BE Computer Engineering (IOE, TU) Probability and Statistics (IOE, SH 602 / ENSH 301) Question Paper 2078

The following frequency distribution shows the lifetime (in hours) of 100 electronic components tested in a quality-control laboratory:

(a) Compute the arithmetic mean, median and mode of the lifetimes. [6]

(b) Calculate the standard deviation and the coefficient of variation, and comment on the consistency of the components. [4]

(c) Using your results from (a), find the Karl Pearson coefficient of skewness and interpret the shape of the distribution. [2]

State the axioms of probability and the theorem of total probability.

Three machines $M_1$, $M_2$ and $M_3$ in a factory produce 25%, 35% and 40% of the total output respectively. The proportion of defective items produced by these machines is 5%, 4% and 2% respectively.

(a) Distinguish between mutually exclusive and independent events with a suitable example. [3]

(b) An item is drawn at random from the total output and is found to be defective. Use Bayes' theorem to find the probability that it was produced by machine $M_2$. [6]

(c) What is the overall probability that a randomly selected item is non-defective? [3]

Define a continuous random variable and state the properties of the normal distribution.

The weekly wages of 1000 workers in a manufacturing plant are normally distributed with a mean of Rs. 70 and a standard deviation of Rs. 5.

(a) State the conditions under which a binomial distribution can be approximated by a normal distribution. [3]

(b) Estimate the number of workers whose weekly wages lie between Rs. 69 and Rs. 72. [5]

(c) Find the lowest wage of the highest-paid 100 workers. [4]

(Use: P(0 < Z < 0.20) = 0.0793, P(0 < Z < 0.40) = 0.1554, P(0 < Z < 1.28) = 0.3997.)

The following data give the number of hours studied ($X$) and the marks obtained ($Y$) by 8 students:

(a) Calculate the Karl Pearson coefficient of correlation between $X$ and $Y$ and interpret the result. [6]

(b) Obtain the two regression equations, $Y$ on $X$ and $X$ on $Y$. [4]

(c) Estimate the marks of a student who studies for 10 hours, and show how the correlation coefficient is related to the two regression coefficients. [2]

A random variable $X$ has the following probability distribution:

(a) Determine the value of the constant $k$. [2]

(b) Find the mean $E(X)$ and the variance $Var(X)$ of the distribution. [4]

In a certain process, 10% of the manufactured screws are defective. A random sample of 8 screws is selected.

(a) Find the probability that exactly 2 screws are defective. [3]

(b) Find the probability that at most 1 screw is defective, and state the mean and variance of the number of defective screws in the sample. [3]

The number of telephone calls arriving at a switchboard follows a Poisson distribution with an average of 3 calls per minute.

(a) Find the probability that no call is received in a given minute. [2]

(b) Find the probability that more than 2 calls are received in a given minute. [4]

(Take $e^{-3} = 0.0498$.)

A random sample of 64 items drawn from a large population has a sample mean of 52 units and a sample standard deviation of 8 units.

(a) Distinguish between a point estimate and an interval estimate. [2]

(b) Construct a 95% confidence interval for the population mean and interpret it. [4]

(Use $z_{0.025} = 1.96$.)

A manufacturer claims that the mean breaking strength of a cable is at least 1800 N. A sample of 50 cables gives a mean breaking strength of 1770 N with a standard deviation of 100 N.

(a) State the null and alternative hypotheses and identify whether the test is one-tailed or two-tailed. [2]

(b) At the 5% level of significance, test whether the manufacturer's claim is justified. [4]

(Use $z_{0.05} = 1.645$.)

The following table shows the observed frequencies of outcomes obtained when a die was rolled 120 times:

Apply the chi-square goodness-of-fit test at the 5% level of significance to determine whether the die is fair. State the degrees of freedom and your conclusion.

(Use $\chi^2_{0.05,\,5} = 11.07$.)

For two events $A$ and $B$, it is given that $P(A) = 0.5$, $P(B) = 0.4$ and $P(A \cup B) = 0.7$.

(a) Find $P(A \cap B)$ and $P(A \mid B)$. [3]

(b) Examine whether the events $A$ and $B$ are independent, and find $P(\bar{A} \cap \bar{B})$. [3]

(a) Define the first four central moments of a distribution. [2]

(b) For a distribution the moments about the value 5 are $\mu_1' = 2$, $\mu_2' = 20$, $\mu_3' = 40$ and $\mu_4' = 250$. Convert these into the central moments $\mu_2$, $\mu_3$ and $\mu_4$, and comment on the skewness of the distribution. [4]