MCQs Probability And Statistics(MTH 302) 50+

1 . Which of the following is a necessary condition for using a t distribution table?

• n is small
• s is known but σ is not
• the population is finite
• (a) and (b) but not (c)

Ans : C

2. If r = 0.6, byx = 1.2 then bxy = ?

• 0.3
• 0.2
• 0.72
• 0.40

Ans : A

3. If a coin is tossed 4 times then the probability of getting at most 3 heads is:

• 15/16
• 10/16
• 12/16
• 5/16

Ans : A

4. In a one-tail test for the population mean, if the null hypothesis is not rejected when the alternative hypothesis is true, then:

• Type I error is committed
• Type II error is committed
• a correct decision is made

Ans : B

5. One problem with hypothesis testing is that a real effect may not be detected. This problem is most likely to occur when

• the effect is small and the sample size is small.
• the effect is large and the sample size is small.
• the effect is small and the sample size is large.
• the effect is large and the sample size is large

Ans : A

6. Which of the following is correct for Chi Square-test forgoodness of fit?

• Sum of the expected frequencies < Sum of the observed frequencies
• Sum of the expected frequencies > Sum of the observed frequencies
• Sum of the expected frequencies = Sum of the observed frequencies
• Sum of the expected frequencies = 2(Sum of the observed frequencies)

Ans : C

7. Find the parameters of the binomial distribution with mean 48 and the standard deviation 4?

• n = 72 and p = 1/3
• n = 144 and p = 1/3
• n = 144 and p = 2/3
• n = 72 and p = 2/3

Ans : D

8. n a locality of 20000 families a sample of 900 families was selected at random. Of these 900 families 250 families were found to have a monthly income of Rs.500 or less. Find the probable limit of proportion for the families having a monthly income of Rs.500 or less?

• 0.20 to 0.29
• 0.129 to 0.411
• 0.359 to 0.288
• 0.27 to 0.73

Ans : A

9. Two samples of sizes 25 and 35 are independently drawn from two normal populations, where the unknown variances are assumed to be equal. The number of degrees of freedom for the equal-variances t-test statistic is:

• 58
• 60
• 62
• 57

Ans : A

10. Which of the following statements correctly describes a difference between the F and t distributions?

• The F distribution is discrete, the t distribution is continuous
• The t distribution is valid only for positive numbers, the F for all real values
• The F distribution is symmetrical, the t distribution is skewed to the right
• The F distribution is valid only for positive numbers, the t for all real values

Ans : D

11. In a random sample of 300 persons of a town, 180 are found to be tea drinkers. In a sample of 500 persons from another town, 200 are found to be tea drinkers. What is the standard error?

• 0.04
• 0.072
• 0.051
• 0.088

Ans : A

12. What should you conclude if the two lines of regression (y on x, and x on y) are perpendicular to each other?

• The correlation coefficient =0
• The correlation coefficient =1
• The correlation coefficient =-1
• No conclusion can drawn

Ans : A

13. Which of the following is not an assumption of Binomial Distribution?

• There are only two disjoint outcomes
• The number of trials must be finite.
• Trials are independent
• Trials are dependent.

Ans : D

14. What is our decision for a goodness-of-fit test with a computed value of chi-square of 1.273 and a critical value of13.388?

• Do not reject the null hypothesis
• Reject the null hypothesis
• Unable to reject or not reject the null hypothesis based on data
• Should take a larger sample

Ans : C

15. Which of the following options is correct? (Here S.D.(X) and Var.(X) are Standard Deviation and Variance in X)

• S.D.(X)= [Var.(X)]|12
• S.D.(X)= [Var.(X)]
• S.D.(X)= [Var.(X)}
• S.D.(X)=[Var (X)]*

Ans : A

16.To fit a non-linear curve Y=a+bX+cX^2 to a data, which of the following sums is not required ?

• Sum of the values of variable ‘X^2
• Sum of the values of variable “Y
• Sum of the values of variable XY
• Sum of the values of variable Y^2

Ans : D

17.If in case of F-test for equality of two population variances, Sy^2 = 16,Sx^2 =4
then the calculated value of the test statistic is

• 1/4
• 4
• 2
• 1/2

Ans : B

18. Which of the following is correct?
i) Probability of Type- error is Level of significance. (i) Probability of Type-ll error is Level of significance.

• option (i) only
• option (ii) only
• both option (i) and (ii)
• None of the option (i) and (ii)

Ans : A

19. Which of the following is correct?
(i) Accepting Ho when it is wrong is Type-I error. (ii) Accepting Ho when it is wrong is Type-II error.

