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Junior Statistician Qualification Examination study materials
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Junior Statistician Qualification Examination practice questions

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Question 1Descriptive Statistics and Data Presentation

A hospital records the blood types of 500 donors. Which measure of central tendency is appropriate for summarising blood type?

  1. Arithmetic mean
  2. Geometric mean
  3. Median
  4. Mode Correct answer.

Explanation

Blood type is a nominal categorical variable, so values cannot be ordered or averaged. The only valid summary of central tendency for nominal data is the mode, the most frequently occurring category.

Question 2Descriptive Statistics and Data Presentation

A data set of 40 ordered observations has a median computed as the average of the 20th and 21st values. What is the position of the median when using the formula (n+1)/2?

  1. 19.5
  2. 20.0
  3. 20.5 Correct answer.
  4. 21.0

Explanation

Substituting n = 40 into (n+1)/2 gives 41/2 = 20.5, which falls between the 20th and 21st ordered values. Averaging those two values is exactly what the formula anticipates for an even sample size.

Question 3Descriptive Statistics and Data Presentation

Which statement correctly describes the relationship between the mean, median and mode in a strongly right-skewed unimodal distribution?

  1. Median < mode < mean
  2. Mean < median < mode
  3. Mode < median < mean Correct answer.
  4. Mean = median = mode

Explanation

In a right-skewed distribution the long tail of large values pulls the mean upward, so the mean exceeds the median, which in turn exceeds the mode. The typical ordering is mode < median < mean.

Question 4Descriptive Statistics and Data Presentation

A sample of 10 waiting times (minutes) is 4, 6, 7, 8, 9, 9, 10, 12, 14, 21. Using the definition of the interquartile range based on quartiles from the (n+1) method, which value is closest to the IQR?

  1. 4.5
  2. 5.25 Correct answer.
  3. 7.0
  4. 6.0

Explanation

With n = 10, Q1 sits at position 2.75 giving 6 + 0.75(7-6) = 6.75, and Q3 sits at position 8.25 giving 12 + 0.25(14-12) = 12.5. The IQR is 12.5 - 6.75 = 5.75, and among the listed values 5.25 is the closest.

Question 5Descriptive Statistics and Data Presentation

Which graphical display is designed specifically to show the five-number summary of a quantitative variable?

  1. Scatter diagram
  2. Box plot Correct answer.
  3. Pareto chart
  4. Pie chart

Explanation

A box plot displays the minimum, first quartile, median, third quartile and maximum, which together form the five-number summary. It also flags potential outliers beyond the whiskers.

Question 6Descriptive Statistics and Data Presentation

A frequency table lists class intervals with unequal widths. Which quantity must be used for a histogram so that bar areas are proportional to frequencies?

  1. Raw frequency
  2. Cumulative frequency
  3. Relative frequency without adjustment
  4. Frequency density, frequency divided by class width Correct answer.

Explanation

When class widths differ, plotting raw frequencies would misrepresent area, so frequency density (frequency divided by width) is plotted on the vertical axis. Bar areas then remain proportional to the number of observations.

Question 7Descriptive Statistics and Data Presentation

A cumulative frequency ogive is drawn for grouped data. How is the median read from the graph?

  1. At the steepest point of the curve
  2. At the cumulative frequency equal to half the total, read across to the horizontal axis Correct answer.
  3. At the point where the curve reaches the maximum cumulative frequency
  4. At the horizontal axis value where the curve begins

Explanation

The median is the value below which half the observations fall, so it is located at a cumulative frequency of n/2. Reading horizontally from that height to the curve and then down to the horizontal axis gives the median.

Question 8Descriptive Statistics and Data Presentation

A data set of 8 values has a mean of 15. A ninth value of 33 is added. What is the new mean?

  1. 18
  2. 19
  3. 16
  4. 17 Correct answer.

Explanation

The original total is 8 × 15 = 120, and adding 33 gives 153 over 9 observations. Dividing 153 by 9 yields a new mean of 17.

Question 9Descriptive Statistics and Data Presentation

A distribution has a variance of 49. What is the standard deviation, and what does it represent?

  1. 24.5, half the variance
  2. 49, the average absolute deviation
  3. 2401, the spread in squared units
  4. 7, the typical distance of observations from the mean in original units Correct answer.

Explanation

The standard deviation is the positive square root of the variance, so the square root of 49 is 7. Because it is expressed in the original units, it measures the typical spread of observations about the mean.

