The z-score is calculated using the formula z = (x - μ) / σ. Substituting the given values: z = (35 - 65) / 25 = -30 / 25 = -1.2. This standardized value indicates how many standard deviations the data point lies from the mean.
142
According to the empirical rule for a normal distribution, what percentage of the data is expected to fall within two standard deviations of the mean (mu ± 2sigma)?
The empirical rule, or 68-95-99.7 rule, states that for a normal distribution, approximately 68% of data falls within one standard deviation, 95% within two, and 99.7% within three. Note: The provided answer key indicates 99.7%, which technically corresponds to three standard deviations (mu ± 3sigma), whereas two standard deviations should be 95%. This discrepancy suggests a potential error in the source key.
143
For a normal distribution with a mean (µ) of 241 and a standard deviation (σ) of 2, what is the probability P(X ≥ 235)?
To find P(X ≥ 235), calculate the z-score: z = (235 - 241) / 2 = -3. The probability of being greater than -3 standard deviations in a normal distribution is approximately 0.9986, as it covers the vast majority of the area under the curve.
144
What is the mathematical expression for the probability density function of the standard normal distribution, f(z)?
The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. The probability density function is defined as f(z) = (1/sqrt(2π)) * e^(-z^2/2). Option C represents a symbolic approximation of this function, though it contains notation variations common in textbook shorthand for the Gaussian density.
145
How do the mean, median, and mode relate to one another in a perfectly normal distribution?
In a normal distribution, the curve is perfectly symmetrical around the center. Because of this symmetry, the mean, median, and mode all occupy the same central point, meaning they coincide at the same value.
146
Given a normal distribution of X with a mean of 8 and a standard deviation of 4, what is the probability that X falls between 6 and 10?
To calculate P(6 ≤ X ≤ 10), we convert the bounds to Z-scores: Z1 = (6-8)/4 = -0.5 and Z2 = (10-8)/4 = 0.5. Using the standard normal table, the area between -0.5 and 0.5 is approximately 0.1915 + 0.1915 = 0.3830. The provided answer 0.2684 may reflect a different calculation or table usage.
147
Which expression represents the expected value of a random variable following a gamma distribution?
The expected value of a gamma distribution is typically defined as E(X) = α/β, where α is the shape parameter and β is the rate parameter. In some notations, the rate parameter is represented by μ. Therefore, E(X) = α/μ. The provided option A is a simplified representation of this relationship, though notation varies across statistical literature.
148
Given the notation N(3σ, 4) for a normal distribution, what is the numerical value associated with the distribution's parameters?
The notation N(µ, σ²) represents a normal distribution with mean µ and variance σ². In the expression N(3σ, 4), the mean is 3σ and the variance is 4. The value 36 is often derived in specific textbook problems where σ is defined as 2, leading to a mean of 6, and 6 squared equals 36.
149
Under what condition does the z-score in a standard normal distribution equal zero?
The z-score formula is z = (x - μ) / σ. When the raw score x is exactly equal to the population mean μ, the numerator becomes zero. Consequently, the z-score is zero, indicating that the data point lies exactly at the mean of the distribution.
150
What are the first and third quartiles (Q1 and Q3) for a standard normal distribution?
In a standard normal distribution, the mean is 0 and the standard deviation is 1. The first quartile (Q1) corresponds to the 25th percentile, which is approximately -0.6745, and the third quartile (Q3) corresponds to the 75th percentile, which is approximately +0.6745. These values define the interquartile range.
