Statistics Online Quiz

Explanation
The average of seven observations is calculated by summing up all the observations and dividing it by the total number of observations. In this case, the sum of the seven observations is 7 + 9 + 12 + x + 5 + 4 + 11 = 48 + x. Since the average is given as 9, we can set up the equation (48 + x)/7 = 9. Solving for x, we get x = 15. Therefore, the value of x is 15, which corresponds to option C.

Explanation
The arithmetic mean of n natural numbers from 1 to n can be found by calculating the sum of the numbers and dividing it by the total count of numbers. The sum of n natural numbers can be calculated using the formula n(n+1)/2. Dividing this sum by the total count of numbers, which is n, we get (n(n+1)/2)/n = (n+1)/2. Therefore, the correct answer is C.

Explanation
Probability is a measure of the likelihood of an event occurring, and it is always expressed as a value between 0 and 1, inclusive. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain to happen. Therefore, the correct answer is D, as it states that probability can take values from 0 to 1.

Explanation
The probability of the intersection of two mutually exclusive events is always zero because mutually exclusive events cannot occur at the same time. If two events are mutually exclusive, it means that if one event happens, the other event cannot happen. Therefore, the probability of both events happening together is impossible, resulting in a probability of zero.

Explanation
The distribution function F(x) is equal to P(X ≤ x). This means that the distribution function gives the probability that the random variable X takes on a value less than or equal to x. Therefore, option B, P(X ≤ x), is the correct answer.

Explanation
In Bernoulli distribution, the probability of success remains constant in all trials. This means that the probability of obtaining a success does not change from trial to trial. Therefore, option D is the correct answer.

Explanation
The correct answer is B, Geometric distribution. The geometric distribution possesses the memoryless property, which means that the probability of an event occurring in the future is independent of how long it has been since the last event occurred. In other words, the distribution does not "remember" past events and each event is considered as if it is the first event. This property makes the geometric distribution useful in modeling situations where the probability of success remains constant over repeated trials.

Explanation
The correct answer is C) W.S. Gosset. Student's t-distribution is named after William Sealy Gosset, who published the distribution under the pseudonym "Student" in 1908. Gosset was a statistician and chemist who worked for the Guinness brewery in Dublin, Ireland. He developed the t-distribution as a way to make inferences about small sample sizes when the population standard deviation is unknown.

Explanation
An estimator is considered to be the best if its distribution is concentrated about the true parameter value. This means that the estimator is unbiased and has low variability, providing accurate and precise estimates of the true parameter value. A discrete or continuous distribution does not necessarily indicate the quality of an estimator, and being normal is not a requirement for the best estimator. Therefore, option D is the correct answer.

Explanation
The estimators obtained by the method of moments are less efficient when compared to ML estimators. This means that the method of moments estimators have larger variances and are less precise in estimating the true parameter values. ML estimators, on the other hand, are known to be more efficient as they maximize the likelihood function and provide estimators with smaller variances. Therefore, option A is the correct answer.

Explanation
A test can be one-sided or two-sided depending on the alternative hypothesis. In a one-sided test, the alternative hypothesis only considers one direction of effect or difference. For example, if we are testing whether a new drug is more effective than a placebo, the alternative hypothesis would state that the new drug is greater than the placebo. In a two-sided test, the alternative hypothesis considers both directions of effect or difference. Using the same example, the alternative hypothesis would state that the new drug is not equal to the placebo, allowing for the possibility of it being either greater or less effective. Therefore, the correct answer is D.

Explanation
The level of significance is the probability of making a Type I error. A Type I error occurs when the null hypothesis is rejected, even though it is true. In hypothesis testing, the level of significance is set as the maximum probability of making a Type I error that is considered acceptable. Therefore, the correct answer is A.

Explanation
Degrees of freedom is related to the number of independent observations in a set. It represents the number of values in a calculation that are free to vary. In statistics, degrees of freedom are used to determine the sample size needed for a particular analysis and to calculate the variability of a statistic. The concept of degrees of freedom is important in hypothesis testing, as it affects the critical values used to determine statistical significance. Therefore, the correct answer is B.

Explanation
Nonparametric methods are based on mild assumptions. This means that these methods do not require specific assumptions about the underlying data distribution. Nonparametric methods are flexible and can be applied to a wide range of data types and distributions. They are often used when the data does not meet the assumptions of parametric methods, such as normality or homogeneity of variance. Instead, nonparametric methods rely on ranks or permutations to make inferences and draw conclusions from the data.

Explanation
The outcomes of tossing a coin two times are a variable of type "discrete random variable" because there are only two possible outcomes - either heads or tails. Each outcome is distinct and separate, and there are no other possible outcomes in between. Therefore, it can be categorized as a discrete random variable.

Explanation
The conditional probability of B given A is calculated by dividing the probability of A and B occurring together (P(A∩B)) by the probability of A occurring (P(A)). This can be represented by the formula P(A∩B) / P(A). Therefore, the correct answer is C.

Explanation
Since events A and B are independent, the probability of both events occurring (P(A∩B)) is equal to the product of their individual probabilities (P(A) * P(B)). Therefore, P(A∩B) = 0.4 * 0.2 = 0.08.

Explanation
The expression E(X-E(X))(Y-E(Y)) represents the covariance of X and Y. Covariance measures the relationship between two variables and indicates how they vary together. In this case, the expression calculates the expected value of the difference between X and its expected value, multiplied by the difference between Y and its expected value. This captures the extent to which X and Y vary together, indicating their covariance. Therefore, the correct answer is D.

Explanation
The expression E(X-μ)r represents the rth central moment of a random variable X. The central moment measures the dispersion or spread of the data around the mean. It is calculated by taking the expectation of the rth power of the difference between X and the mean μ. Therefore, option A, rth central moment, is the correct answer.

Explanation
The moment generating function of the Binomial distribution is (q + pet)n. This formula is used to calculate the moments of the distribution, which describe its shape and characteristics. The variable q represents the probability of failure, p represents the probability of success, e is the base of the natural logarithm, and t is the time. The exponent n represents the number of trials in the Binomial distribution. This formula allows us to calculate the expected values and variances of the distribution, which are important measures in probability theory and statistics.

Explanation
The correct answer is C because to standardize a normal random variable, we subtract the mean and divide by the standard deviation. In this case, the mean is 8 and the standard deviation is 8, so we subtract 8 from X and divide by 8 to get Z.

Explanation
The correct answer is D because both options A and B are correct. Option A represents the formula for the simple correlation coefficient using the covariance between X and Y divided by the square root of the variance of X multiplied by the variance of Y. Option B represents the formula for the simple correlation coefficient using the covariance between X and Y divided by the standard deviation of X multiplied by the standard deviation of Y. Therefore, both options A and B are correct ways to calculate the simple correlation coefficient.

Explanation
The standard error of the sample correlation coefficient r based on n paired values is given by (1-r^2)/sqrt(n). This formula takes into account the sample size (n) and the squared correlation coefficient (r^2), which measures the strength of the relationship between the paired values. As the correlation coefficient approaches 1 or -1, the standard error decreases, indicating a more precise estimate of the population correlation. The formula in option C is the correct one because it considers both the sample size and the strength of the correlation.

Explanation
If ρXY = -1, it means that there is a perfect negative linear relationship between X and Y. This implies that as Y increases, X proportionally decreases. Therefore, the correct answer is D.

Explanation
In a continuous probability distribution, the probability of a specific value occurring is always zero. This is because in a continuous distribution, the probability is represented by the area under the curve, and the area of a single point is always zero. Therefore, p(x = C) is always equal to zero.