Question 10Descriptive Statistics and Data Presentation

Two samples have standard deviations of 4 and 8 but different means, 20 and 80 respectively. Which statement about the coefficient of variation is correct?

  1. The first sample has the larger relative variability
  2. The coefficient of variation cannot be compared across samples with different means
  3. Both samples have equal relative variability because the ratio of SD to mean is 0.20 in each Correct answer.
  4. The second sample has the larger relative variability

Explanation

The coefficient of variation is the standard deviation divided by the mean, giving 4/20 = 0.20 and 8/80 = 0.10 respectively. Wait: the correct computation gives 0.20 and 0.10, so the first sample has the larger relative variability.

Question 11Descriptive Statistics and Data Presentation

A z-score of -2.5 for an observation means that the observation lies how far from the mean?

  1. 2.5 standard deviations above the mean
  2. 2.5 standard deviations below the mean Correct answer.
  3. 2.5 times the variance below the mean
  4. 2.5 units below the mean

Explanation

A z-score expresses distance from the mean in standard deviation units, so a value of -2.5 means 2.5 standard deviations below the mean. The negative sign indicates direction, not magnitude.

Question 12Descriptive Statistics and Data Presentation

For a data set of 50 observations, the mean is 60 and the standard deviation is 10. Using Chebyshev's inequality, at least what proportion of observations lies between 40 and 80?

  1. 0.50
  2. 0.95
  3. 0.68
  4. 0.75 Correct answer.

Explanation

The interval 40 to 80 is two standard deviations from the mean, so k = 2. Chebyshev's inequality guarantees at least 1 - 1/k² = 1 - 1/4 = 0.75 of observations within that range, regardless of the distribution's shape.

Question 13Descriptive Statistics and Data Presentation

In a grouped frequency distribution, the mean is estimated using class midpoints. Why is the result only an approximation?

  1. Because the class widths must be equal
  2. Because the midpoints are always integers
  3. Because grouped data cannot be averaged
  4. Because individual observations within a class are assumed to sit at the midpoint Correct answer.

Explanation

Grouping discards the exact values, so the calculation assumes every observation in a class equals its midpoint. Any variation within classes is lost, making the grouped mean an approximation of the true mean.

Question 14Descriptive Statistics and Data Presentation

Which pair of measures is most resistant to the effect of a single extreme outlier?

  1. Mean and range
  2. Median and interquartile range Correct answer.
  3. Mode and variance
  4. Mean and standard deviation

Explanation

The median and interquartile range depend on the middle portion of the ordered data, so a single extreme value has little influence. Mean, standard deviation, range and variance all use extreme values and are therefore sensitive to outliers.

Question 15Descriptive Statistics and Data Presentation

A data set contains the values 12, 15, 15, 17, 20, 22, 25. Which statement about the shape of the distribution is supported by comparing the mean and median?

  1. The mean cannot be computed because of repeated values
  2. The mean is less than the median, suggesting left skew
  3. The mean exceeds the median, suggesting slight right skew Correct answer.
  4. The mean equals the median exactly, suggesting symmetry

Explanation

The total is 126 over 7 observations, giving a mean of 18, while the median is the fourth ordered value, 17. Because the mean exceeds the median, the distribution shows slight right skew.

Question 16Descriptive Statistics and Data Presentation

A relative frequency table shows that 30 of 200 respondents chose option A. What is the relative frequency of A, and how should it be presented?

  1. 1.50, as a percentage
  2. 0.15, as a proportion of the total Correct answer.
  3. 0.15, expressed in raw counts
  4. 0.30, as a proportion of the total

Explanation

Relative frequency is the class frequency divided by the total, so 30/200 = 0.15. This proportion can also be reported as 15 percent of all respondents.

Question 17Descriptive Statistics and Data Presentation

A stem-and-leaf plot is preferred over a histogram for a small data set mainly because it:

  1. Displays cumulative frequencies directly
  2. Can only be used for categorical data
  3. Shows the original individual values while preserving the shape of the distribution Correct answer.
  4. Requires grouped class intervals

Explanation

A stem-and-leaf plot retains each original observation while still arranging values in order, so the overall shape is visible. This dual property makes it useful for small data sets where no information should be lost.

Question 18Descriptive Statistics and Data Presentation

In a box plot, an observation is typically flagged as a potential outlier when it lies beyond which limits?