• option (i) only
• option (ii),only
• both option (i) and (ii)
• None of the option (i) and (ii)

Ans : B

20. If X is a binomial variate with parameter ‘p’ such that n=6 and P(X=0)}=0.5 and P(X=1)=0.25, then ‘q=?

• 12/13
• 1/13
• 1/6
• 1/7

Ans : A

21. If a coin is tossed 4 times then the probability of getting at most 3 heads is:

• 15/16
• 10/16
• 12/16
• 5/16

Ans : A

22. . If F(x)=x^3 is the distribution function and f(x) is the p.d.f. of a continuous random variable ‘X in the interval [0,1], then f(x)

• 3X^2
• X^4/4
• X^2
• Non of these

Ans: A

23. A random variable ‘x’ has the following probability functions :

• 2/3
• 5/6
• 1
• 0

Ans : B

24. If P(A)=p, P(B)=q,P(AnB) =r, thenP(AUB’) =

• 1+p+q-r
• 1-r-q
• 1-r+q
• 1+-q

Ans : D

25.Which of the following statements is/are correct?
(i) If events A and B are mutually exclusive then P(AnB) = P(A)P(B)
(ii) If events A and B are mutually exclusive then P(AUB) = P(A) + P(B)
(i) If events A and B are mutually independent then P(A n B) = P(A)P(B).

• option (i) is correct only
• both options (i) and (i) are correct only
• option (1) is correct only.
• option (i) is correct only

Ans : B

26. If a problem in Statistics is given to three students whose chances of solving it are 1/3, 2/3, 1/2 respectively.
Then the probability that the problem is solved if all of the three students try independently is

• 1/2
• 3/8
• 1/9
• 8/9

Ans : C

27. If P(B) = 0.4, P(A|B’) = 0.6, P(AIB) = 0.7 then P(A) =?

• 0.36
• 0.64
• 0.54
• 0.48

Ans : D

28. If the variance for the random variable x is 19 and  the second moment about the origin is 100 then the first moment about the origin is-

• 9
• 2
• 6
• 1

Ans : A

29. In an examination 40 percent students have failed in mathematics, 25 percent of the students have failed in chemistry and 10 percent have failed in both mathematics and chemistry. A student is selected at random. What is the probability that the student has failed in mathematics if it is known that he has failed in chemistry?

• 0.50
• 0.20
• 0.40
• 0.60

Ans : C

30. In a Binomial distribution with 6 independent trials, the probabilities of 3 and 4 successes are found to be 0.2457 and 0.0819 respectively. Find the parameter p of the binomial?

• 4/13
• 5/9
• 4/9
• 8/13

Ans : A

31.If a random variable follows Poisson distribution such that P(X=1) = P(X=2). Find the mean and variance of the distribution?

• Mean = 2 and Variance = 4
• Mean = 2 and Variance = 2
• Mean = 4 and Variance = 6
• Mean = 4 and Variance = 4

Ans : B

32. The average number of customers who appear at a counter of a certain bank per minute is two. Find the probability that during a given minute one or more than one customers appear? (Given e-2 = 0.135)

• 0.27
• 0.865
• 0.676
• 0.15

Ans : A

33. which one of the statements below is correct? If the significance level is equal to 0.1, it implies that we are

• 10% confident that the population mean will lie between the confidence limits
• If the sample size increases the confidence interval becomes wider.
• As the population standard deviation increases, the confident interval becomes narrower.
• A 90% confidence interval for the population mean is narrower than a 95% confidence interval for the population mean

Ans : D

34. In a sample of 9 observations, the sum of square of items from mean was 80.96. In another sample of 10 observations , the value was found to be 108.9. Test whether the difference in variance is significant at 5% level. Given F(8,9) = 3.18, F(9,8) = 1.15.

• Ho is accepted that the difference is not significant.
• Ho is rejected that the difference is not significant.
• Ho is accepted that the difference is significant.
• Ho is rejected that the difference is significant

Ans :A

35. In a sample of 8 observations, the sum of square of items from mean was 84.4 and mean is 72. In another sample of 10 observations , the value was found to be 102.6 and mean 68. Test whether the difference in mean is significant at 5% level. Given table value of t at 16 d.o.f is 3.07

• Ho is accepted that the difference is not significant.
• Ho is rejected that the difference is not significant.
• Ho is accepted that the difference is significant.
• Ho is rejected that the difference is significant

Ans : A

36. To fit a non-linear curve Y=a+bX+cX2 to a data, which of the following sums is not required?

• Sum of the values of variable ‘X^2’
• Sum of the values of variable ‘Y’
• Sum of the values of variable ‘XY’
• Sum of the values of variable ‘Y^2’.

