Questions and Answers #1 Measurement

Questions and Answers Sheet 1 Measurement Question #1 What will be the time if the clock covered a distance of −4𝜋/3 radians? What is the measure of the angle formed in degree? Answer: Step 1 Given: The clock covered a distance of −4𝜋/3 radians. Step 2 A circle has a 360-degree angle or 2𝜋 radian. So, the hand of a clock also travels as complete 360 degrees or 2𝜋 radian to complete 1 hour. So, the angle in between each hour is Convert − −4×180 3 4𝜋 3 360∘ 12 = 30∘ , angle into degree. = −240∘ The measured angle formed in degree is −240∘ . The negative sign indicated direction counterclockwise. To find the time in hours required to angle −240∘ between each hour is 240 30 = 8 hours . The clock is usually based on the 12-hour category of time. The angle is counterclockwise so, subtract 8 from 12, 12-8=4 So, the time will be 4'o-clock. Question #2 A piece of 14.0 in. by 18.0 in. metal duct is a rectangular prism. If it is 4.00 ft long, what is the lateral surface area (in squa inches) assuming the duct is open on both ends? What is its volume (in cubic inches)? (Hint: Convert to inches. Use the rule working with measurements to give your answers to the appropriate accuracy and/or precision.) lateral surface area square in 2 3 volume square in Answer: Since you have asked multiple question, we will solve the first question for you. If you want any specific question to be solved then please specify the question number or post only that question, Given set Measurements ( 47.21 cm; 48.23 cm; 47.92 cm; 47.81) we are to find the mean of the Measurements 𝑀𝑒𝑎𝑛 = 𝑀𝑒𝑎𝑛 = 𝑀𝑒𝑎𝑛 = Sum of terms Number of terms 47.21+48.23+47.92+47.81 191.17 4 4 = 47.79 cm Question #3 A sample of 12 measurements has a mean of 24 and a standard deviation of 3.75. Suppose that the sample is enlarged to 14 measurements, by including two additional measurements having a common value of 24 each. A. Find the mean of the sample of 14 measurements. Mean = square B. Find the standard deviation of the sample of 14 measurements. Standard Deviation = square Answer: Given, Mean of 12 measurements = 24 = sum of 12 measurements 12 = 24 sum of 12 measurements = 24 ⋆ 12 = 288 Standard deviation of 12 measurements=3.75 a) Mean of 14 measurements=? = = = sum of 14 measurements 14 sum of 12 measurements+24+24 288+24+24 14 14 = 24 b) Standard deviation of 14 measurements=? Sum of squares for additional measurements = (24 − mean)2 + (24 − mean)2 = (24 − 24)2 + (24 − 24)2 =0 Thus, SD of 14 measurements=SD of 12 measurements=3.75 Question #4 Data for head injury measurements (in HIC, which is a standard head injury criterion) from samples of small, midsize, and large cars involved in the standard crash test were used to produce the accompanying ANOVA table. Using a 0.05 significance level, test the claim that the different size categories have the same mean head injury measurement. Does the size of a car appear to affect head injuries? ANOVA table Dialog content starts Between Groups Within Groups Total Sum of Squares 73630.272 435546.443 509176.715 What are the hypotheses to test? 𝐻0 : 𝐻1 : The test statistic is df 2 19 21 Mean Square 36815.136 22923.497 F Sig. 1.606 0.227 (Round to two decimal places as needed.) The P-value is (Round to three decimal places as needed.) ___________ 𝐻0 . Since the P-value is __________ than the significance level, there ___________ sufficient evidence to refute the claim that the different size categories have the same mean head injury measurement. It __________ that the size of a car affects head injuries. Answer: We have to interpret the result of ANOVA. We have to test that if the different size categories have the same mean head injury measurement. That is 𝐻0 :different size categories have the same mean head injury measurement 𝐻1 :different size categories have at least on different mean head injury measurement From the table The test statistic is 1.61 The P-value is 0.227 Fail to reject H_0 . Since the P-value is more than the significance level, there not sufficient evidence to refute the claim that the different size categories have the same mean head injury measurement. It dont suggest that the size of a car affects head injuries. Question #5 Determining the class standing (1𝑠𝑡 , 2𝑛𝑑 ,and so on) for the graduating seniors at a high school would involve measurement on a(n) _____ scale of measurement. Answer: Introduction: Scales of measurement: Nominal: When the possible observations are tags or categories or names, without any natural ordering, the scale of measurement is nominal; arithmetic operations cannot be performed even if the values are numerical. Ordinal: When the possible observations are tags or categories or names, with a natural