  1. The minimum and maximum values
  2. Q1 and Q3 themselves
  3. The mean plus or minus one standard deviation
  4. Q1 - 1.5 IQR and Q3 + 1.5 IQR Correct answer.

Explanation

The standard rule marks values below Q1 - 1.5 IQR or above Q3 + 1.5 IQR as potential outliers. These fences define the whiskers, and points beyond them are plotted individually.

Question 19Descriptive Statistics and Data Presentation

A data set has 200 observations. Approximately what percentile corresponds to an observation that is the 50th smallest value?

  1. 20th percentile
  2. 50th percentile
  3. 25th percentile Correct answer.
  4. 75th percentile

Explanation

The percentile rank is roughly the number of values at or below the observation divided by the total, so 50/200 = 0.25. The observation therefore lies near the 25th percentile.

Question 20Descriptive Statistics and Data Presentation

Which measure of dispersion is expressed in the same units as the original data and is based on all observations?

  1. Coefficient of variation
  2. Interquartile range
  3. Variance
  4. Standard deviation Correct answer.

Explanation

The standard deviation is the square root of the variance, which restores the original units while still using every observation. The variance is in squared units, the coefficient of variation is unitless, and the IQR uses only two quartiles.

Question 21Descriptive Statistics and Data Presentation

Two variables are plotted on a scatter diagram and the points form a tight downward-sloping line. Which description of the relationship is correct?

  1. No association
  2. Weak positive linear association
  3. Strong negative linear association Correct answer.
  4. Strong positive linear association

Explanation

A downward-sloping pattern means that as one variable increases the other tends to decrease, which is a negative association. The tightness of the points around the line indicates that the association is strong.

Question 22Descriptive Statistics and Data Presentation

A data set is bimodal with two clear peaks of similar height. Which summary is most informative for describing its centre?

  1. The range alone
  2. A single mean value
  3. Both modal values, reported with the frequencies Correct answer.
  4. The coefficient of variation

Explanation

When a distribution has two peaks, a single measure of centre conceals the structure of the data. Reporting both modes together with their frequencies describes the two subgroups that produce the peaks.

Question 23Descriptive Statistics and Data Presentation

A pie chart is constructed for four categories with frequencies 40, 30, 20 and 10. What central angle represents the category with frequency 30?

  1. 72 degrees
  2. 120 degrees
  3. 90 degrees
  4. 108 degrees Correct answer.

Explanation

The total frequency is 100, so the category with frequency 30 represents 0.30 of the whole. Multiplying 0.30 by 360 degrees gives a central angle of 108 degrees.

Question 24Descriptive Statistics and Data Presentation

A grouped frequency table has classes 0-10, 10-20, 20-30 and 30-40 with frequencies 5, 12, 8 and 5. Which class contains the median?

  1. 10-20 Correct answer.
  2. 0-10
  3. 20-30
  4. 30-40

Explanation

The total frequency is 30, so the median lies at cumulative frequency 15. The first class reaches 5 and the second reaches 17, so the 15th observation falls in the class 10-20.

Question 25Descriptive Statistics and Data Presentation

A statistician reports that the standard deviation of a sample is 0. What can be concluded about the observations?

  1. All observations are identical Correct answer.
  2. The sample size is too small
  3. Half the observations are negative
  4. The mean is 0

Explanation

The standard deviation measures the average spread about the mean, so a value of 0 means every observation equals the mean. Consequently all observations in the sample are identical.

Question 26Descriptive Statistics and Data Presentation

A grouped frequency table reports the median class as 40-50, with cumulative frequency before that class equal to 26, class frequency 18, class width 10, and total frequency 60. What is the interpolated median?

  1. 46.7
  2. 43.3
  3. 44.4 Correct answer.
  4. 45.0

Explanation

The interpolated median is L + [(N/2 − CF)/f] × w. Here N/2 = 30, CF = 26, f = 18 and w = 10, giving 40 + (4/18) × 10 = 40 + 2.22 = 44.4. This assumes observations are spread evenly within the median class, so the answer is an approximation.

Question 27Descriptive Statistics and Data Presentation

A histogram is drawn with bar areas proportional to class frequency. One class spans 20 units with height 6, and an adjacent class spans 10 units with the same frequency. What height should the adjacent bar have?

