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12th Business Maths Monthly Test - 1 ( Probability Distributions )

St. Britto Hr. Sec. School - Madurai

12th Business Maths Monthly Test - 1 ( Probability Distributions )-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

In 8 throws of a die, 1 or 3 is considered a success. Then the mean number of successes is

\(\frac83\)
\(\frac43\)
\(\frac23\)
\(\frac53\)

A statistical analysis of long-distance telephone calls indicates that the length of these calls is normally distributed with a mean of 240 seconds and a standard deviation of 40 seconds. What proportion of calls lasts less than 180 seconds?

0.214
0.094
0933
0.067

The probability that a normal variate X lies in the interval (\(\mu - \sigma, \mu + \sigma\)) is

0.0027
0.9973
0.6826
0.9544

Which of the following cannot generate a Poisson distribution?

The number of telephone calls received in a ten-minute interval
The number of customers arriving at a petrol station
The number of bacteria found in a cubic feet of soil
The number of misprints per page

An experiment succeeds twice as often as it fails. The chance that in the next six trials, there shall be at least four successes is

240/729
489/729
496/729
251/729

The sum of the mean and variance of a binomial distribution for 6 trial is 2.16. Then p =_________

0.4
0.6
0.8
0.2

In a parametric distribution the mean is equal to variance is :

binomial
normal
poisson
all the above

For a normal distribution if the mean is m, mode is n and median is m, then

m > m₁ > m₀
m < m₁ < m₀
m \(\neq\) m₁ x m₀
m = m₁ = m₀

An experiment succeeds twice as often as it fails. The chance that in the next six trials, there shall be at least four successes is

\(\frac { 240 }{ 729 } \)
\(\frac { 489 }{ 729 } \)
\(\frac { 496 }{ 729 } \)
\(\frac { 251 }{ 729 } \)

For the standar-d normal distribution

A B

1) Area under the curve (a) 99.74%

2) Area between z = - 1 and z = 1 (b) 95.44%

3) Area between z = - 2 and z = 2 (c) 68.26%

4) Area between z = - 3 and z = 3 (d) 1

The correct match is

1-a, 2-b, 3-c, 4 - d
1 - b, 2 - c, 3 - d, 4 - a
1-d, 2 - c, 3 - b, 4 - a
1-c, 2-d, 3-a, 4 - b

The starting annual salaries of newly qualified chartered accountants (CA's) in South Africa follow a normal distribution with a mean of Rs.1,80,000 and a standard deviation of Rs.10,000. What is the probability that a randomly selected newly qualified CA will earn between Rs.1,65,000 and Rs.1,75,000 per annum?

0.819
0.242
0.286
0.533

If Z is a standard normal variate, the proportion of items lying between Z = -0.5 and Z = -3.0 is

0.4987
0.1915
0.3072
0.3098

If X ~N(μ, σ²), the maximum probability at the point of inflexion of normal distribution is

\({ \left( \frac { 1 }{ \sqrt { 2\pi } } \right) }^{ { e }^{ \frac { 1 }{ 2 } } }\)
\({ \left( \frac { 1 }{ \sqrt { 2\pi } } \right) }^{ { e }^{ \left( -\frac { 1 }{ 2 } \right) } }\)
\({ \left( \frac { 1 }{ \sigma \sqrt { 2\pi } } \right) }^{ { e }^{ \left( \frac { 1 }{ 2 } \right) } }\)
\({ \left( \frac { 1 }{ \sqrt { 2\pi } } \right) }\)

In a binomial distribution if the mean is 8 and the variance is 6, then the number of trials is

32
48
16
12

If the area to the left of a value of z (z has a standard normal distribution) is 0.0793, what is the value of z?

-1.41
1.41
-2.25
2.25

Part - B

Generic 2 - Answer All Question

Determine the binomial distribution for which the mean is 4 and variance 3. Also find P(X=15)

Determine the binomial distribution for which the mean is 4 and variance 3. Also find P(X=15)

Derive the mean and variance of poisson distribution.

In a packet of 50 pens, 10 are defective, 10 pens are selected at random. What is the probability that atleast one is defective.

If electricity power failures occur according to a Poisson distribution with an average of 3 failures every twenty weeks, calculate the probability that there will not be more than one failure during a particular week.

Suppose X is a binomial variate X \(\sim \) B (5, p) and P(X = 2) = P(X = 3), then find p.

Assuming that a fatal accident in a factory during the year is 1/1200, calculate the probability that in a factory employing 300 workers there will be atleast two fatal accidents in a year. (given e^–0.25 = 0.7788)
Define Normal distribution.

