Statistics

Simple univariate statistics

1. In the city park there are 40 trees: 12 of them are maple trees, 7 mulberry trees, 15 ginkgos and 6 oaks.
   

   (a)   Represent the data using a diagram.

   (b)   Find the mode.

   Solutions:    (b)  Mode: ginkgo
2. The heights of the players of a basketball team are: 185, 190, 193, 199, 201, 204, 205, 206, 208 cm.
   

   (a)   Find the average (the mean).

   (b)   Find the median.

   Solutions:    (a)  Mean: 199 cm;     (b)  median: 201 cm
3. The headmistress conducted a survey. Among many other questions she asked the students about their households. She obtained the following results: 3 students live in a household of 2 members, 6 students live in a household of 3 members, 2 students live in a household of 4 members, 2 students live in a household of 5 members, 2 students live in a household of 6 members.
   

   (a)   Find the mean.

   (b)   Find the median.

   Solutions:    (a)  Mean: 3.6 members;     (b)  median: 3 members
4. A statistician checked all the apartments in Penny Lane. He discovered that there are 6 apartments occupied by only 1 inhabitant, 12 apartments occupied by 2 inhabitants, 8 apartments occupied by 3 inhabitants, 7 apartments occupied by 4 inhabitants, 6 apartments occupied by 5 inhabitants and 1 apartment occupied by 6 inhabitants.
   

   (a)   Represent the data using different diagrams.

   (b)   Find the mode, the median and the mean.

   Solutions:    (b)  Mode: 2,   median: 3,   mean: 2.95
5. A teacher asked his students how much time did they spend learning on the previous day. Here are their answers  —  Andrew: 50 minutes, Benjamin: 35 minutes, Claude: 1 hour and 30 minutes, Dolores: 1 hour and 20 minutes, Esmeralda: 2 hours, Frank: 55 minutes, Gloria: 1 hour and 50 minutes, Hans: 20 minutes. Find the average and the median.
   Solutions:    Average or mean: 70 minutes,   median: 67.5 minutes
6. Several contestants participated in a pancake eating contest. Alice ate 12 pancakes, Boris ate 14 pancakes, Connie ate 13 pancakes, Deborah ate 12 pancakes, Edgar ate 14 pancakes, Fernando ate 8 pancakes, George ate 15 pancakes, Helen ate 9 pancakes, Irene ate 15 pancakes, Jeremy ate 11 pancakes, Kim ate 12 pancakes and Louis ate 10 pancakes.
   

   (a)   Find the median and quartiles.

   (b)   Represent the results using box and whisker plot.

   Solutions:    (a)  Median: \(M=12\),   quartiles: \(Q_1=10.5,~ Q_2=M=12,~ Q_3=14\)
7. Draw the box-and-whisker plot for the following data set representing the masses of tomatoes (in grams):
   82, 88, 90, 95, 95, 99, 100, 104, 108, 111, 112, 115, 119, 120.
   Solutions:    Median: \(M=102\),   quartiles: \(Q_1=95,~ Q_2=M=102,~ Q_3=112\)
8. In the Ultrasoft Company the chief executive has a monthly salary of 35 000 €. There are two assistant managers who have a salary of 10 000 €. Four engineers have a salary of 3 000 €. Six foremen have a salary of 1 000 €. The remaining twelve employees earn 500 € monthly.
   

   (a)   Find the mode, the mean and the median.

   (b)   Represent the results using box and whisker plot.

   Solutions:    (a)  Mode: 500 €,   mean: 3 160 €,   median: 1 000 €;     (b)  quartiles: \(Q_1=500,~ Q_2=M=1000,~ Q_3=3000\)
9. A farmer measured the lengths of several cucumbers growing in his garden. He obtained the following results (in cm):
   25, 27, 27, 28, 29, 29, 29, 30, 30, 31, 31, 31, 31, 32, 33, 34, 34, 34, 35, 35.
   

   (a)   Write the table of cumulative frequencies.

   (b)   Draw the cumulative frequency diagram.

