Probability and Statistics

Consider the package weight data summarized with the following normal probability plot. Suppose there is a lower specification at 0.985 kg. Probability Plot of Weight Normal – 95% CI 99 95 Mean SID N AD P-value 0.9968 0,02167 15 0.223 0.492 90 Percent 70 60 50 40 30 5 1 0.92 0.94 0.96 1.04 1.06 1.08 0.98 1.00 1.02 Ex8-13Wt What percentage of the packages produced by this process is estimated to be below the specification limit? Use the equation: 3 =pr{z

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A process is under control with x – double bar = 45 and s-bar = 2. The process specifications are 80 + – 8. The sample size is n = 5. Estimation of the real capacity of the process (Cpk). Use three decimals and round. (0.XXX)

A process is under control with x – double bar = 45 and s-bar = 2. The process specifications are 80 + – 8. The sample size is n = 5. Estimation of the real capacity of the process (Cpk). Use three decimals and round. (0.XXX) EXPERT ANSWER

A process is in statistical control X (=) = 75 and S’ = 2. The control bar uses a sample size of n = 5. Specifications are at 80 ± 8. The quality characteristics are normally distributed

A process is in statistical control X (=) = 75 and S’ = 2. The control bar uses a sample size of n = 5. Specifications are at 80  ± 8. The quality characteristics are normally distributed a) Estimate the Cp and Cpk. Appreciate answer with workings EXPERT ANSWER

The manager of a store recorded the annual sales S (in thousands of dollars) of a product over a period of 7 years, as shown in the table, where t is the time in years, with t = 6 corresponding to 2006. t s 6 5.4 7 6.9 8 11.5 9 15.5 10 19.0 11 22.0 12 23.6 a. Use the regression features of a graphing calculator to find a model of the form S = a + bť + ct+d for the data above. (2 points) b. Use a graphing calculator to plot the data and sketch the model. (2 points) c. Use calculus and the model to find the time t when sales were increasing at the greatest rate. (4 points) d. Do you think the model would be accurate for predicting future sales? Explain. (2 points)

EXPERT ANSWER a) Putting the values in desmos, we get the estimated model to be the following: b) The graph of the data is the following: c) Observe that when the sales was increasing as greatest rate, then Hence the sales at that time would be d) No, the model would not be great for …

The manager of a store recorded the annual sales S (in thousands of dollars) of a product over a period of 7 years, as shown in the table, where t is the time in years, with t = 6 corresponding to 2006. t s 6 5.4 7 6.9 8 11.5 9 15.5 10 19.0 11 22.0 12 23.6 a. Use the regression features of a graphing calculator to find a model of the form S = a + bť + ct+d for the data above. (2 points) b. Use a graphing calculator to plot the data and sketch the model. (2 points) c. Use calculus and the model to find the time t when sales were increasing at the greatest rate. (4 points) d. Do you think the model would be accurate for predicting future sales? Explain. (2 points) Read More »

Question 2 – A bucket starts with r = 1 red ball and b = 1 black ball at time 0. An experiment consists of repeating the following: • select a ball uniformly at random from the bucket and note the colour • return the ball to the bucket, and add another ball of the same colour, we observe the entire sequence of ball selections. Let Bi be the event that the ith ball selected is black, Rį be the event that the first time that a red ball is selected is on the ith selection, and let T denote the first time that a red ball is selected (so T > 1). Question 3 Suppose that we conduct the same experiment as in Question 2 except that instead we start with r=2 red balls and b= 106 black balls in the bucket initially. (a) Show that P(B2) = P(B1). (b) Determine whether E(T) is finite or infinite.

EXPERT ANSWER a. P(B1) = 106/(106+2) P(B2) = P(R1)P(B2|R1) + P(B1)P(B2|B1) = 2/(106+2)*106/(106+3) + 106/(106+2)*(106+1)/(106+3) = 106(106+3)/(106+2)(106+3) = 106/(106+2) So, P(B1) = P(B2) b. P(T=1) = 2/(106+2) P(T=2) = P(B1)P(R2|B1) = 106/(106+2)*2/(106+3) = 2*106/(106+2)(106+3) P(T=3) = P(B1)P(B2|B1)P(R3|B2,B1​​​​​) = 106/(106+2)*(106+1)/(106+3)*2/(106+4) = 2*106(106+1)/(106+2)(106+3)(106+4) P(T=i) = 2*106*(106+1)*…*(106+i-2)/(106+2)(106+3)….(106+i+1) Therefore, E(T)