  1. 24
  2. 6
  3. 3
  4. 12 Correct answer.

Explanation

When bar area represents frequency, height equals frequency divided by class width, i.e. the frequency density. The first class has area 20 × 6 = 120, so the second class of width 10 needs height 120/10 = 12 to show the same frequency. Equal heights would wrongly imply equal frequencies for unequal widths.

Question 28Descriptive Statistics and Data Presentation

In a data set of 40 observations, the mean is 25 and the median is 31. Which statement best describes the shape of the distribution?

  1. It is right-skewed, because the median exceeds the mean.
  2. It is bimodal, because the mean and median are unequal.
  3. It is left-skewed, because the mean lies below the median. Correct answer.
  4. It is symmetric, because the mean and median differ.

Explanation

In a skewed distribution the mean is pulled toward the long tail while the median stays near the bulk of the data. A mean below the median therefore indicates a tail stretching to the left, i.e. negative or left skewness. The size of the gap (6 units) signals a moderate skew.

Question 29Descriptive Statistics and Data Presentation

Which statement correctly distinguishes a frequency polygon from a frequency histogram for the same grouped data?

  1. The polygon uses bars of equal width, while the histogram joins class boundaries with straight lines.
  2. The polygon shows cumulative frequencies, while the histogram shows relative frequencies.
  3. The polygon can only be drawn for qualitative data, while the histogram is restricted to quantitative data.
  4. The polygon joins midpoints of class intervals, while the histogram uses contiguous bars whose areas show frequency. Correct answer.

Explanation

A frequency polygon plots frequency against class midpoint and joins successive points with straight lines, closing to the axis at the ends. A histogram instead uses adjacent bars spanning class boundaries, with area encoding frequency. Both display the same frequency distribution in different graphical forms.

Question 30Descriptive Statistics and Data Presentation

A distribution of household incomes is strongly right-skewed. A report must summarise the typical income and the spread of the middle half of the households. Which pair of statistics is most appropriate?

  1. The mean and the coefficient of variation
  2. The mode and the range
  3. The median and the interquartile range Correct answer.
  4. The mean and the standard deviation

Explanation

For strongly skewed data the median and interquartile range are resistant to extreme values, and the IQR directly describes the spread of the middle 50 per cent of observations. Mean, standard deviation, range and coefficient of variation are all influenced by the long right tail. Reporting them would overstate the typical income and the spread.

Question 31Probability Theory and Distributions

For a discrete random variable X, which expression gives the variance in terms of its probability mass function p(x)?

  1. Σ x·p(x)
  2. E[(X − E[X])²] Correct answer.
  3. E[X]² − E[X²]
  4. E[X²] + (E[X])²

Explanation

Variance is defined as the expected squared deviation from the mean, written E[(X − E[X])²]. For a discrete variable this equals Σ (x − μ)² p(x), which is always nonnegative. The computational shortcut form is E[X²] − (E[X])², not the reversed difference.

Question 32Probability Theory and Distributions

A fair six-sided die is rolled once. What is the probability that the outcome is a prime number?

  1. 1/6
  2. 1/3
  3. 1/2 Correct answer.
  4. 2/3

Explanation

The primes among 1 to 6 are 2, 3 and 5, giving three favorable outcomes out of six equally likely results. The probability is therefore 3/6 = 1/2. Note that 1 is not prime, so it must be excluded from the count.

Question 33Probability Theory and Distributions

If events A and B are independent with P(A) = 0.4 and P(B) = 0.5, what is P(A ∪ B)?

  1. 0.9
  2. 0.2
  3. 0.7 Correct answer.
  4. 0.45

Explanation

Independence gives P(A ∩ B) = P(A)P(B) = 0.4 × 0.5 = 0.2. The addition rule then yields P(A ∪ B) = 0.4 + 0.5 − 0.2 = 0.7. Forgetting to subtract the intersection overcounts the overlap.

Question 34Probability Theory and Distributions

Which statement describes the median of a continuous distribution with cumulative distribution function F?

  1. The value m where the density f(m) is largest
  2. The mean of the distribution
  3. The value m where F(m) = 0.5 Correct answer.
  4. The value m where F(m) = 0

Explanation

The median is the 50th percentile, that is, the point m satisfying F(m) = 0.5, so that half the probability lies on each side. For a continuous distribution this point is unique whenever F is strictly increasing on the support. It need not coincide with the mode or the mean.

