Calculating the Number of Washing Machines and Correlation Coefficient

Extract

[...] We are looking such that P(T soit : By solving this equation we find = = 9.495 years 4. For an exponential law with parameter,is the expected value so it is 13.699 years in our case. 5. On pose p = P(T 10) and q = 1-p = 0.518 If we pass to the complement of the event A = "on the n washing machines there is at least one that exceeds 10 years", this gives the event = « the n washing machines have all lasted less than 10 years. [...]

[...] Using the table of the cumulative distribution function of the standard normal distribution, we find i. P(X ≤ 18) = P(Z ≤ 0.6) = 0.7257 ≤ = P(Z ≤ = 0.841 ii. P(X = 18) = 0 because the normal law is a continuous law iii. P(X [...]

[...] A statistical study says that after 7 years of washing machines are still in operation. This means that P(T > = 0.6, so P(T ≤ = 0.4 Or we know that the distribution function of an exponential law is given by So we have 0.4 = soit = 0.073 2. a. The probability that a washing machine has a lifespan of less than 6 months is: P(T [...]

[...] We adapt the previous method to calculate the standard deviation: = 195.570 hours 5. We note D the theoretical lifespan, so D follows a normal law with a mean of 724 and a standard deviation of 196. We note Z the reduced normal law. First, we will calculate P(300 ≤ D ≤ 500) and P(500 ≤ D ≤ 700) using the table of the reduced normal distribution function: = P(Z ≤ -1.14) - P(Z ≤ -2.16) then we use the symmetry of the standard normal distribution = P(Z ≤ 2.16) - P(Z ≤ 1.14) then we use the table of the distribution function = 0.112 By applying the same method, we find: P(500 ≤ D ≤ 700) = 0.325 Finally, to find the theoretical number of lamps in each of these two intervals, we multiply the probabilities above by the total number of lamps: theoretical number of lamps in [300; 500[ = 0.112 * 400 = 44.8 theoretical number of lamps in [500; 700[ = 0.325 * 400 = 130 These results are well consistent with the empirical values. [...]

[...] Z has a mean of 0 and a standard deviation of 1 b. The graph representing the density of c. The standard normal distribution is symmetric and integrable therefore: P(Z = 0.5 d. Using the table of the cumulative distribution function of the standard normal distribution, we find: i. P(Z ≤ 1.38) = 0.916 ii. P(Z 1.38) = 1 - P(Z ≤ 1.38) = 0.084 iii. P(Z ≤ = or Z is symmetric so = 0.159 iv. P(Z -0.8) = P(Z ≤ 0.8) = 0.788 2. [...]