Ans : B

37. If the letters of the word ‘VOWELS’ be arranged at random, what is the probability that there are exactly two letters between E and O?

• 1/4
• 1/3
• 1/6
• 1/5

Ans : D

38. A fair die is rolled. The probability of getting an even face given that face is less than 5 is given by:

• 0
• 0.5
• 0.6
• 0.8

Ans : C

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63. When the possible outcomes of an experiment are equally likely to occur, this we apply:

Ans: Classical probability

64. A number between 0 and 1 that is use to measure uncertainty is called:

Ans: Probability

65. Probability can be expressed

• Ration
• Fraction
• Percentage
• All of the above

Ans: (d) All of the above

66. The probability based on the concept of relative frequency is called:

• Empirical probability
• Statistical probability
• Both (a) and (b)
• Neither (a) nor (b)

Ans: (c) Both (a) and (b)

67. The probability of an event cannot be:

Ans: Less than zero

68. A measure of the chance that an uncertain event will occur:

Ans: A probability

69. A graphical device used to list all possibilities of a sequence of outcomes in systematic way is
called:

Ans: Tree diagram

70. A random experiment contains:
Ans: At least two outcomes

71. The probability of all possible outcomes of a random experiment is always equal to:

Ans: One

72. The outcome of tossing a coin is a:

Ans: Simple event

73. The result of no interest of an experiment is called:

Ans: Failure

74. A set of all possible outcomes of an experiment is called:

Ans: Sample space

75. The numbers of counting rules that are useful in determining the number of outcomes in an
experiment are:

Ans: Three

76. The events having no experimental outcomes in common is called:

Ans: Mutually exclusive events

77. A set of outcomes formed after some additional information is called:

Ans: Reduced sample space

78. The probability associated with the reduced sample space is called:

Ans: Conditional probability

79. An arrangement of objects without regard to order is called:

80. When each outcome of a sample space is as likely to occur as any other, the outcomes are called:

Ans: Equally likely

81. When certainty is involved in a situation, its probability is equal to:

Ans: One

82. If we toss a coin and P(H) = 2P(T), then probability of head is equal to:

Ans: 2/3

83. A letter is chosen at random from the word “Statistics”. The probability of getting a vowel is:

Ans: 3/10

84. Two books are to be selected at random without replacement out of four books. Then number of possible
selections are:

Ans: 6

85. Three books of different colours are to be arranged in a book-shelf. The possible arrangements are:

Ans: 6

86. When a die and a coin are rolled together, all possible outcomes are:

Ans: 12

87. Which of the following test is the test for equality of variance?

• Z-test (single mean)
• t-Test
• Paired t-test
• F-test

Ans: (d) F-test

88. A sample of 900 members has a mean of 3·4 and s.d. of 2·61. To test the sample taken from a large population of mean 3·25 and s.d. 2·61, the alternative hypothesis will be

• μ=3.25
• μ>3.25
• μ<3.25
• μ≠3.25

Ans: (a) μ=3.25

89.

Which of the following is the test for Goodness of fit?

• χ2 test
• Z test
• t test
• F test

Ans: (a) χ2 test (chi-square test)

90. Compute the coefficient of regression of X on Y from the following data:

x = 3 years, y = 32 years, x = 6 years, y = 8 years, r = +0.8

• 0.8
• 0.4
• 0.3
• 0.6

Ans: (d) 0.6

91. If sample size is large, then which of the following is true? (σ=population variance, s=sample variance)

σn≈sn

σn=2sn

σn≠sn

None of these

Ans:

92. The population mean and SD are estimated as 1000 and 257.298. But, from a sample of size 5, collected from population gives the sample mean as 885 and SD as 21. What is the standard error?

• 105.56
• 120.67
• 115.067
• 110

Ans:

93. What is our decision for a goodness-of-fit test with a computed value of chi-square of 1.273 and a critical value of 13.388?

• Do not reject the null hypothesis
• Reject the null hypothesis
• Unable to reject or not reject the null hypothesis based on data
• Should take a larger sample

Ans: (a) Do not reject the null hypothesis

94. A sample of 44 observations is selected from a normal population. The sample mean is 24, and the population standard deviation is 3. Conduct the following test of the hypothesis using the 0.05 significance level: H0: µ ≤ 23 versus H1: µ > 23. What is the value of the test statistic? (Round your answer to 2 decimal places)

• 4.32
• 5.22
• 2.21
• 1.10

Ans: (c) 2.21

Given information:

Hypotheses are ;

(a)

Alternative hypothesis shows that the test is one tailed (right-tailed). 

