Name of Student : Lecturer: Chye Mei Sian, Pang Aik YenSubject : Class :

Time : Date :

Practice examination 1Time: 1 hour 15 minutes. Answer all the questions.The use of an electronic calculator is expected, where appropriate.

1. The random variable X has the distribution N(40,25), and X denotes the mean of a random sample of 10 observations of X. Find P(X<42). [4] [Ans:0.897]

2. Statistical investigations may involve random sampling. (i) Explain what is meant by the term ‘random sampling’, and state why random sampling is used. [3](ii)A random sample of 5 students is to be taken from the 763 students who attend a college. Explain

briefly how this could be done by using random numbers. [3]

3. A student is investigating the shape of a certain type of shell found on the seashore. She makes several measurements for each shell and combines the results into a single ‘shape index’, x. The resulting values from a random sample of 208 shells are summarized by ∑ x=633.36 ,∑ x2=2640.4612.(i) Calculate unbiased estimates of the population mean and variance for the shape index of shells of

this type. [3](ii)Obtain a symmetric 90% confidence interval for the population mean. [4][Ans: (i)3.045, 3.439; (ii)[2.833,3.257] ]

2

4. The continuous random variable X has probability density function given by

f ( x )={k (3−x )2 for 0≤x ≤3 ,0otherwise ,

Where k is a constant.

(i) Show that k=19 . [2]

(ii) Find P(1≤x ≤2). [2]

(iii) Find E(X). [3] [Ans: (ii)727 ; (iii)

34 ]

3

5. It is thought that more baby boys than baby girls are being born. A test of the null hypothesis p=0.5 against the alternative hypothesis p>0.5 is carried out, where p denotes the probability of a randomly chosen baby being male. For the test, a random sample of 10 babies is taken and the null hypothesis is rejected if 8, 9 or 10 of them are male.(a) Calculate the probability of a Type I error in this test. [3](b) Calculate the probability of a Type II error if the true value p is 0.6. [4][Ans: (a)0.0547; (b)0.833]

4

6. The accident and emergency department at a city hospital keeps records of the numbers of cases arriving between the hours of 2200 and 2300 throughout the week. The numbers, each day, have independent Poisson distributions with mean 4.3 for Monday to Friday and mean 6.2 for Saturday and Sunday.(i) Calculate the probability that, on a particular Wednesday, at least 4 cases will arrive between 2200

and 2300 hours. [3]

The total number of cases that arrive between 2200 and 2300 hours during one randomly chosen week is denoted by T. (ii) State the probability distribution of T. [2](iii) Use a suitable approximation to find P(30≤T ≤40). [4][Ans: (i) 0.623; (ii)P0 (33.9 ); (iii) 0.647]

5

7. In a certain population, the weights in kg of men and women have independent normal distributions with means and standard deviations as follows.Men: mean 75, standard deviation 6.4.

Women: mean 54, standard deviation 4.9.One man and one woman are chosen at random.

(i) Find the probability that their total weight is greater than 140kg.(ii)Find the probability that the woman’s weight is less than half the man’s weight.[Ans: (i)0.0861; (ii)0.0024]

6