   Solutions:
   
   \( \begin{array}{|c|c|} \hline grade & cum.~frq. \cr\hline 25 & 1 \cr 27 & 3 \cr 28 & 4 \cr 29 & 7 \cr 30 & 9 \cr 31 & 13 \cr 32 & 14 \cr 33 & 15 \cr 34 & 18 \cr 35 & 20 \cr \hline \end{array} \)
10. The following table shows the results of a math test:
             \( \begin{array}{|c|c|} \hline grade & frequency \cr\hline 1 & 2 \cr 2 & 1 \cr 3 & 3 \cr 4 & 5 \cr 5 & 7 \cr 6 & 8 \cr 7 & 4 \cr \hline \end{array} \)
    

    (a)   Write the table of cumulative frequencies.

    (b)   Draw the cumulative frequency diagram.

    Solutions:
    
    \( \begin{array}{|c|c|} \hline grade & cum.~frq. \cr\hline 1 & 2 \cr 2 & 3 \cr 3 & 6 \cr 4 & 11 \cr 5 & 18 \cr 6 & 26 \cr 7 & 30 \cr \hline \end{array} \)
11. Students measured the shoe sizes in a group of 20 persons. The results are written in the following table:
             \( \begin{array}{|c|c|} \hline size & frequency \cr\hline 37 & 5\% \cr 38 & 10\%\cr 39 & 15\% \cr 40 & 25\% \cr 41 & 30\%\cr 42 & 15\%\cr \hline \end{array} \)
    

    (a)   Write the table of cumulative frequencies.

    (b)   Draw the cumulative frequency diagram.

    Solutions:
    
    \( \begin{array}[t]{|c|c|} \hline grade & cum.~frq. \cr\hline 37 & 5\% \cr 38 & 15\% \cr 39 & 30\% \cr 40 & 55\% \cr 41 & 85\% \cr 42 & 100\% \cr \hline \end{array} \)
12. A statistician measured the weights od 15 students (in kg). the results are written bellow:
    55, 57, 58, 58, 60, 60, 60, 62, 65, 65, 69, 71, 71, 73, 75.
    

    (a)   Draw the cumulative frequency graph.

    (b)   Find the median, the quartiles and the interquartile range.

    Solutions:    (b)  Median: \(M=62\);   quartiles: \(Q_1=58,~ Q_2=M=62,~ Q_3=71\);   interquartile range: \(IQR=13\)
13. The following tables shows the distribution of the shoe sizes in two groups of children (labeled Group A and Group B):
    Group A: \( \begin{array}{|l|c|c|c|c|c|} \hline shoe~size & 20 & 21 & 22 & 23 & 24 \cr\hline frequency & 2 & 5 & 10 & 7 & 1 \cr\hline \end{array} \)          Group B: \( \begin{array}{|l|c|c|c|c|c|} \hline shoe~size & 20 & 21 & 22 & 23 & 24 \cr\hline frequency & 8 & 3 & 2 & 5 & 7 \cr\hline \end{array} \)
    

    (a)   Find the mean (for each group separately).

    (b)   Find the variance and standard deviation (for each group separately).

    Solutions:    (a)  Mean: \(\overline{x}=22\) (for A and for B);     (b)  Group A: variance: \(Var(x)=0.96\), standard deviation: \(\sigma\approx0.980\);     Group B: variance: \(Var(x)=2.72\), standard deviation: \(\sigma\approx1.65\)
14. Contestants can win up to 4 points in the Spelling Wasp contest. The following table shows how many contestants won certain number of points:
             \( \begin{array}{|l|c|c|c|c|c|} \hline number~of~points & 0 & 1 & 2 & 3 & 4 \cr\hline number~of~contestants & 6 & 7 & 8 & 5 & 4 \cr\hline \end{array} \)
    

    (a)   Find the mean.

    (b)   Find the variance and standard deviation.

    Solutions:    (a)  Mean: \(\overline{x}=1.8\);     (b)  variance: \(Var(x)\approx1.69\), standard deviation: \(\sigma\approx1.30\)
15. In a candy factory candies are produced and packaged in bags. Each bag should contain 30 candies. Internal inspection checked 20 bags: one bag contained only 28 candies, two bags contained 29 candies, two bags contained 31 candies and all the other bags contained 30 candies.
    

    (a)   Find the mean.

    (b)   Find the standard deviation.

