Stats MCQ

Here are the questions along with their correct answers:

1. What is the formula for calculating the standard deviation?

(b) Square root of the variance

2. What is the difference between a sample and a population?

(a) A sample is a subset of a population

3. A survey of 700 freshmen in OSU found that 60% got their first item program. The 60% represent

(a) Statistic

4. What is the mode of the data set?

5. Compute the mean of the data set.

(d) 35.27

6. Find the median of the data set.

7. What is the first quartile (Q1) of the data set?

(b) 24

8. What is the third quartile (Q3) of the data set?

(b) 41

9. Compute the inter-quartile range (IQR) of the data set.

(d) 20

10. What is the range of the data set?

(a) 25

11. What is the 90th percentile of the data set?

(b) 44

12. What is the mode of the given dataset?

13. Calculate the median of the dataset.

(d) 30

14. Calculate the mean of the dataset.

(b) 37.06

15. Identify the minimum and maximum values of the data set.

(b) Min: 2, Max: 10

16. Identify the dataset’s lower and upper quartiles (LQ, UQ).

17. Find the interquartile range (IQR) of the dataset.

18. The sampling technique that groups objects into homogeneous groups is referred to as......

19. Which of the measures of central tendency is considered not to be unique......

(a) Mode

20. Which of the following summary statistics is heavily influenced by outliers?

c. Mean

21. You have collected the weights (in kilograms) from a sample of 5 students. You find the standard deviation of weights is equal to 0. What does this tell you?

b. All of the values in the dataset are the same.

22. Which of the following is NOT a discrete random variable?

a. Diameter of a piston, rounded to the nearest millimeter.

23. Suppose \( P(A) = 0.60 \), \( P(B) = 0.20 \), and \( P(A \text{ or } B) = 0.68 \). Based on these numbers, we can conclude that A and B are:

b. Not mutually exclusive and independent.

24. Suppose \( P(A|B) = 0.25 \), \( P(B) = 0.50 \), and \( P(A) = 0.25 \). Based on these numbers, we can conclude A and B are:

b. Not mutually exclusive and independent.

25. Suppose \( X \) is a continuous random variable with cumulative distribution function \( F(X) \) and \( C \) is a value in the domain. \( F(C) \) represents:

d. The probability that \(X \leq C\)

26. Suppose \( X \) is a continuous random variable with cumulative distribution function \( F(X) \). To compute \( P(X>8) \), we would find:

a. 1 - \(F(8)\)

27. Consider the probability density function for \( 0 < X < \infty \). What integration technique would find the mean of \( X \)?

a. Integration by parts

28. Which of the following is a requirement of a binomial random variable?

b. Number of successes fixed in advance.

29. For a binomial random variable with a small value of \( n \) and \( p > 0.50 \), the probability distribution of \( X \) will be:

b. Left skewed

30. Suppose \( X \) is Normally distributed with \( \mu = 80 \). Suppose that 99.7% of observations fall within 65 to 95. From this, we can conclude the standard deviation is roughly:

a. 5

31. Suppose for a given dataset the mean is 40 and the median is 70. From this we can conclude:

b. The distribution is left skewed

32. Find the area under the normal curve outside \( Z = -1.51 \) to \( Z = -0.75 \). Round to four decimal places:

a. 0.1611

33. Let \( Z \sim N(0,1) \). Find a constant \( C \) for which \( P(-C < Z < C) = 0.2139 \):

d. C = 0.271

34. If \( X \) and \( Y \) are independent random variables with means \( \mu_1 = 9.5 \) and \( \mu_2 = 6.8 \), and standard deviations \( \sigma_1 = 0.4 \) and \( \sigma_2 = 0.1 \), find the mean and standard deviation of \( X + 4Y \):

d. 36.7, 0.56

35. A chemical process is run 15 times, and the yield is measured each time. This process is:

a. Tangible

36. The population consists of all the registered voters in the state. This population is:

a. Tangible

37. True or false: A simple random sample is guaranteed to reflect exactly the population from which it was drawn.

b. False

38. To determine whether a sample should be treated as a simple random sample, which is more important?

b. A good knowledge of the process that generated the data

39. A vendor converts the weights on packages from pounds to kilograms (1 kg ≈ 2.2 lb). How does this affect the mean and standard deviation?

