Statistics and Probability (STA301)

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

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

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

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

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

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

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

5. 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

6. 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

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

8. 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

9. 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

10. 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

11. 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

12. 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. \(\int\limits_a^b \int\limits_c^d f(x,y) \,dy \,dx\)

13. 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

14. 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.

15. 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

16. 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

17. Atmosphere pressure is the example of:

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

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

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

19. 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

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

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

21. 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

22. 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

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

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

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

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

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

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

26. In a binomial experiment successive trials are:

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

27. 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

28. 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

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. 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

31. 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

32. 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

33. 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

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

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

35. 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

36. In normal distribution, β₂ = ________:

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

37. 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

38. 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

39. 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

40. Value of harmonic mean is lower than ________.

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

41. 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

42. 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

43. E(4X + 5) = ________

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

44. 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

45. 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

46. The shape of the Poisson distribution is:

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

47. 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

48. 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

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

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

50. 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

51. 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

52. 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

53. 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)

54. 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

55. 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

56. 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

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

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

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

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

60. The value of x² can never be

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

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

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

62. 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

63. Hypergeometric probability distribution has

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

64. The probability of an event always lies between:

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

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

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

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

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

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

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

68. 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

69. 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

70. 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

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

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

72. 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

73. 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

74. Positive square root of variance is known as:

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

75. 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

76. 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

77. For positively skewed distribution
    Mean ________ Median ________ Mode

    1. ≠
    2. =
    3. <
    4. >

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

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

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

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

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

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

81. 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)

82. In normal distribution, β₁ = ________:

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

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