Statistics and Probability (STA301)

1. If f(x, y) is bivariate probability density function of continuous random variables X and Y then marginal density function of Y i.e. h(y) is:

   1. \(\int\limits_{-\infty}^\infty f(x,y)\,dx\)
   2. \(\int\limits_{-\infty}^\infty f(x,y)\,dy\)
   3. \(\int\limits_{-\infty}^\infty \int\limits_{-\infty}^\infty f(x,y) \,dx \,dy\)
   4. \(\int\limits_a^b \int\limits_c^d f(x,y) \,dy \,dx\)

2. If P(X=x)=0, then f(x) is a ________ probability function.

   1. Univariate
   2. Bivariate
   3. Continuous
   4. Discrete

3. From the table given below, how many students obtained marks 80 or above:
   

   Marks	f
   50-59	5
   60-69	8
   70-79	7
   80-89	5
   90-100	3

   1. 15
   2. 8
   3. 5
   4. 3

4. For a particular data set the Pearson's coefficient of skewness is greater than zero. What will be the shape of distribution?

   1. Negatively skewed
   2. J-shaped
   3. Symmetrical
   4. Positively skewed

5. In a set of 10 values all the values are 5, what will be the P₅₀?

   1. 2
   2. 5
   3. 10
   4. 20

6. When the population is normally distributed, population standard deviation is unknown and the sample size is n = 15, the confidence interval for the population mean is based on

   1. the z (standard normal) distribution
   2. the t distribution
   3. the Binomial distribution
   4. the Poisson Distribution

7. The number of parameters in a Poisson distribution is (are):

   1. 1
   2. 0
   3. 2
   4. 3

8. In a binomial experiment successive trials are:

   1. Dependent
   2. Independent
   3. Both dependent and independent
   4. None of the given

9. Which of the following is a characteristic of the normal distribution:

   1. It is a skewed distribution
   2. It is bell-shaped
   3. It is not asymptotic
   4. It is leptokurtic

10. Which type of the curve is represented by the following shape?
    

    1. Bell shape curve
    2. semi-symmetrical curve
    3. Positively skewed curve
    4. Negatively skewed curve

11. The degrees of freedom for a t-test with sample size 14 is:

    1. 14
    2. 13
    3. 7
    4. 0

12. A fair coin is tossed three times. What is the probability that at least one head appears?

    1. 1/8
    2. 7/8
    3. 4/8
    4. 6/8

13. A ________ is a range of numbers inferred from the sample that has a certain probability of including the population parameter over the long run.

    1. Hypothesis
    2. Lower limit
    3. Confidence interval
    4. Probability limit

14. From the table given below, how many students obtained marks between 60 and 69?
    

    Marks	f	Mid-Points
    50-59	5	54.5
    60-69	7	64.5
    70-79	8	74.5
    80-89	5	84.5

    1. 64.5
    2. 12
    3. 60
    4. 7

15. Positive square root of variance is known as:

    1. Mean deviation
    2. Range
    3. Standard Deviation
    4. All of the given

16. Serious disadvantage of using range as a measure of dispersion is that it is based on only:

    1. Minimum Values
    2. Maximum Values
    3. Both Minimum and Maximum values
    4. None of the given

17. An experiment was conducted to estimate the mean yield of a new variety of wheat. A sample of 20 plots gave a mean yield of 2.9, and a 95% confidence interval of (2.48, 3.32). This means:

    1. We are sure the true mean yield of this new variety is between 2.48 and 3.32
    2. We are 95% confident that the true mean yield of this variety is 2.9
    3. We are 95% confident that the true mean yield of this variety is between 2.48 and 3.32
    4. None of the given

18. For test hypothesis Ho:μ 1 = μ 2 and H1:μ1 < μ2, the critical region at 0.05 level of significance and n > 30

    1. Z less and equal 1.96
    2. Z> 1.96
    3. Z>1.645
    4. Z<-1.645

19. If a significance level of 5% is used rather than 1%, the null hypothesis is:

    1. Less likely to be rejected
    2. Just as likely to be rejected
    3. More likely to be rejected
    4. None of the given

20. Which of the following is correct option to find P(X + Y ≤ 1) ?

    1. f(0, 0) + f(0, 1) + f(1, 2)
    2. f(2, 0) + f(0, 1) + f(1, 0)
    3. f(0, 0) + f(1, 1) + f(1, 0)
    4. f(0, 0) + f(0, 1) + f(1, 0)

21. If S.D(X) = 5 then S.D(\(2X + 5 \over 2\)) = ________

    1. 5
    2. 10
    3. 15
    4. 7.5

22. The proportion of males in Pakistan is at least 0.48, the alternative hypothesis H₁ is

    1. P ≤ 0.48
    2. P = 0.48
    3. P < 0.48
    4. P ≥ 0.48

23. In a cumulative frequency polygon, the cumulative frequency of each class is plotted against:

    1. Mid-point
    2. Lower class boundary
    3. Upper class boundary
    4. Upper class limit

24. If total probability of joint probability distribution f(x, y) is ONE then what will be the total probability of its marginal pdf h(y)?