c. Mean and standard deviation both divided by 2.2

40. The type of data that consists of labels or names and cannot be ranked is:

(a) Nominal

41. Temperature measurements taken from a city each day are an example of:

(b) Interval data

42. The number of students in a class is an example of:

(d) Ratio data

43. What type of sampling technique divides a population into groups and randomly selects some groups?

44. The sampling technique that groups objects into heterogeneous groups is referred to as......

(b) Cluster sampling

45. Which measure of central tendency is most affected by extreme values?

(a) Mean

46. If a data set has two modes, it is:

(b) Bimodal

47. Which measure of central tendency is best for categorical data?

48. The difference between the highest and lowest values in a data set is called:

(b) Range

49. The square root of variance is ...

(a) Standard deviation

50. A high standard deviation in a data set implies:

(b) Data points are spread out from the mean.

51. Which of the following measures spread out the data in a data set?

(d) Standard deviation

52. The probability of an impossible event is:

(A) 0

53. The sum of probabilities of all possible outcomes in an experiment is:

(B) 1

54. If two events cannot occur at the same time, they are:

55. In a deck of 52 cards, the probability of drawing an ace is:

(A) \(\frac{1}{13}\)

56. From this plot, the median weight is the same as

(b) Q2

57. If the box spans from 15 to 30 grams, what percentage of the data is contained within this range?

(b) 50%

58. Are there any potential outliers in this dataset? Justify your answer.

(a) Yes; values extend beyond the whiskers.

59. How would the box plot change if a new data point of 70 were added?

(d) A new outlier would appear at 70.

60. What does the length of the left whisker (extending to 10 grams) indicate about the data distribution?

61. If the median = mean = mode, is the data symmetric, skewed left, or skewed right? Explain.

62. Which category has the smallest portion in the pie chart?

63. If the total data is 100 students, what percentage of them prefer burgers?

(a) 30

64. The 95% confidence interval for the mean of a process was (85.71, 97.93). If the estimate of the mean \( \bar{X} \) was increased, what would happen to the center of the confidence interval and the width of the interval?

d. The center would increase, and the width would remain unchanged.

65. An engineer constructed a 95% confidence interval for the mean temperature of a furnace to be \( (a, b) \). Give the correct interpretation of the interval.

b. The method used to get the interval, when used over and over, produces intervals that include the true population mean 95% of the time.

66. A teacher conducted a z-test of the hypotheses \( H_0 : \mu = 22 \) vs \( H_A : \mu > 22 \), and found a p-value of 0.03. This means:

b. If \( \mu = 22 \), the probability of obtaining a sample mean of \( \bar{X} \) or larger is 0.03.

67. Suppose a two-tailed alternative hypothesis has been set up for a 1% level of significance, and the test statistic is found to be \( Z = -1.7 \). What will the p-value be?

c. 0.0892

68. When computing a 90% confidence interval using the student t-distribution with a sample size of 20, the t-critical value is:

d. 1.729

69. To increase the probability of rejecting a false null hypothesis, we could:

a. Increase \( n \); decrease Type II error rate (\(\beta\))

70. If a random variable comes from a normal distribution with \(\sigma\) unknown, the sampling distribution of the sample mean for a sample of size 15 will be:

d. Following a t-distribution.

71. Based on this quantile plot, we can conclude the population that generated the plot:

d. Does come from a normal distribution due to the linear pattern.

72. Committing a Type I error here would be:

c. Concluding the average life is not 1.5 years when it actually is.

73. Committing a Type II error here would be:

d. Concluding it is plausible the average life is 1.5 years when it is not.

74. If an engineer is trying to prove that a new method of measuring speed is more effective than a traditional one, he/she will conduct a:

a. One-tailed test.

