Statistics and Probability Answers

a) Let the random variable X have the following distribution:
P( and X = )0 = P(X = )2 = p
P(X = )1 =1− 2 p where
2
1
0 ≤ p ≤
for what value of p is the ) var(X a maximum? Justify. (3)
b) In a binomial distribution consisting of 5 independent trials, probabilities of 1 and 2
successes are 0.4096 and 0.2048 respectively. Find the parameter ‘ p ’ of the
distribution. (3)
c) Fit a Poisson distribution to the following data

a) For a distribution, the mean is 10, variance is 16, coefficient of skewness is +1 and
coefficient of kurtosis is 4. Obtain the first four moments about the origin i.e., zero.
Comment upon the nature of distribution. (5)
b) Calculate the correlation coefficient for the following heights (in inches) of father
(X ) and their sons ) (Y : (5)

X : 65 66 67 67 68 69 70 72
Y : 67 68 65 68 72 72 69 71

Which of the following statements are true or false? Justify your answer.
i) For normal distribution, mean deviation about mean is greater than quartile
deviation.
ii) In uniform distribution, the percentile points are equi-spaced.
iii) A high positive value of coefficient of correlation between the increase in cigarette
smoking and increase in lung cancer establishes that cigarette smoking is responsible
for lung cancer.
iv) In the case of Poisson distribution with parameter λ, x is sufficient for λ .
v) The sum of two independent Poisson variates is also a Poisson variate.
vi) If Var(X ) = Var ) (Y and 2X + Y and X − Y are independent, then X and Y are
independent.
vii) Mutually exclusive events are not independent.
viii) For any two events A and ) B, P(A∩ B cannot be less than either ) P(A or ) P(B .
ix) If 0 P(A) = , then A = φ.
x) If P(A∩ B) = ,2/1 P(A ∩ B) = 2/1 and 2 , then the value of P(A) = P(B) = p p is
4/1 .

a) If 6n tickets numbered 1 ,1,0 ,2 K, 6n − are placed in a bag and three are drawn out,
show that the probability that the sum of the numbers on them is equal to 6n is
3n /(6n − 6()1 n − )2 . (5)
b) From a bag containing 3 white and 5 black balls, 4 balls are transferred into an empty
vessel. From this vessel a ball is drawn and is found to be white. What is the
probability that out of four balls transferred 3 are white and 1 is black?

If µ equals 42, what percentage of all possible sample means are greater than or equal to 42.95? Since we have actually observed a sample mean of Picture = 42.95, is it more reasonable to believe that (1) µ equals 42 and we have observed a sample mean that is greater than or equal to 42.95 when µ equals 42, or (2) that we have observed a sample mean that is greater than or equal to 42.95 because µ is greater than 42? Explain. What do you conclude about whether customers are typically very satisfied with the XYZ Box video game system? (Round your answer to 2 decimal places. Do not round your intermediate calculations. Input your answer to percent without percent sign.)

Grade 11 students enrolled in schools in the LSB jurisdiction have had an average score of 63% on the exam with a standard deviation of 11%. Using the data from a clustered random sample of students from amongst the school board (see table below), determine if this year’s performance (2012) on the math exam was significantly lower than the historical average. Conduct a complete hypothesis test using a confidence level of 95%.
A complete hypothesis test includes:
a) The null and alternative hypotheses, Ho and Ha
b) The critical value (confidence coefficient)
c) The test statistic [2] (including sample mean and standard deviation
d) The p-value

student scores:
72, 52, 85, 60, 41, 39, 50, 48, 89, 70, 75, 61, 62, 40, 64, 79

Grade 11 students enrolled in schools in the LSB jurisdiction have had an average score of 63% on the exam with a standard deviation of 11%. Using the data from a clustered random sample of students from amongst the school board (see table below), determine if this year’s performance (2012) on the math exam was significantly lower than the historical average. Conduct a complete hypothesis test using a confidence level of 95%.
A complete hypothesis test includes:
a) The null and alternative hypotheses, Ho and Ha
b) The critical value (confidence coefficient)
c) The test statistic (including sample mean and standard deviation

Student Score
A 72 I 52
B 85 J 60
C 41 K 39
D 50 L 48
E 89 M 70
F 75 N 61
G 62 O 40
H 64 P 79





Determine the direction of the hypothesis test (one-sided left, one-sided right, bidirectional)
 Determine the test statistic (z* or t*) and the p-value for each of the following situations and
 Determine if they would cause the rejection of the null hypothesis if the confidence level was set
at 95% in each case. (Hint: be wary of the sample size) [2 points each]:
a) Ho: μ = 50 mL, Ha: μ ≠ 50 mL, sample mean = 48.1 mL, sample standard deviation = 5, n = 40
b) Ho: μ ≤ 8.4 m3
, Ha: μ > 8.4 m3
, sample mean = 10 m3
, s = 3.5, n = 25
c) Ho: μ ≥ 20oC, Ha: μ < 20oC , sample mean = 17.1oC, s =4.6
oC, n = 12
d) Ho: μ = 357 s, Ha: μ ≠ 380 s, sample mean = 410 s, s = 75, n = 40
e) Ho: μ ≤ 46 units, Ha: μ > 46 units, sample mean = 50 units, s = 9.5, n = 41

Based on studies conducted in various regions across the country, the average cost of pumpkins for consumers is $3.18 per kg.
From a random sample of 15 farmer’s markets in the Montreal-area you determine that the average price for pumpkins is $4.25 per kg with a standard deviation of $1.90 per kg

Construct a 98% confidence interval for the average price of pumpkins in the Montreal-area and determine if it differs significantly from that of the rest of the country.

2. a) The distribution of marks obtained by 500 candidates in a particular exam is given
below:

Marks more than: 0 10 20 30 40 50
Number of candidates 500 460 400 200 100 30

Calculate the lower quartile marks. If 70% of the candidates pass in the exam, find
the minimum marks obtained by a pass candidate. (5)
b) An analysis of monthly wages paid to the workers of two firms A and B belonging
to the same industry gives the following results:

Firm A Firm B
Number of workers 500 600
Average daily wages ` 186 ` 175
Variance of distribution of wages 81 100

i) Which firm, A or B, has a large wage bill?
ii) In which firm, A or B, is there greater variability in individual wages