    1. One
    2. Less than one
    3. Anywhere between zero and one
    4. Depend on given values

25. β is the probability of:

    1. Accept H0/H0 is false
    2. Reject H0/H0 is true
    3. Accept H0/H0 is true
    4. Reject H0/H0 is false

26. In hyper geometric distribution formula, N= number of units in the population, n = number of units in the sample, and k is defined as:

    1. number of failures in the sample
    2. number of failures in population
    3. number of successes in sample
    4. number of successes in population

27. Dividing mean deviation by mean we get a pure number known as

    1. Co-efficient of mean deviation
    2. Co-efficient
    3. Co-efficient of quartile deviation
    4. None of the given

28. A diagram in which each circle is divided into different sectors is called

    1. Simple Bar Chart
    2. Multiple Bar Chart
    3. Component Bar Chart
    4. Pie Chart

29. ________ is a range of numbers inferred from the sample that has a certain probability of including the population parameter over the long run.

    1. Hypothesis
    2. Lower limit
    3. Confidence interval
    4. Probability limit

30. For positively skewed distribution
    Mean ________ Median ________ Mode

    1. ≠
    2. =
    3. <
    4. >

31. What is the graphical shape of the chi-square distribution?

    1. Positively skewed
    2. Negatively skewed
    3. Uniformly distributed
    4. Norrmally distributed

32. In sampling from a large population with sigma = 35, the standard error of the mean is found to be 4 next. The size of the sample used is:

    1. 56
    2. 77
    3. 25
    4. 15

33. If X_m is maximum and X₀ is minimum value in a data set then formula for co-efficient of dispersion is as:

    1. \( {X_m - X_0 \over X_0 - X_m} \)
    2. \( {X_m - X_0 \over X_m + X_0} \)
    3. \( {X_0 + X_m \over X_0 - X_m} \)
    4. \( {X_m + X_0 \over X_m - X_0} \)

34. A good way to get a small standard error is to use a ________.

    1. Repeated sampling
    2. Small sample
    3. Large sample
    4. Large population

35. Dividing interquartile range by 2 we get:

    1. Co-effient of range
    2. Mid range
    3. Semi Inter Quatile Range
    4. None of the given

36. To determine a sample size which test we use ________.

    1. z
    2. t
    3. F
    4. None of the given

37. The total number of samples when sampling is done without replacement is equal to:

    1. \(N^n\)
    2. \(C_{n}^N\)
    3. \(N - n \over N - 1\)
    4. 1

38. In normal distribution, β₁ = ________:

    1. 0
    2. 1
    3. 2
    4. 3

39. The conditional probability function f(x|1) = ________

    1. f(1,1)
    2. f(x,1)
    3. \(f(x,1) \over h(1)\)
    4. \(f(x,1) \over h(x)\)

40. Which one of the following is not a type of frequency curve?

    1. The symmetrical frequency curve
    2. The extremely skewed frequency curve
    3. The U-shaped frequency curve
    4. Frequency polygon

41. What is the median for the following data:
    
    1, 1, 2, 2, 5, 5, 7, 7, 7

    1. 1
    2. 2
    3. 5
    4. 7

42. For Platykurtic distribution, b2 (moment ratio) will be:

    1. Greater than 3
    2. Less than 3
    3. Equal to 3
    4. Equal to zero

43. If a significance level of 1% is used rather than 5%, the null hypothesis is:

    1. More likely to be rejected
    2. Less likely to be rejected
    3. Just as likely to be rejected
    4. None of the given