75. The value added and subtracted from a point estimate in order to develop an interval estimate of the population parameter is known as the:

b. Margin of error.

76. A single numerical value obtained as an estimate for a population parameter is known as:

b. Point estimate.

77. If the p-value is less than \(\alpha\), when comparing the means of 2 sets of quantitative data, your conclusion would be:

c. Reject the null hypothesis.

78. A numerical estimate of a sample feature, such as a sample mean, is known as:

d. A statistic.

79. Evaluate the integral: \(\int_{-\infty}^{\infty} f(x) \, dx\) is always equal to:

b. 1

80. A continuous random variable \(X\) has p.d.f. \(f(x) = e^{-x}\), for \(x \geq 0\). Find \(P(X > 1)\) (3 decimal places).

Answer: 0.368

81. Let \(Z = 5X - 2Y\), where \(\text{Var}(X) = 0.2\), \(\text{Var}(Y) = 0.5\). Find the variance of \(Z\).

Answer: 7.0

82. Let \(Y\) have distribution:

Answer: 1.714

83. Let \(X\) be a discrete random variable with pmf:

\begin{array}{c|cccc}

X & 0 & 1 & 2 & 3 \\

\hline

P(X) & \frac{1}{8} & \frac{1}{2} & \frac{1}{8} & \frac{1}{4} \\

\end{array}

Find \(P(X \leq 1)\).

Answer: 0.625

84. The area under the standard normal curve is:

b. 1

85. Which of the following is true for a probability mass function (PMF)?

d. The sum of the probabilities of all values is 1

86. For a continuous probability density function, the probability of any single point is:

a. 0

87. For continuous random variables, the probability at a single point is:

a. Zero

88. One property of a probability density function is:

c. The integral is equal to 1

89. Find the probability that the donor has blood type \( O \).

90. Find the probability that the donor has blood type \( A \) or type \( O \).

(d) 19/26

91. Find the probability that the donor has blood type \( B \) or is Rh negative.

(a) 43/65

92. Find the probability that the donor is Rh positive given that he has blood type \( B \).

93. Find the probability that the donor has blood type \( O \) given that he is Rh negative.

94. A random variable \( Z \) has:

\begin{array}{c|cccc}

Z & 1 & 2 & 3 & 4 \\

\hline

P(Z) & 0.1 & 0.2 & 0.3 & 0.4 \\

\end{array}

Compute the mean and standard deviation of \( Z \).

Answer: Mean = 3.0, Standard deviation ≈ 1.0

95. Let \( p(X) = K(X-1) \) for \( X = 3,4,5 \). Determine the value of \( K \) (3 decimal places).

Answer: 0.167

96. Let the p.d.f. be \( f(x) = cxe^{-x}, x > 0 \). Find the value of \( c \).

Answer: 1

97. Given:

\begin{array}{c|ccccc}

x & 0 & 2 & 4 & 6 & 8 \\

\hline

P(x) & 0.17 & 0.34 & k & 0.20 & 0.05 \\

\end{array}

Find:

- The value of \( k \): 0.24

- The mean: 3.14

- The variance: 5.04

- \( P(X > 4) \): 0.49

- \( P(X \leq 2) \): 0.51

- \( P(X < 6) \): 0.75

- \( P(2 < X \leq 6) \): 0.54

98. For the p.d.f. \( f(x) = \frac{k - x}{4} \) for \( 1 \leq x \leq 3 \), find:

- The value of \( k \): 3

- \( P(X \leq 2.5) \): 0.8125

- Mean: 1.833

- Variance: 0.3056

99. Let \( f(x) = \frac{3}{10}(3x - x^{2}) \), for \( 0 \leq x \leq 2 \). Find the mean and variance of the continuous random variable \( X \).

Answer: Mean = 1.0, Variance ≈ 0.2

100. If the probability density of \( X \) is given by \( f(x) = 2(1 - x) \) for \( 0 < X < 1 \), find \( E(2X + 1) \).

Answer: 1.6667