Extra Homework Problems/Practice Problems. Note: The solutions to these problems will not be posted. Come see me during office hours to discuss them.

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1 Note: The solutions to these problems will not be posted. Come see me during office hours to discuss them. Chapter 1 1. True or False: a. A measurement with high precision (i.e., low precision error) has high accuracy. b. Uncertainty is merely an estimate of the Error in a measurement. c. In determining the weight of an object, Newton s law, F = ma, is invoked, where a = g. d. Precision Error is also called random or statistical error. e. One pound-mass is equal to one pound-force on Earth. 2. Weighing yourself with clothes on (vs. weighing yourself without clothes on) introduces an error to your measurement. This error is called: a. reading error d. instrument error b. bias error e. absolute error c. precision error 3. We discussed in class that there are two general categories of error (bias and precision) but three practical sources of error encountered in a measurement. Name them. 4. What are the definitions of (a) the newton in the SI system, (b) the pound-force in the slug system, and (c) the pound-force in the pound-mass system? Problem 1-A. The Reynolds number Re is a dimensionless number used in fluid mechanics and is defined as ρ VD Re =, µ where ρ is the fluid density, V is the fluid velocity, D is some characteristic length of the body immersed in the fluid, and µ is the fluid absolute viscosity. a. Calculate the Reynolds number for the following properties: ρ = 1.16 kg/m 3, V = 0.30 km/hr, D = m, and µ = N s/m 2. SHOW ALL CONVERSIONS (CONVERSION FACTORS), and use only the most basic conversions (e.g., do not convert kg/hr to m/s directly). b. Convert the properties listed in part (a) to English Units (pound-mass system): lbm/ft 3, ft/s, ft, lbf s/ft 2. Show all work. c. Calculate the Reynolds number based on the English-unit properties calculated in part (b). d. What can you conclude about the (dimensionless) Reynolds number s dependence on unit system?

2 Chapter 2 1. True or False: a. The sum of all frequencies in a frequency distribution equals 1. b. Relative frequency is the same as probability. 2. Which statement about the 3 rd quartile is true? a. The 3 rd quartile is the range of values below the 75 th percentile. b. The 3 rd quartile is the range of values above the 75 th percentile. c. The 3 rd quartile is the range of values between the 50 th and 75 th percentile. d. The 3 rd quartile is the value such that 75 percent of the observations are smaller and 25 percent are larger. 3. Sixteen measurements of temperature range from 66.5 to 81.8 ºF. Select the most appropriate bin assignment for this data from those below: a. 65.0, 67.5, 70.0, 72.5, 75.0, 77.5, 80.0, 82.5 b , , , , , , , c. 66.5, 70.5, 74.5, 78.5, 82.5 d. 66, 70, 74, 78, 82 e. None of the above Problem 2-A. Consider the measurement of the temperature of hot gas flowing in a duct. The relative frequency distribution (or at least part of it) is depicted below. Answer the following questions: a. What is the probability of obtaining a measurement between 1090< T 1105 C? b. Three measurements fell within the range 1110 <T 1115 ºC. Are any measurements missing from this graph, and if so, how many? Relative Frequency Temperature (C)

3 Chapter 3 1. Choose the correct equation (by letter) that you would apply to the following scenarios. zσ zσ a. x = µ ± zσ b. µ = x ± zσ c. µ = x ± d. x = µ ± n n i. The boats on Disneyland's It's a Small World hold 15 passengers each. Using the weight parameters of the population (mean, standard deviation), what combined passenger weight would you expect 95% of full boats to have? ii. The temperature in this room is measured 50 times in a particular location. The population standard deviation is well-known. Estimate the true temperature. iii. The heights of people have well-known parameters (mean, standard deviation). You are designing a doorway. What range of heights comprise 99% of the population? iv. You are attempting to measure the temperature of a water bath. But you are using only one thermocouple, in one fixed location, and the water temperature varies from location to location. In a previous tests, however, you measured the variation of temperature at many locations within the bath, and the standard deviation was about 1.5 ºC. How well does your single-location measurement estimate the mean temperature of the entire volume? 2. True or False: a. All random measurements are normally distributed. b. Using the z value in confidence intervals presumes that the population standard deviation is known. c. Approximately 92% of all data in a normal distribution lie within ±1.75 standard deviations from the population mean. 3. When predicting the mean of a population based on the mean of a sample (the population standard deviation is known), what happens to the confidence interval when the sample size approaches infinity? Circle all that apply: a. the confidence interval approaches a finite value b. the confidence interval approaches zero c. the z value approaches 1.96 d. the confidence interval approaches infinity x 4. Evaluate the integral: f ( x) = e dx a b a b. Hint: e + = e e. Do so without any integration functions on your calculator. 0.10