ordering, the scale of measurement is ordinal; arithmetic operations cannot be performed even if the values are numerical. Interval: When the possible observations are numerical values that can be added or subtracted, but not divided or multiplied, the scale of measurement is interval; zero is not absolute, but relative. Ratio: When the possible observations are numerical values that can be added or subtracted, divided, or multiplied, the scale of measurement is ratio; zero is absolute. Explanation: The class standings or ranks are given as 1𝑠𝑡 , 2𝑛𝑑 , 3𝑟𝑑 , etc. These give the relative positions of the students with respect to the other students in the class. Evidently, there is a natural ordering in the data. However, it is meaningless to add, subtract, multiply, or divide these numbers denoting the positions in the standings, 1, 2, 3, and so on. For example, if the rank of one student is 2 and that of the other one is 6, then the difference, (6 - 2) would not give any meaningful value, as the difference in ranks does not, in any way, indicate the difference in their scores or grades. Similarly, the division (6 ÷ 2) is also meaningless, as it gives no idea about the ratio of their actual scores. Hence, it is an ordinal scale of measurement. Question #6 Is it possible for a circle to have the following approximate measurements? Diameter: 1.8 mm Circumference: 5.6 mm Explain. A. Yes, a circle can have those measurements. The ratio of the circumference to the diameter is approximately 3.1, which is close to the value of 𝜋. B. No, a circle cannot have those measurements. Millimeters are too small of a unit to measure, so it would be impossible to measure the diameter and circumference of a circle of that size. C. Yes, a circle can have those measurements. So long as the circumference is greater than the diameter, a circle can have any measurements for diameter and circumference. D. No, a circle cannot have those measurements. The ratio of the circumference to the diameter is approximately 0.32, which is not close to the value of 𝜋. Answer: For a circle we have to see the measurements of the circumference and the diameter by calculating their ratio. Option A is correct as the ratio of circumference to diameter is 𝜋. and option A is giving the value of 𝜋. Question #7 Is it possible for a circle to have the following approximate measurements? Explain. Radius: 0.91 mi Circumference: 2.72 mi A. Yes, a circle can have those measurements. The ratio of the circumference to the radius is approximately 3, which is close to the value of 𝜋. B. Yes, a circle can have those measurements. So long as the circumference is more than twice the radius, a circle can have any measurements for radius and circumference. C. No, a circle cannot have those measurements. The ratio of the circumference to the diameter would be approximately 1.5, which is not close to the value of 𝜋𝜋. D. No, a circle cannot have those measurements. Miles are too large of a unit to measure, so it would be impossible to measure the radius and circumference of a circle of that size. Answer: Given data is : Radius r=0.91 mi Circumference C=2.72 mi Now calculate the circumference, 𝐶 = 2𝜋𝑟 = 2 × 𝜋 × 0.91 mi =5.72 mi Thus, the given value of circumference is not correct. Now calculate the ratio of circumference to the radius, 𝐶 𝑟 = 5.72 0.91 𝐶 diameter =2𝜋 = 𝜋𝑑 𝑑 =𝜋 Thus, it the not possible for the circle to have given measurement. Hence, the correct option is (C) Question #8 To convert a degree measurement to a radian measurement, you multiply by square and divide by square. Note: If you need to type T just type pi Answer: According to the question, we need to convert a degree measurement to a radian measurement. The relationship between degree measurement and radian measurement is given as: Radian measurement = ( 𝜋 180 ) ⋆ (Degree measurement). Thus, to convert a degree measurement to radian measurement, multiply by 180𝑜 and divide it by pi. Question #9 What is the relationship between the model (right measurement) and the actual object (left measurement) in each of these examples? 10 ft : 24 in square 1 mi : 5,280 ft square 20 cm : 1,000 mm square Answer: We can find the relationship between model measurements and actual measurements as below using the unit conversation method In the 1st example it's given that 10ft:24inch It means 10ft measurements of model is equal to 24In or 2ft measurements of actual object. (since 1ft=12inch) In the 2nd example it's given that 1mi:5280ft It means 1mi measurements of model is equal to 5280ft measurements of actual object. In the 3rd example it's given that 20cm:1000mm It means 20cm measurements of model is equal to 1000mm or 100cm measurements of actual object.(since 1cm=10mm)