Measurement #3 - Questions and Answers

Question 1 How is the correlation coefficient independent of units of measurement? Answer Correlation: Correlation is a measure that indicates the "go-togetherness" of two data sets. It can be denoted as 𝑟 r. The value of the correlation coefficient lies between − 1 −1 and + 1 +1. A positive 𝑟 r indicates that the two data sets are perfect and both are in the same direction. A negative 𝑟 r indicates that the two data sets are perfect and both are in opposite directions. 𝑟 = 0 r=0 indicates no relationship between the data sets. Correlation coefficient 𝑟 r: The Pearson's product-moment correlation coefficient measures the strength of a linear association between two variables and is denoted by 𝑟 r or 𝑟 𝑥 𝑦 r xy ​ . The formula is given by: 𝑟 𝑥 𝑦 = 𝑛 ∑ 𝑥 𝑖 𝑦 𝑖 − ( ∑ 𝑥 𝑖 ) ( ∑ 𝑦 𝑖 ) [ 𝑛 ∑ 𝑥 𝑖 2 − ( ∑ 𝑥 𝑖 ) 2 ] [ 𝑛 ∑ 𝑦 𝑖 2 − ( ∑ 𝑦 𝑖 ) 2 ] r xy ​ = [n∑x i 2 ​ −(∑x i ​ ) 2 ][n∑y i 2 ​ −(∑y i ​ ) 2 ] ​ n∑x i ​ y i ​ −(∑x i ​ )(∑y i ​ ) ​ This formula indicates that the correlation coefficient 𝑟 r: Does not consider dependent or independent variables. Treats all variables equally. A change of scale or origin does not affect the value of 𝑟 r. When constructing the correlation coefficient, the units of measurements cancel out. Hence, 𝑟 r is independent of units of measurement. Question 2 a) Birth years of your family are being analyzed. Are these data quantitative or qualitative? Are these data discrete, continuous, or neither? What is the highest level of measurement of birth years? (Nominal, Ordinal, Interval, or Ratio?) b) Survey responses to "What is the gender of your first child?" Are these data quantitative or qualitative? Are these data discrete, continuous, or neither? What is the highest level of measurement of gender? Answer Step 1: Data in statistics can be divided into two categories: Quantitative data: Numerical values. Qualitative data: Used to categorize (e.g., gender). Quantitative variables are subdivided into continuous (can take any value) and discrete (finite countable values). Levels of measurement: Nominal: Categories or tags without order. Ordinal: Ordered categories without measurable differences. Interval: Measured with equal differences, but no absolute zero. Ratio: Includes equal differences and an absolute zero. Step 2a: Quantitative or qualitative: Birth years are numerical. Hence, quantitative. Discrete or continuous: Birth years are finite and countable. Hence, discrete. Level of measurement: Since birth years are numerical with an absolute zero, the level is interval. Step 2b: Quantitative or qualitative: Gender data are non-numerical. Hence, qualitative. Discrete or continuous: Gender cannot take numerical values. Hence, neither. Level of measurement: Gender categories (male, female) lack order. Hence, nominal. Question 3 A measurement using a ruler marked in centimeters is reported as 6 cm. What is the range of values for the actual measurement? Answer Step 1: The ruler has markings in centimeters. The given reading is 6 cm. Step 2: The range of actual measurements depends on the precision of the ruler. Values between 5.5   cm 5.5cm and 6.5   cm 6.5cm would be recorded as 6 cm. Result: The range of actual measurement is 6 ± 0.5   cm . 6±0.5cm. Question 4 What are quantitative data measurements? a) Measurements that are appropriate for any type of variable b) Measurements of categorical variables and can be displayed as types or descriptions c) Measurements that are appropriate in all experiments d) Measurements of numerical variables and are displayed as numerical values Answer: Step 1: Quantitative variable: Displays numerical values. Arithmetic operations are performed. Categorical variable: Classifies into groups based on attributes. Includes nominal and ordinal scales. Step 2: Quantitative data are measurements of numerical variables. Result: Option (D) is correct. Question 5 Which of the following quantities represents a volume measurement: 15   𝑚 2 15m 2 , 2.5 × 1 0 2   𝑚 3 2.5×10 2 m 3 , 5.77   𝐿 5.77L, or > 𝑠 ? >s? Answer: Volume: Measured in units like liters (L), cubic meters ( 𝑚 3 m 3 ), etc. Analysis: 15   𝑚 2 15m 2 : Area (not volume). 2.5 × 1 0 2   𝑚 3 2.5×10 2 m 3 : Valid volume measurement. 5.77   𝐿 5.77L: Valid volume measurement. > 𝑠 >s: Invalid. Result: 2.5 × 1 0 2   𝑚 3 2.5×10 2 m 3 represents volume.