Measurement #2 - Questions and Answers

Question 1 A data set consists of the ages at death for each of the 41 past presidents of United States. Is this set of measurement a population or a sample? What is the variable being measured? What measurement scale is appropriate for the data? Answer Step 1 In statistics, various types of variables are studied. There are two types of variables that are mainly focused in statistics. These are: Qualitative variable Quantitative variable On the other hand, there are 4 types of scales that belong to the above 2 variables. These are: Ordinal Interval Ratio Nominal These could be defined as: Quantitative variable: Variables with numeric values assigned are classified as quantitative variables. These values can be discrete or continuous. Example: number of students scoring marks in 0–20, 20–40, 40–60, 60–80, etc. Qualitative variable: These are categorical variables, subdivided into nominal and ordinal types. Nominal level: Labels variables without order or ranking. Ordinal level: Uses order or ranking but does not measure absolute differences. In the interval level of measurement, the difference between observations has a significant meaning but does not include an absolute zero. The ratio level of measurement is similar to the interval level but includes a natural zero point. Step 2 The data contains about all 38 deceased U.S. presidents. Therefore, the set of measurements is a population. Step 3 The variable being measured is age at death of president. Step 4 The measurement scale is ratio scale of measurement. Question 2 Juan makes a measurement in a chemistry laboratory and records the result in his lab report. The standard deviation of lab measurements made by students is 𝜎 = 10   milligrams. σ=10milligrams. Juan repeats the measurement 3 times and records the mean 𝑥 ˉ x ˉ of his 3 measurements. What is the standard deviation of Juan’s mean result? (That is, if Juan kept making sets of 3 measurements and averaging them, what would be the standard deviation of all his 𝑥 ˉ x ˉ 's?) How many times must Juan repeat the measurement to reduce the standard deviation of 𝑥 ˉ x ˉ to 5? Explain to someone who knows nothing about statistics the advantage of reporting the average of several measurements rather than the result of a single measurement. Answer Step 1 Given: The standard deviation of lab measurements made by students is 𝜎 = 10 σ=10. Sample size is 𝑛 = 3 n=3. Now, the standard deviation of the sampling distribution is given by: 𝜎 𝑥 ˉ = 𝜎 𝑛 σ x ˉ ​ = n ​ σ ​ Step 2 a) 𝜎 𝑥 ˉ = 10 3 ≈ 5.7735 σ x ˉ ​ = 3 ​ 10 ​ ≈5.7735 b) To reduce the standard deviation to 5, 5 = 10 𝑛 5= n ​ 10 ​ 𝑛 = 10 5 = 2    ⟹    𝑛 = 2 2 = 4 n ​ = 5 10 ​ =2⟹n=2 2 =4 Juan must repeat the measurement 4 times to reduce the standard deviation of 𝑥 ˉ x ˉ to 5. Reporting the average of several measurements has the advantage of being closer to the true mean of the distribution. Averaging reduces random errors and increases the accuracy of the result. Question 3 Luka and Anja each measured the height of their friend three times. Their friend is 59 inches tall. They recorded their measurements as shown. Luka: 59 in, 58 in, 58 in. Anja: 59.3 in, 59.6 in, 58.2 in. Which statement is true? Luka’s measurements are more precise and more accurate. Anja’s measurements are more precise and more accurate. Luka’s measurements are more precise, but Anja’s measurements are more accurate. Luka’s measurements are more accurate, but Anja’s measurements are more precise. Answer Step 1 Define accuracy and precision of values. An accurate value is closer to the true value. Precision measures consistency between repeated values. Step 2 The average of Luka’s measurements is: 𝐴 = 59 + 58 + 58 3 = 58.33   in. A= 3 59+58+58 ​ =58.33in. The average of Anja’s measurements is: 𝐴 = 59.3 + 59.6 + 58.2 3 = 59.03   in. A= 3 59.3+59.6+58.2 ​ =59.03in. Anja’s measurements are closer to the true value, making them more accurate. Luka’s measurements are more consistent, making them more precise. Result: Option (C) is correct. Question 4 Which of the levels of measurement (nominal, ordinal, interval, ratio) describe data that cannot be used with the methods of rank correlation? Explain. Answer Nominal data cannot be used in rank correlation methods. Reasoning: Nominal level: Data cannot be arranged in specific order (e.g., colors, names). It only classifies or categorizes. Ordinal level: Data has order or ranks but no degree of difference. Interval level: Ordered with equal differences but no absolute zero. Ratio level: Ordered with equal differences and absolute zero. Rank correlation methods require data with meaningful order. Nominal data lacks order and cannot be converted into ranks, making it unsuitable. Question 5 Levels of Measurement: Provide examples of each level of measurement that are not included in Clear-Sighted Statistics. What is the Nominal level of measurement? What is the Ordinal level of measurement? What is the Interval level of measurement? Answer Nominal level: Data classified with no inherent order. Example: Jersey numbers in football. Ordinal level: Data with an ordered relationship. Example: Socio-economic status (low, middle, high). Interval level: Ordered data with equal differences. Example: Temperature in Celsius.