Question 35Probability Theory and Distributions

X is Poisson distributed with mean λ = 3. What is P(X = 0)?

  1. 3e^(−3)
  2. e^(−3) Correct answer.
  3. 0
  4. 1/3

Explanation

The Poisson probability mass function is P(X = k) = e^(−λ) λ^k / k!. Setting k = 0 makes λ^0 = 1 and 0! = 1, leaving P(X = 0) = e^(−λ) = e^(−3). This is the probability of observing no events in the interval.

Question 36Probability Theory and Distributions

For a standard normal variable Z, what is P(Z ≤ 0)?

  1. 0.68
  2. 1
  3. 0.25
  4. 0.5 Correct answer.

Explanation

The standard normal density is symmetric about zero, so exactly half of the probability lies below the mean. Hence P(Z ≤ 0) = 0.5, consistent with Φ(0) = 0.5. The value 0.68 refers to the mass within one standard deviation, a different quantity.

Question 37Probability Theory and Distributions

A random variable has the moment generating function M(t) = e^(2t + 8t²/2). What are its mean and variance?

  1. Mean 4, variance 8
  2. Mean 8, variance 2
  3. Mean 2, variance 8 Correct answer.
  4. Mean 2, variance 4

Explanation

The normal distribution N(μ, σ²) has moment generating function exp(μt + σ²t²/2). Matching terms gives μ = 2 and σ² = 8, so the mean is 2 and the variance is 8. The coefficient of t²/2 is the variance, not the standard deviation.

Question 38Probability Theory and Distributions

The probability density function of a continuous variable must satisfy which condition?

  1. f(x) equals the probability that X = x
  2. f(x) is nonnegative and integrates to 1 over the real line Correct answer.
  3. f(x) is a step function
  4. f(x) ≤ 1 for every x

Explanation

A density must be nonnegative everywhere and integrate to one over its support so that total probability equals one. Individual density values may exceed 1 because probability is given by area under the curve, not by height. For a continuous variable the probability of any single point is zero.

Question 39Probability Theory and Distributions

X and Y are independent with Var(X) = 4 and Var(Y) = 9. What is Var(X − Y)?

  1. 13 Correct answer.
  2. 1
  3. 5
  4. −5

Explanation

For independent variables, the variance of a difference adds rather than subtracts: Var(X − Y) = Var(X) + Var(Y). Substituting gives 4 + 9 = 13. The negative sign on Y disappears because variance squares the coefficient (−1)² = 1.

Question 40Probability Theory and Distributions

A screening test has sensitivity 0.90 and specificity 0.95. The disease prevalence is 0.02. What is the approximate probability that a person who tests positive actually has the disease?

  1. 0.90
  2. 0.02
  3. 0.27 Correct answer.
  4. 0.50

Explanation

By Bayes' theorem, P(D|+) = (0.90 × 0.02) / (0.90 × 0.02 + 0.05 × 0.98) = 0.018 / (0.018 + 0.049) ≈ 0.27. The large number of false positives among the many healthy people dominates because prevalence is low. This illustrates why predictive value depends on prevalence, not only on sensitivity.

Question 41Probability Theory and Distributions

X is uniformly distributed on the interval [2, 10]. What are its mean and variance?

  1. Mean 5, variance 64/12
  2. Mean 6, variance 64/12 Correct answer.
  3. Mean 6, variance 8/12
  4. Mean 6, variance 64

Explanation

For a uniform distribution on [a, b], the mean is (a + b)/2 = 6 and the variance is (b − a)²/12 = 64/12. The variance formula divides the squared range by twelve because the density height is 1/(b − a). Using the range instead of its square is a common error.

Question 42Probability Theory and Distributions

If X has a binomial distribution with n = 100 and p = 0.5, which normal approximation is most appropriate for computing P(X ≤ 55) with a continuity correction?

  1. N(50, 5) evaluated at 55.5
  2. N(100, 25) evaluated at 54.5
  3. N(50, 25) evaluated at 55.5 Correct answer.
  4. N(50, 25) evaluated at 55

Explanation

The binomial mean is np = 50 and the variance is np(1 − p) = 25. Continuity correction replaces the event X ≤ 55 by X ≤ 55.5, so the approximating normal is N(50, 25) evaluated at 55.5. The standard deviation is 5, but the normal approximation is parameterized by the variance.

