The z-score is calculated as (x - mean) / standard deviation. Since x (120) is greater than the mean (75), the numerator (120 - 75 = 45) is positive. Assuming a positive standard deviation, the resulting z-statistic must be positive.
1202
According to the empirical rule for a normal distribution, which interval captures approximately 95.45% of the data?
The empirical rule (or 68-95-99.7 rule) states that for a normal distribution, approximately 68.27% of data falls within one standard deviation of the mean, 95.45% falls within two standard deviations (μ ± 2σ), and 99.73% falls within three standard deviations. Thus, the interval μ ± 2σ covers 95.45% of the observations.
1203
Given the linear transformation y = bx + c, what is the relationship between the range of y and the range of x?
The provided answer key suggests 'Quartile of X', which is mathematically unusual for a range transformation. Typically, the range of y = |b| * range of x. This answer may be incorrect or based on a specific context not provided. We retain the source answer as requested but flag it for potential conflict.
1204
Calculate the standard normal variable (z-score) given a sample mean of 70, a population mean of 15, and a standard deviation of 20.
The z-score is calculated using the formula z = (x - μ) / σ. Substituting the given values: z = (70 - 15) / 20 = 55 / 20 = 2.75. This value represents how many standard deviations the observed value is from the mean.
1205
According to the empirical rule for a normal distribution, what interval centered around the mean covers approximately 99.7% of the data?
The empirical rule, also known as the 68-95-99.7 rule, states that for a normal distribution, approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three standard deviations. Therefore, the interval μ ± 3σ encompasses nearly all the data points in a perfectly normal distribution.
1206
If a dataset of 40 observations has a variance of 50, what will the variance be if every observation is increased by 20?
Variance is a measure of dispersion, which is invariant to changes in origin (adding or subtracting a constant). When every observation in a dataset is increased by a constant value, the spread of the data remains unchanged, so the variance remains 50.
1207
Which measure of dispersion is considered unsuitable for open-end distributions?
The range is defined as the difference between the maximum and minimum values. In open-end distributions, the exact values of the extreme classes are unknown, making it impossible to calculate the range accurately. Therefore, it is considered a poor measure of dispersion in such cases.
1208
What is the definition of the positive square root of the mean of the squared deviations of observations from their mean?
The standard deviation is the square root of the variance. While the definition provided in the question describes the standard deviation, the provided answer key indicates option D. This represents a conflict between standard statistical definitions and the provided key.
1209
What term describes the value used to measure the distance between the mean and a random variable in units of standard deviation?
The z-value, or standard score, indicates how many standard deviations an observation is from the mean. It is calculated by subtracting the mean from the value and dividing by the standard deviation. This transformation allows for the comparison of scores from different distributions by standardizing them to a common scale with a mean of zero and a standard deviation of one.
1210
If the standard deviation is divided by the coefficient of variation, what statistical measure is obtained?
The coefficient of variation (CV) is defined as the ratio of the standard deviation (σ) to the arithmetic mean (μ), expressed as CV = σ / μ. Therefore, rearranging this formula, the arithmetic mean is equal to the standard deviation divided by the coefficient of variation.
