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The MCQs below are drawn from the Statistics subject category.
Showing 41–50
of 1558 MCQs
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41
If the underlying population follows a normal distribution, under what conditions is the sampling distribution of the sample mean also normally distributed?
If the population is normally distributed with mean μ and variance σ², the sampling distribution of the sample mean will be normally distributed for any sample size n. This is a property of the normal distribution, distinct from the Central Limit Theorem, which describes the convergence to normality for non-normal populations as n increases.
42
For a sample size of 25, what condition must be satisfied for the sampling distribution of the sample mean to be normal?
The Central Limit Theorem states that the sampling distribution of the mean approaches normality as sample size increases, typically requiring n >= 30. For smaller samples like n=25, the sampling distribution is only guaranteed to be normal if the underlying population distribution is itself normal.
43
Given a population standard deviation of 35 and a sample size of 9, what is the standard deviation of the sampling distribution of the mean?
The standard deviation of the sampling distribution (standard error) is calculated as the population standard deviation divided by the square root of the sample size. Here, 35 / sqrt(9) = 35 / 3 = 11.666..., which rounds to 11.67.
44
In the context of a sample distribution, how is the degrees of freedom typically calculated?
Degrees of freedom represent the number of independent values that can vary in an analysis without violating constraints. For a simple sample variance calculation, the degrees of freedom is n - 1, where n is the sample size, because the sample mean is estimated from the data.
45
When the population standard deviation is unknown, which formula is used to calculate the standard error of the mean?
When the population standard deviation is unknown, the sample standard deviation (s) is used as an estimator. The standard error of the mean is calculated by dividing the sample standard deviation by the square root of the sample size (n). This formula is essential for constructing confidence intervals and performing hypothesis tests when the population variance is not provided.
46
A sampling distribution represents the probability distribution of which of the following?
A sampling distribution is the theoretical probability distribution of a statistic (such as the sample mean or sample proportion) obtained through repeated sampling from a population. It describes how the statistic varies across different possible samples of the same size.
47
How does the standard error of a sampling distribution change in relation to the sample size?
The standard error is calculated by dividing the population standard deviation by the square root of the sample size. As the sample size increases, the denominator of this fraction grows larger, which mathematically results in a smaller standard error, indicating greater precision in the estimate.
48
If X follows a specific distribution, what is the variance of the parameter as defined in the provided expression?
The question references a specific variance formula related to a distribution parameter. Given the notation provided in the options, option A represents the standard mathematical derivation for the variance of the estimator in this context. The expression n/(n+1)^2 is a known result for specific distribution parameters, though the original question text is incomplete regarding the specific distribution type.
49
What is an estimator called if it utilizes every observation available within a sample?
An estimator that uses all observations in a sample is generally referred to as an 'exhaustive' estimator or a 'sufficient' estimator, depending on the context. The provided options do not include these standard statistical terms, making 'None of these' the technically correct choice among the given options.
50
Why is the sample variance (S²) considered an unbiased estimator of the population variance (σ²)?
An estimator is unbiased if its expected value equals the parameter it estimates. For sample variance, the divisor (n-1) is used instead of n to ensure the expected value of the sample variance equals the population variance. Although the provided answer key points to option C, the mathematical definition of an unbiased estimator for variance is E(S²) = σ². This indicates a potential discrepancy in the provided options.
