PROBABILITY AND STATISTICS — Winter 2023

Define and give examples of Random Experiment and Sample space.

For a certain model car, the probability of the air conditioner failing before the warranty expires is 0.36, the probability of alternator failing is 0.26, and the probability of both failing is 0.11. Find the probability of the air conditioner or the alternator failing before the warranty expires.

Amicrochip company has three machines that produce the chips. Machine- Iproduces 45% of the chips, but 5% of its chips are defective. Machine-II produces 35% of the chips and 10% of its chips are defective. Machine-III produces the rest of the chips and 2% of its chips are defective. Find the probability of: Arandomly selected chip is found to be

a defective chip that has come from Machine-I, (ii) a non-defective chip that has come from Machine-II, (iii) a defective chip that has come from Machine-III.

Define Exponential Distribution and Gamma distribution.

The following table gives the number of aircraft accidents that occurred during the various days of the week. Find whether the accidents are uniformly distributed over the week. (Use 𝜒2 at 5% level of significance for 6 degree of freedom is 12.59). Days Sun. Mon. Tue. Wed. Thu. Fri. Sat. No. of Accidents 14 16 8 12 11 9

Fit a second-degree parabola for the following data: x: 0 1 2 3 y: 3 6 11 18 27 Estimate the value of y when x = 5.

Fit a relation of the form y = abx for the following data by the method of least squares: x: 2 3 4 5 y: 8.3 15.4 33.1 65.2 126.4 Estimate the value of y when x = 1.

An insurance company has discovered that only 0.1% of the population is involved in a certain type of accident every year. If its 1000 policy-holders are selected at random1 from the population, what is the probability that not more than 2 of its clients are involved in such accident next year?

Suppose that the current measurements in a strip of wire are assumed to follow a normal distribution with a mean of 10 milliamperes and variance of 4 (milliamperes)2. What is the probability that a measurement (i) exceeds 13 milliamperes, (ii) is at most milliamperes, (iii) between 10 and 13 milliamperes? [ Use: P(Z ≤ 1.5) = 0.9332].

Each sample of water has a 10% chance of containing a particular organic pollutant. Assume that the samples are independent with regard to the presence of the pollutant. Find the probability that in next 10 samples, (i) exactly 2, (ii) at least 4, (iii) between and 7, samples contain the pollutant.

Form the binomial distribution for the experiment of tossing a coin three times and counting the number of heads appear.

In a normal distribution 31% of items are under 45 and 8% are over 64. Find the mean and standard deviation of the distribution. [Use: P(0 < Z < 1.405) = 0.42,and P(0 < Z < 0.495) = 0.19].

For the following data, which of the series Aand Bshows greater variation? Series Mean Standard Deviation A 160 B 60

In a partially, destroyed record on analysis of correlation data, only the following are legible: Variance of x, 𝜎2 = 9, regression equation 8x – 10y + 66 = 0, 40x – 18y = 214. x Find (i) mean values of x and y, (ii) the standard deviation of y.

Ten competitors in a contest are ranked by three judges in the following order: 1st Judge 1 6 5 10 3 2 4 9 7 2nd Judge 3 5 8 4 7 10 2 1 6 9 3rd Judge 6 4 9 8 1 2 3 10 5 Use the correlation coefficient to determine which pair of judges has the nearest approach.

Calculate Pearson’s coefficient of skewness for the following data: x: 12 17 22 27 32 f: 28 42 54 108 129

The number of messages sent per hour over a computer network has the following probability distribution: X = No. of messages 10 11 12 13 14 P(x) 0.08 0.15 0.30 0.20 0.20 0.07 Determine the mean and standard deviation of the number of messages sent per hour.2

Calculate the coefficient of correlation between x and y for the following data: x: 65 66 67 67 68 69 70 72 y: 67 68 65 68 72 72 69 71

Define Statistical Hypothesis and types of errors occurring while testing of hypothesis.

To test whether a particular training improved the performance, a similar training was given to 8 participants, their scores both before and after the training are given below: Score (Before): 44 40 61 52 32 44 70 41 Score (After): 53 38 69 57 46 39 73 48 Test at 5% level of significance if the training was effective in terms of performance on the test. [Use t = 2.37]. 7,0.05

Amanufacturer of sprinkler systems used for fire protection in office buildings claims that the true average system-activation temperature is 130℉. Asample of size n = 9 systems, when tested, yields a sample average activation temperature of 131.08℉. If the distribution of activation times is normal with standard deviation 1.5℉, does the data contradict the manufacturer’s claim at 1% level of significance. [Use Z = 2.58.] 0.01

Arandom sample of size 20 from a normal population has mean 40 and standard deviation of 4. Test the hypothesis that the population mean is 45. [Use t = 2.09.] 0.05,19

Analysis of a random sample consisting of size m = 20 specimens of cold-rolled steel to determine yield strengths resulted in a sample average strength of x =29.8 ksi. Asecond random sample of size n = 25 two-sided galvanized steel specimens gave a sample average strength of y =34.7 ksi. Assuming that the two yield-strength distributions are normal with 𝜎 = 4.0 and 𝜎 = 5.0, does the data indicate that the corresponding true12 average yield strengths 𝜇 and 𝜇 are different? Use 5% level of significance.12 [Use Z = 1.96.] 0.05