GTU 3130006 Probability and Statistics (PS) Summer 2024 Paper Solution & PDF

In a bolt factory, three machines A, B and C produce 25%, 35% and 40% of total output respectively. It was found that 5%,4% and 2% are defective bolts in the production by machines A, B, C respectively. A bolt is chosen at random from the total output and is found to be defective. Find the probability that it is manufactured from (i) Machine A(ii) Machine B (iii) Machine C.

A bag contains 3 red and 4 white balls. Two draws are made without replacement. What is the probability that both balls are red.

The probability that a student A solves a mathematics problem is 2 5 and the probability that a student 𝐵 solves it is 2

Verify that the following function 𝐹(𝑥) is a distribution function. 𝐹(𝑥) = { 0; 𝑥 < 0 1 − 𝑒−𝑥 4; 𝑥 ≥ 0 Also, find the probabilities 𝑃(𝑋 ≤ 4), 𝑃(𝑋 ≥ 8), 𝑃(4 ≤ 𝑋 ≤ 8).

Find the constant 𝑘 such that the function 𝑓(𝑥) = {𝑘𝑥2; 0 < 𝑥 < 3 0; 𝑜𝑡ℎ𝑒𝑤𝑖𝑠𝑒 is a probability density function,

𝑋 1 2 3 4 5 6 𝑃(𝑋 = 𝑥) 1 36 3 36 5 36 7 36 9 36 11 36 Find (i) mean (ii) variance.

Compute Karl Pearson’s coefficient of correlation between 𝑋 and 𝑌 for the following data: 𝑥 10 14 18 22 26 30 𝑦 18 12 24 6 30 36

Explain the term related to testing of hypothesis: (i) Null hypothesis (ii) Alternate hypothesis and (iii) Level of Significance

A dice is tossed 960 times and it falls with 5 upwards 184 times. Is the dice unbiased at a level of significance of 0.01? (|𝑍0.01| = 2.58)

A set of five similar coins is tossed 320 times and result is obtained as follows No of heads 0 1 2 3 4 5 Frequency 6 27 72 112 71 32 Test the hypothesis that the data follow a binomial distribution. (Critical value 𝜒0.05 2 = 11.07)

𝑥 1 2 3 4 6 8 𝑦 2.4 3 3.6 4 5

𝑥 1 2 3 4 5 6 7 8 𝑦 1 1.2 1.8 2.5 3.6 4.7 6.6 9.1

The lifetime of a certain kind of batteries has a mean life of 400 hours and the standard deviation as 45 hours. Assuming the distribution of lifetime to be normal, find (i) the percentage of batteries with a lifetime of at least 470 hours (ii) the proportion of batteries with a lifetime between 385 and 415 hours, and (iii) the minimum life of the best 5% of batteries. (𝑃(0 < 𝑧 < 0.33) = 0.1293)