(b)

Since population standard deviation is known so single sample z test will be used. For 0.05 significance level, the critical value of z-test is 1.645. Rejection rule:

If z > 1.645, then reject the null hypothesis. 

(c)

The test statistics will be 

(d)

Since test statistics, value is greater than 1.645 so we reject H0. 

(e)

Since the test is right-tailed so P-value of the test is 

P(z > 2.21) = 0.0136

95. When the correlation coefficient between x and y is positive, then as variable x decreases, variable y remains ____________

• Increases
• Decreases
• Changes linearly
• does not change

Ans: (b) Decreases

96. In a sample of 9 observations, the sum of square of items from mean was 80.96. In another sample of 10 observations , the value was found to be 108.9. Test whether the difference in variance is significant at 5% level. Given F(8,9) = 3.18, F(9,8) = 1.15.

• Ho is accepted that the difference is not significant.
• Ho is rejected that the difference is not significant.
• Ho is accepted that the difference is significant.
• Ho is rejected that the difference is significant.

Ans:

97. A random sample of 100 recorded deaths in a city in India during the last year showed an average life span of 71.8 years. Does the average life span today is greater than 70 years. Is it a……

• One tail test, left tailed test
• One tail test, right tailed test
• two tailed test
• cannot be determined

Ans: (b) One tail test, right tailed test

Calculate the test statistic value.

From the standard z table values, observe that the critical value of z for the one tail test at the 5% level of significance is 

From the rejection region, observe that the Null hypothesis is rejected.

Therefore, conclude that the mean life span today is greater than 70 years.

98. If for the n=4 data,

∑x = 8 ∑y = 15 ∑X2 = 10 ∑Y2 = 14.75 and

the coefficient of correlation is 0.9881

then ∑XY =?

• 12
• 24
• 20
• its not available in the options

Ans: 24

99. The calculated t value is 9.10 whereas the table value is given as 2.262 for 5% level of significance and 9 degrees of freedom. The conclusion is drawn as null hypothesis is accepted. Then which of the following is true?

• Type 2 error
• Type 1 error
• No error is there
• No conclusion

Ans: Type 2 error

100. The Z statistic range is given as

• 0 to 1
• -1 to 0
• -1 to 1
• −∞ to ∞

Ans: −∞ to ∞

101. The SD of sample 1 and sample 2 are respectively 100 and 50. What is the value of F-statistic?

• 100
• 50
• 5000
• 200

Ans:

102. The sample mean is a

• consistent estimator of population mean
• biased estimator of population mean
• unbiased estimator of population mean
• both unbiased and consistent estimator of population mean

Ans: (d) both unbiased and consistent estimator of population mean

103. From the 12 data, we obtain xˉ=106 and s = 8.83. with hypothesis mean μ = 100 Then the calculated t-statistic value is:

• 2.30
• 2.45
• 2.85
• 2.35

Ans: (d) 2.353394 (https://www.assignmentexpert.com/homework-answers/mathematics/statistics-and-probability/question-212720)

104. If the Cov(X, Y)=3, Var(X)=4.5, and Var(Y)=5.5, then what is the value of the correlation coefficient?

• 0.203
• 0.403
• 0.503
• 0.603

Ans: (d) 0.603

105. The value of the correlation coefficient ranges between

• 0 to 1
• -1 to 0
• -1 to 1
• -2 to 2

Ans: (c) -1 to 1

106. If the two sets of data (X and Y) are independent, then the regression line will be

• parallel to x-axis
• perpendicular to x-axis
• at a slope of
• tan⁡π4
• None of these

Ans: (b) perpendicular to x-axis

107. For a positive correlation between two sets of data, which of the following is true?

• r<0
• r=0
• r>0
• r=1

Ans: (d) r=1

108. The regression coefficients are calculated as 0.67 and 0.2. What is the value of the correlation coefficient?

• 0.37
• -0.37
• 0.21
• -0.21

Ans: (a) 0.37

109. The test statistic for a two-sided significance test for a population mean is z = -2.12. The hypotheses are H0: μ=10 versus H1:μ>10. What is the corresponding p-value?

• 0.983
• 0.887
• 0.175
• 0.589

Ans: (a) 0.0983 (not sure)

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