4 Problem 3-A. A voltmeter is used to measure a known voltage of 100 V. Forty percent of the readings are within 0.5 V of the mean value. a. Assuming a normal distribution for the error, estimate the standard deviation for the meter. b. What is the probability that the mean of 10 readings will have an error greater than 0.75 V? Problem 3-B. On October 2, 2005, a tour boat named the Ethan Allen capsized on Lake George, in New York. Twenty of the 47 passengers on board died. The maximum weight capacity of the boat is estimated to be 7500 pounds-force which, based on decades-old passenger weight statistics, would have allowed the 47 passengers to ride safely. The latest statistics from the Centers for Disease Control show that the mean weight of American adults (men and women combined) is 167 pounds, with a standard deviation of 35 pounds. (Assume, for simplicity, that the combined weights are normally distributed; they are not. Why?) Given the new weight statistics, what is the probability that 47 adult passengers would exceed the maximum weight requirement of the boat? Problem 3-C. The Robert E. Kennedy Library has an elevator with a stated weight capacity of 2500 pounds. The latest statistics from the Centers for Disease Control show that the mean weight of American adults (men and women combined) is 167 pounds, with a standard deviation of 35 pounds. You are to determine the maximum safe number of passengers. What is the maximum number that will ensure that, at least 99.5% of the time, a full load of passengers does not exceed the design weight? (Hint: the 16-person limit listed on the elevator is not correct!) Problem 3-D. When predicting the combined effect of several measurements (that is, we write it as n x and then for x we use x=μ±z σ / n. n x i ), i=1 Show that, alternatively, you could achieve the same result (and same uncertainty) beginning with the individual x prediction, x=μ±z σ (note that the uncertainty is developed differently). NOTE: Never do this alternate approach. We will find out why in Chapter 4.

5 Chapter 4 1. True or False: a. Using the t distribution presumes that the population is normally distributed. b. At the same level of confidence, the Student-t value is always less than or equal to the z value. c. The t table is used in place of the z table when the population standard deviation is unknown. d. The shape of the t distribution depends on the number of measurements in the sample. 2. Fill in the missing values. SKETCH the z or t distribution, AND graphically demonstrate the probabilities and ranges. a. P( - z + ) = 0.75 (both unknowns have same magnitude) b. P(-2.35 z ) = c. P( t ) = 0.75 (50 degrees of freedom) d. P( t ) = (12 degrees of freedom) Problem 4-A next page

6 Problem 4-A. You are designing an automatic coin counter that stacks 50 pennies at a time to be placed in coin wrappers. You plan on measuring the height of the stack to determine the number of coins, but are concerned as to how accurate the method will be. You measure the thickness of 30 pennies, with the data listed below. Thickness, x (mm) 2 ( x x) No Sums a. Based on this data, estimate the height of a stack of 50 pennies. How accurate is your estimate, at 95% confidence? b. What percentage of pennies from the population have thicknesses of at least 1.34 mm?

7 Chapter 5 1. Using the t distribution in confidence intervals requires the population to be normally distributed; however, it is approximately valid for non-normal populations as long as what conditions are met? 2. Comment on the following statement: You can (and should) reject outliers from a set of data as long as the statistical models identify them as outliers.

8 Chapter 6 1/ 3 d 1. Consider the equation H =. If the uncertainty of each variable is the same fraction of 1/ 5 10k their respective nominal value, what variable s uncertainty has the largest effect on the uncertainty in H? a. b. d c. k d. H / 7 2. Consider the function F = AB C 2 / / D, where A, B, C, and D are measured variables. If all these variables have the same uncertainty (as a percentage of the nominal value), which variable affects the uncertainty in the function F the most? a. A b. B c. C d. D Problem 6-A. Earth s gravity varies along its surface, and a large portion of this variation is from centripetal acceleration due to the rotation of the earth. The centripetal acceleration at the surface of the Earth can be calculated by a 2 2 π = cos 2 φ, T R R E where T = period of rotation of the Earth = 8.64x10 4 s (negligible uncertainty) R E = radius of Earth = 6.37x10 6 m ± 1500 m φ = Latitude on Earth (radians) a. If our latitude on Earth is 35 degrees, 17 minutes ± 1 degree (35.3 ± 1.0 ), determine our local centripetal acceleration and its uncertainty. Use dimensional uncertainty propagation. (Hint: and recall that 180 = π radians) b. If the local gravity (without rotation) is m/s 2, what is the local effective gravity, and its uncertainty? (In doing so, make an assumption about the uncertainty in the value of gravity without rotation, g = m/s 2.)

9 Chapter 7 1. True or false: a. When choosing an appropriate model (equation) for a curve-fit, you should pick the equation that yields an R 2 value closest to 1. b. The correlation coefficient, r, is a measure of the degree of linear correlation, but the R 2 value applies to any curve-fit function. c. When choosing an appropriate model (equation) for a curve-fit, you should pick the equation that minimizes the residuals. 2. In performing statistical analysis of curve fits, we assume that the data are scattered normally about the curve-fit line. Why? 3. The following are plots of data and their predicted curve fits. Which values of the correlation coefficient are most likely to be incorrect? (Choose all that apply) a. r = 0.8 b. r = 0.9 c. r = d. r = 0.9

10 Chapter 8 1. True or false: a. A set of n x-y data pairs can be fit with a polynomial curve up to an order of n-1. b. If a 6 th order polynomial curve is fit to 5 data points, the resulting R 2 value will be equal to exactly Initially, what factor determines the choice of an appropriate curve fit model (sinusoidal, linear, polynomial, etc.) to a set of experimental data? 3. What role does statistics play in refining a curve-fit model (for example, when choosing between a 3 rd -order and a 4 th -order polynomial fit)? 4. Name two strategies for dealing with outliers in a curve-fitted set of data (aside from discarding them).