44. In hypothesis testing, suppose the critical value is |Z| > 2.33. This value shows that the applied test is ________?

    1. One tailed test
    2. Two tailed test
    3. Right tailed test
    4. Left tailed test

45. The shape of the Poisson distribution is:

    1. Negatively skewed
    2. Positively skewed
    3. U shaped
    4. J shaped

46. In normal distribution, β₂ = ________:

    1. 1
    2. 2
    3. 3
    4. 0

47. In a one-way ANOVA:

    1. The interaction term has (c - 1)(n - 1) degrees of freedom
    2. An interaction term is given
    3. An interaction effect can be tested.
    4. There is no interaction term

48. Hypergeometric probability distribution has

    1. (n, k) parameter
    2. (N) parameter
    3. (N, n, N − k) parameter
    4. (N, n, k) parameter

49. The mean of a distribution is 30, the mode is 24 and the standard deviation is 4, then the coefficient of skewness will be:

    1. Less than zero
    2. Equal to zero
    3. Greater than zero
    4. None of the given

50. For α = 0.01, the critical values of z for two tailed test are equal to:

    1. −2.58 and +2.58
    2. −2.33 and +2.33
    3. −1.645 and +1.645
    4. −1.96 and +1.96

51. Which one is the measure of central tendency:

    1. Variation of the distribution
    2. Average of the distribution
    3. Scatterness of the distribution
    4. Dispersion of the distribution

52. In how many ways, a team of 11 players can be chosen from a total of 16 players?

    1. 4368
    2. 2426
    3. 5400
    4. 2680

53. Suppose that a sample space S consists of four simple events: A, B, C, and D. That is S = {A, B, C, D}. If P(A) = .4, P(B) = .3, P© = .2, what is P(D)?

    1. 0.2
    2. 0.9
    3. 0.1
    4. None of the given

54. Which of the following is an unbiased estimator of population σ²?

    1. \(\sum (x - \bar x)^2 \over n\)
    2. \(\sum (x - \bar x)^2 \over n-2\)
    3. \(\sum (x - \bar x)^2 \over n(n-1)\)
    4. \(\sum (x - \bar x)^2 \over n-1\)

55. Which of the following is a measure of absolute dispersion?

    1. Skewness
    2. Mean Deviation
    3. Coefficient of variation
    4. Kurtosis

56. The weights of 7 students were measured and recorded as follows:
    53.0, 54.5, 57.2, 51.7, 57.4, 50.5, 52.8
    The stem part is:

    1. 5, 4, 5, 8, 7
    2. 523, 50, 51, 54, 52, 57, 58
    3. 53, 54, 57, 51, 50, 52
    4. 0, 5, 2, 7, 4, 5, 8

57. The value of x² can never be

    1. Zero
    2. Less than 1
    3. Greater than 1
    4. Negative

58. Which of the following statements is NOT true?

    1. The t distribution is symmetric about zero
    2. The t distribution is more spread out than the standard normal distribution
    3. As the degrees of freedom get smaller, the t-distribution's dispersion gets smaller
    4. The t distribution is mound-shaped

59. Which of the following is not a characteristic of the arithmetic mean?

    1. It is influenced by extreme values
    2. Fifty percent of the observations are larger than the median
    3. The sum of the squared deviations from the mean is always minimum
    4. The sum of the observations from the mean is zero

60. When we draw the sample with replacement, the probability distribution to be used is:

    1. Binomial
    2. Hypergeometric
    3. Binomial & hypergeometric
    4. Poisson

61. A judge can acquit a guilty person is the example of

    1. Type I error
    2. Type II error
    3. Correct decision
    4. No information regarding this

62. Which scale of measurementis more suitable in the following example?
    The number of response given in a specified time period.

    1. Ordinal Scale
    2. Interval Scale
    3. Ratio Scale
    4. Nominal Scale

63. What is 'f_m' in the formula of mode:

    1. First Frequency
    2. Last Frequency
    3. Maximum Frequency
    4. Minimum Frequency

64. Atmosphere pressure is the example of:

    1. Constant
    2. Qualitative variable
    3. Quantitative variable
    4. None of the given

65. The parameter of the chi-square distribution is ________.

    1. v
    2. v − 1
    3. v − 2
    4. v − p

66. E(4X + 5) = ________

    1. 16 E(X)
    2. 16 E(X) + 5
    3. 12 E(X)
    4. 4 E(X) + 5

67. What factor determines the shape of the t-distribution?

    1. Degree of freedom
    2. Critical value
    3. Frequency of data
    4. Probability

68. In case of a frequency distribution, the 7th decile is given by the formula:

    1. l+ h/f (n/10-10c)
    2. l+ h/f (7n/10-c)
    3. l+ 7h/h (n/10-c)
    4. l+ f/h (7n/10-c)

69. The probability of an event always lies between:

    1. 0 and −∞
    2. −1 and +1
    3. −∞ and + −∞
    4. 0 and 1

70. Value of harmonic mean is lower than ________.

    1. Arithmetic Mean
    2. Geometric Mean
    3. Both arithmetic mean & geometric mean
    4. None of the given

71. An automobile is running, during the first 60 Km, at the rate of 10 Km/hr. During the second 60 Km at the rate of 30 Km/hr, while during the third 60 Km, its speed was 40 Km/hr. What method is more appropriate to calculate the average speed?