Among 28 professors of a certain department, 18 drive foreign cars and 10 drive local made cars. If 5 of these professors are selected at random, what is the probability that atleast 3 of them drive foreign cars?

If 5% of the items produced turn out to be defective, then find out the probability that out of 10 items selected at random there are
(i) exactly three defectives
(ii) atleast two defectives
(iii) exactly 4 defectives
(iv) find the mean and variance

In a distribution 30% of the items are under 50 and 10% are over 86. Find the mean and standard deviation of the distribution.

Assume that a drug causes a serious side effect at a rate of three patients per one hundred. What is the probability that atleast one person will have side effects in a random sample of ten patients taking the drug?

Forty percent of business travellers carry a laptop. In a sample of 15 business travelers,
(i) what is the probability that 3 will have a laptop?
(ii) what is the probability that 12 of the travelers will not have a laptop?
(iii) what is the probability that atleast three of the travelers have a laptop?

Students of a class were given an aptitude test. Marks were found to be normally distributed with mean 60 and S.D. 5. Find the percentage of students who scored more than 60 marks.

The average number of customers, who appear in a counter of a certain bank per minute is two. Find the probability that during a given minute
(i) No customer appears
(ii) three or more customers appear.

The mean of a binomial distribution is 5 and standard deviation is 2. Determine the distribution.

Part - C

Generic 3 - Answer All Question

The average daily sale of 550 branch offices was Rs.150 thousand and standard deviation is Rs. 15 thousand. Assuming the distribution to be normal, indicate how many branches have sales between
(i) Rs. 1,25,000 and Rs. 1, 45, 000
(ii) Rs. 1,40,000 and Rs. 1,60,000

The mean of Binomials distribution is 20 and standard deviation is 4. Find the parameters of the distribution.

If x is a binomially distributed random variable with E(x) =2 and van (x)=4/3 Find P(x=5)

If X is a normal variate with mean 30 and SD 5. Find the probabilities that
(i) 26≤X≤40
(ii) X > 45
If on the average rain falls on 9 days in every thirty days, find the probability that rain will fall on atleast two days of a given week.

A bank manager has observed that the length of time the customers have to wait for being attended by the teller is normally distributed with mean time of 5 minutes and standard deviation of 0.6 minutes. Find the probability that a customer has to wait
(i) for less than 6 minutes
(ii) between 3.5 and 6.5 minutes

If a random variable X follows Poisson distribution such that P(X = 2) = 9. P(X = 4) + 90 P(X = 6) then find the mean and variance.

Suppose A and B are two equally strong table tennis players. Which of the following two events is more probable:
(a) A beats B exactly in 3 games out of 4 or
(b) A beats B exactly in 5 games out of 8 ?

The average daily procurement of milk by village society in 800 litres with a standard deviation of 100 litres. Find out proportion of societies procuring milk between 800 litres to 1000 litres per day.

Suppose that the amount of cosmic radiation to which a person is exposed when flying by jet across USA is a random vertical. having a normal distribution with mean of 4.35m rem and a standard deviation of 0.59m rem.
What is the probability that a person will be exposed to more than 5.20 m rem of cosmic radiation of such a flight?

Verfy the following statement:
The mean of a Binomial distribution is 12 and its standard deviation is 4.

Part - D

Generic 5 - Answer All Question

Marks in an aptitude test given to 800 students of a school was found to be normally distributed 10% of the students scored below 40 marks and 10% of the students scored above 90 marks. Find the number of students scored between 40 and 90?
If the height of 300 students are normally distributed with mean 64.5 inches and standard deviation 3.3 inches find the height below which 99% of the student lie?

One fifth percent of the the blades produced by a blade manufacturing factory turn out to be defective. The blades are supplied in packets of 10. Use Poisson distribution to calculate the approximate number of packets containing no defective, one defective and two defective blades respectively in a consignment of 1,00,000 packets (e^–0.2 =.9802)

20% of the bolts produced in a factory are found to be defective. Find the probability that in a sample of 10 bolts chosen at random exactly 2 will be defective using
(i) Binomial distribution
(ii) Poisson distribution (e^-2 = 0.1353)

The mean weight of 500 male students in a certain college is 151 pounds and the S.D is 15 pounds. Assuming the weights are normally distributed, find how many students weight
(i) between 120 and 155 pounds
(ii) more than 185 pounds.

Assuming one in 80 births is a case of twins, calculate the probability of 2 or more sets of twins on a day when 30 births occur.

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