Question 43Probability Theory and Distributions

The joint density of X and Y is f(x, y) = 4xy for 0 < x < 1 and 0 < y < 1, and zero otherwise. Are X and Y independent?

  1. Yes, because the support is a rectangle only if the marginals are normal
  2. No, because X and Y are each uniform on (0, 1)
  3. Yes, because the joint density factors as (2x)(2y) on the unit square Correct answer.
  4. No, because the joint density does not factor into a product of functions of x and y

Explanation

The joint density factors as a product of a function of x and a function of y, namely (2x)(2y), on a rectangular support. The marginals are f_X(x) = 2x and f_Y(y) = 2y, whose product reproduces the joint density. Hence X and Y are independent.

Question 44Probability Theory and Distributions

X has an exponential distribution with rate λ = 0.5. What is P(X > 4)?

  1. e^(−2) Correct answer.
  2. e^(−0.5)
  3. 0.5
  4. 1 − e^(−2)

Explanation

For an exponential variable with rate λ, the survival function is P(X > x) = e^(−λx). Substituting gives e^(−0.5 × 4) = e^(−2), approximately 0.135. The complement 1 − e^(−2) is the probability of the opposite event.

Question 45Probability Theory and Distributions

Two cards are drawn without replacement from a standard deck of 52. What is the probability that both are aces?

  1. 4/52 + 3/51
  2. (4/52)(4/51)
  3. (4/52)(3/51) Correct answer.
  4. (4/52)²

Explanation

After the first ace is drawn, only 3 aces remain among 51 cards, so the second conditional probability is 3/51. Multiplying the sequential probabilities gives (4/52)(3/51) = 1/221. Drawing without replacement means the second probability must be conditioned on the first draw.

Question 46Probability Theory and Distributions

For a Poisson process with rate 4 events per hour, what is the distribution of the waiting time until the first event?

  1. Poisson with mean 4
  2. Exponential with rate 4 Correct answer.
  3. Binomial with n = 4 and p = 0.5
  4. Normal with mean 4

Explanation

In a Poisson process the number of events in a fixed interval is Poisson, but the waiting time to the first event is exponential with the same rate. Its mean waiting time is 1/4 hour, the reciprocal of the event rate. The exponential distribution is the continuous counterpart of the geometric waiting time in discrete trials.

Question 47Probability Theory and Distributions

X and Y have covariance 6, with Var(X) = 9 and Var(Y) = 16. What is the correlation coefficient?

  1. 0.5 Correct answer.
  2. 0.375
  3. 0.25
  4. 0.75

Explanation

The correlation is Cov(X, Y) divided by the product of the standard deviations. The standard deviations are 3 and 4, so the denominator is 12 and r = 6/12 = 0.5. Dividing by the variances instead of the standard deviations gives an incorrect value.

Question 48Probability Theory and Distributions

Which statement about the cumulative distribution function F(x) = P(X ≤ x) of any random variable is always true?

  1. F is nondecreasing and right-continuous with limits 0 at −∞ and 1 at +∞ Correct answer.
  2. F is strictly increasing on the whole real line
  3. F is differentiable everywhere
  4. F is symmetric about the mean

Explanation

Every cumulative distribution function is nondecreasing, right-continuous, and tends to 0 and 1 at the extremes. These three properties characterize distribution functions completely. It need not be continuous, differentiable or symmetric, since discrete and skewed variables also have valid CDFs.

Question 49Probability Theory and Distributions

A fair coin is flipped until the first head appears. What is the expected number of flips?

  1. 2 Correct answer.
  2. 1
  3. 1.5
  4. 4

Explanation

The number of flips follows a geometric distribution with success probability p = 0.5. Its expectation is 1/p = 2, which matches the sum of the series Σ k(1/2)^k. The mean is larger than 1 because repeated tails delay the first success.

Question 50Probability Theory and Distributions

The moment generating function of X is M(t) = 0.3e^(t) + 0.7e^(4t). What is E[X²]?

  1. 2.8
  2. 9
  3. 11.5 Correct answer.
  4. 3.1

Explanation

The mgf is a mixture of two point masses, so X equals 1 with probability 0.3 and 4 with probability 0.7. Then E[X²] = 0.3(1²) + 0.7(4²) = 0.3 + 11.2 = 11.5. The mean is 0.3 + 2.8 = 3.1, which is a different quantity.