Analyzing functions #2 - Questions and Answers

Question 1 Which of the following statements is not true? Multiple Choice A statistic is a single measure (usually numerical) that is calculated from a sample. For day-to-day business data analysis, most firms rely on a large staff of expert statisticians. A statistical test may be significant yet have no practical importance. Statistics is the science of collecting, organizing, analyzing, interpreting, and presenting data. Answer The following statement is not true: The majority of businesses rely on a sizable staff of knowledgeable statisticians for daily business data analysis. Explanation: We are aware that most businesses rely on a small team of highly skilled statisticians to analyze business data on a daily basis. We know that there are lot of advance statistical softwares are available. So due to technological enhancement of statistical analysis, most firms needs a small staff of expert statisticians Other remaining statements are true Statements. The sample statistic is known to be a function of the sample observations. Statistics is the branch of science which deals with the data collection, organization, analysis, interpretation, and presentation of the data. Additionally, although we are aware that some statistical tests may be significant, their practical significance is not known. Question 2 Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider f(t) = g(t) - h(t), where g(t) and h(t) are the position functions of the two runners. Answer Step 1 f(t) = g(t) - h(t) - the difference positions of two runners. g(t), h(t) are continuons and differentiable. So, f(t) is too continuons and differentiable. We want to show that there was no velocity difference between the runners at some point during the race. Find the first derivative of the equation f(t) = g(t) - h(t). f'(t) = g'(t) - h'(t) Consider the intreval (t₀, t₁), where is the start time t₀ and the finish time t₁. Is given that runners start a race at same time and finis in a tie. So, g(t₀) = h(t₀) and g(t₁) = h(t₁). We have: f(t₀) = 0 and f(t₁) = 0 We can apply the Rolle's theorem, because we have: f(t₀) = f(t₁) and f(t) is continuons and differentiable. So, there exists some c that such f'(c) = 0 f'(c) = g'(c) - h'(c) = 0 ⇒ g'(c) = h'(c) For c = t, t ∈ (t₀, t₁) ⇒ g'(t) = h'(t) Answer Apply the Rolle's theorem. Question 3 When functions are defined by more than one equation, they are called ___ functions. Answer Functions are referred to as piecewise functions when they are defined by many equations. examples, Let f(x) = {x , x > 0 x - 1, 0 ≤ x ≤ 5 x + 2, x > 5 this function has three equations in seperate intervals. Hence, these type of functions are called piecewise function. Question 4 A geologist has collected 10 specimens of basaltic rock and 10 specimens of granite. The geologist instructs a laboratory assistant to randomly select 15 of the speciems for analysis What the pmf of the number of granite specimens selected for analysis? What is the probability that the number of granite specimens selected for analysis is greater than 7? Answer Total number of specimen is 20 (10+10). So, the probability that a randomly selected specimen is granite is 10/20 = 1/2 = 0.5. Consider a random variable X which represents the number of granite specimen in the sample. X will follow binomial distribution with parameters n = 15 and p = 0.5 The p.m.f of X can be defined as: P(X=x) = (15 choose x) * (0.5)^x * (1-0.5)^(15-x) x = 0,1,2,...,15 The probability that number of granite selected for the analysis is greater than 7 can be calculated as: P(X>7) = 1 - P(X ≤ 7) = 1 - ∑(x=0 to 7) (15 choose x) * 0.5^x * (1-0.5)^(15-x) = 1-0.5 = 0.5 So, the required probability is 0.5 Question 5 Find an nth-degree polynomial function with real coefficients satisfying the given conditions. Verify the real zeros and the given function value. n = 3, 2 and 2 - 3i are zeros, f(1) = -10 Answer The coefficients of the polynomial are real numbers. Therefore, the complex zeros are conjugates of each other. If 2-3i is a zero, then 2 + 3i is also a zero. There are only three zeros: 2-3i, 2+3i, 2. f(x) = k(x-2)(x-(2-3i))(x-(2+3i)), where k is a non-zero real number. F(x) = k(x-2)(x²-4x+13) = k(x³-6x²+21x-26) Given that f(1) = -10, k(1³-6 × 1²+21-26) = k(1-6+21-26) = -10k -10k = -10 k = 1, f(x) = x³-6x²+21x-26. BECAUSE DEGREE IS 3, WE HAVE ONLY 3 ZEROS (CONJUGATES): 2-3i; 2+3i; 2. f(x)= k(x-2)(x-2+3i) (x-2-3i), WHERE K is A NON-ZERO REAL NUMBER. f(x) = k(x³-6x²+21x-26) if f(1)=-10, THEN k (1³-6*1²+21-26) = k(1-6+21-26)= =-10k -10k = -10 k = 1, THEN f(x)=x³-6x²+21x-26).