    1. Median
    2. Arithematic mean
    3. Harmonic mean
    4. Geometric mean

72. The covariance of a random variable with itself is:

    1. Zero
    2. One
    3. Its variance
    4. Its correlation

73. If mean of X² distribution is k then variance will be:

    1. k²
    2. 2k
    3. 1/k
    4. k

74. From the given data, which one of the following chart we can draw:

    Year	Turnover
    1965	350000
    1966	42000
    1967	43500
    1968	48000
    1969	48500

    1. Simple bar chart
    2. Multiple bar charts
    3. Component bar chart
    4. Pie chart

75. Total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is equal to

    1. 60
    2. 120
    3. 720
    4. 520

76. The degrees of freedom for a t-test with sample size 6 is:

    1. 1
    2. 3
    3. 5
    4. 7

77. Which statement is true for a discrete Uniform distribution?

    1. It is absolutely symmetrical distribution.
    2. Sometimes it becomes symmetrical.
    3. Its graphical shape of the data is like "U".
    4. Cannot say anything without observing the actual data.

78. The data which has not undergone any statistical treatment is called:

    1. Discrete data
    2. Secondary data
    3. Primary data
    4. Qualitative data

79. By definition f(y | x) = ________

    1. f(y)
    2. f(x,y)
    3. \(f(x,y) \over h(x)\)
    4. \(f(x,y) \over h(y)\)

80. An urn contains 4 red balls and 6 green balls. A sample of 4 balls is selected from the urn without replacement. It is the example of:

    1. Binomial distribution
    2. Hypergrometric distribution
    3. Poisson distribution
    4. Exponential distribution

81. Number of Hours	Frequency
    0 - 9	40
    10 - 19	50
    20 - 29	70
    30 - 39	40

    What is its class interval?

    1. 9
    2. 10
    3. 11
    4. 5

82. For any two estimators T1 and T2, if VAR(T1) < VAR(T2), then T1 is:

    1. Consistent
    2. Unbaised
    3. Sufficient
    4. Efficient

Select a course code for Objective Questions:

ACC311 ACC501 BIF101 BIF401 BIF501 BIO101 BIO201 BIO202 BIO203 BIO204 BIO301 BIO302 BIO303 BIO401 BIO503 BNK603 BT101 BT201 BT301 BT302 BT401 BT402 BT403 BT404 BT405 BT406 BT501 BT504 BT505 BT604 CHE201 CHE301 CS001 CS101 CS201 CS202 CS204 CS301 CS302 CS304 CS311 CS401 CS402 CS403 CS408 CS501 CS502 CS504 CS506 CS508 CS601 CS602 CS604 CS605 CS607 CS609 CS610 CS614 CS707 CS710 ECO401 ECO402 ECO403 ECO404 EDU302 EDU401 EDU402 EDU501 EDU601 ENG101 ENG201 ENG301 ENG501 FIN611 FIN624 FIN625 ISL201 IT430 MCM101 MCM301 MGMT623 MGT101 MGT211 MGT301 MGT401 MGT402 MGT501 MGT502 MGT503 MGT510 MGT601 MGT603 MKT501 MKT610 MKT630 MTH101 MTH202 MTH302 MTH501 MTH601 MTH634 PAK301 PHY101 PHY301 PSY101 PSY403 PSY502 PSY512 SOC101 SOC301 SOC401 STA301 STA630 STA641 ZOO102 ZOO401 ZOO501 ZOO502 ZOO503 ZOO504 ZOO505 ZOO506 ZOO510 ZOO630

Select a course code for Subjective Questions:

ACC311 CS001 CS101 CS201 CS301 CS607 CS701 CS702 CS703 CS704 CS707 CS708 CS710 CS711 CS718 CS721 CS724 ECO401 ECO403 ENG101 ENG301 MGT703 MTH501 STA301