of "no effect" or "no difference"

3. Rejected when the p-value is high

4. The same as the alternative hypothesis

Show me the answer

Answer: 2. Always a statement of "no effect" or "no difference"

Explanation:

• The null hypothesis ($H_0$) typically represents a default position, a statement of no change, no effect, or no difference (e.g., $\mu = \mu_0$).

• The alternative hypothesis ($H_1$ or $H_a$) is what the researcher aims to support, indicating a change, effect, or difference.

• A low p-value (typically < 0.05) provides evidence against $H_0$, leading to its rejection.

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15. The p-value in hypothesis testing is:

1. The probability that the null hypothesis is true

2. The probability of obtaining the observed results if the null hypothesis is true

3. The significance level

4. The power of the test

Show me the answer

Answer: 2. The probability of obtaining the observed results if the null hypothesis is true

Explanation:

• The p-value is the probability, assuming the null hypothesis ($H_0$) is true, of obtaining a test statistic at least as extreme as the one actually observed.

• A small p-value (≤ α, the significance level) suggests the observed data is inconsistent with $H_0$, leading to its rejection.

• It is not the probability that $H_0$ is true.

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Regression

16. In a simple linear regression model $y = \beta_0 + \beta_1 x + \epsilon$, $\beta_1$ represents:

1. The y-intercept

2. The slope of the regression line

3. The error term

4. The dependent variable

Show me the answer

Answer: 2. The slope of the regression line

Explanation:

• In the model $y = \beta_0 + \beta_1 x + \epsilon$:

  • $y$ is the dependent (response) variable.

  • $x$ is the independent (predictor) variable.

  • $\beta_0$ is the y-intercept (the value of y when x = 0).

  • $\beta_1$ is the slope (the change in y for a one-unit change in x).

  • $\epsilon$ is the random error term.

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17. The coefficient of determination, $R^2$, measures:

1. The correlation between x and y

2. The proportion of variance in y explained by the linear regression model

3. The significance of the slope

4. The intercept of the regression line

Show me the answer

Answer: 2. The proportion of variance in y explained by the linear regression model

Explanation:

• $R^2$ is a key output of regression analysis. Its value ranges from 0 to 1.

• An $R^2$ of 0 means the model explains none of the variability of the response data.

• An $R^2$ of 1 means the model explains all the variability.

• $R^2 = 1 - \frac{SS_{res}}{SS_{tot}}$, where $SS_{res}$ is the sum of squares of residuals and $SS_{tot}$ is the total sum of squares.