    Solutions:    (a)  Mean: \(\overline{x}=29.85\approx29.9\);     (b)  standard deviation: \(\sigma\approx0.654\)

16. The following table shows the results of a math test:
             \( \begin{array}{|c|c|} \hline grade & frequency \cr\hline 1 & 1 \cr 2 & 2 \cr 3 & 2 \cr 4 & 5 \cr 5 & 7 \cr 6 & 9 \cr 7 & 5 \cr \hline \end{array} \)
    

    (a)   Find the mean, the standard deviation and the variance.

    (b)   Find the median and the quartiles.

    Solutions:    (a)  Mean: \(\overline{x}=5\), standard deviation: \(\sigma\approx1.57\), variance: \(\sigma^2\approx2.45\);     (b)  median: \(Med=5\), quartiles: \(Q_1=4, Q_2=Med=5, Q_3=6\)
17. At gym several students were trying to do as many push ups as possible in one minute. The following table shows their results:
             \( \begin{array}{|c|c|} \hline push~ups & frequency \cr\hline 21 & 4 \cr 22 & 3 \cr 23 & 8 \cr 25 & 10 \cr 27 & 4 \cr 28 & 1 \cr \hline \end{array} \)
    

    (a)   Find the mean.

    (b)   Find the standard deviation and the variance.

    Solutions:    (a)  Mean: \(\overline{x}=24\);     (b)  standard deviation: \(\sigma\approx1.98\), variance: \(\sigma^2\approx3.93\)
18. Policemen controlled the traffic on the main street in their town. They were measuring speeds of cars passing by. The following table shows the results they obtained (in km/h):
             \( \begin{array}{|c|c|} \hline speed & frequency \cr\hline 45 & 3 \cr 47 & 8 \cr 48 & 15 \cr 49 & 6 \cr 50 & 32 \cr 51 & 12 \cr 52 & 9 \cr 76 & 1 \cr 132 & 1 \cr \hline \end{array} \)
    

    (a)   Find the mean.

    (b)   Find the standard deviation.

    Solutions:    (a)  Mean: \(\overline{x}\approx50.7\);     (b)  standard deviation: \(\sigma x\approx9.36\)

Binomial distribution and normal distribution

19. A standard fair playing die is rolled 12 times. The random variable \(X\) represents the number of sixes. Calculate:
    

    (a)   \(P(X=1)\) and \(P(X=2)\).

    (b)   \(P(X\leqslant 3)\) and \(P(X\gt 3)\).

    (c)   \(E(X)\) and \(\sigma\).

    Solutions:    (a)  \(P(X=1)\approx0.269,~ P(X=2)\approx0.296\);    (b)  \(P(X\leqslant 3)\approx0.875,~ P(X\gt 3)\approx0.125\);    (c)  \(E(X)=np=2,~ \sigma=\sqrt{np(1-p)}\approx1.29\)
20. Random variable \(X\) is the number of heads obtained when you toss a coin twenty times. Calculate:
    

    (a)   \(P(X=7)\), \(P(X\leqslant7)\) and \(P(X\lt7)\).

    (b)   \(P(7\leqslant X\leqslant 13)\) and \(P(7\lt X\lt 13)\).

    (c)   \(E(X)\) and \(\sigma\).

    Solutions:    (a)  \(P(X=7)\approx0.0739,~ P(X\leqslant 7)\approx0.132,~ P(X\lt 7)\approx0.0577\);    (b)  \(P(7\leqslant X\leqslant 13)\approx0.885,~ P(7\lt X\lt 13)\approx0.737\);    (c)  \(E(X)=np=10,~ \sigma=\sqrt{np(1-p)}\approx2.24\)
21. The Singsing company produces LCD screens for personal computers. It's known that 1.25% of their LCD screens are defective. The Gertrude Cox Middle School buys 160 of their LCD screens.
    

    (a)   Find the expected number of defective LCD screens.

    (b)   Find the probability that there are 3 defective LCD screens.

    (c)   Find the probability that the number of defective LCD screens is less or equal 4.

    (d)   Find the probability that the number of defective LCD screens is greater or equal 5.

    Solutions:    (a)  \(E(X)=2\);     (b)  \(P\approx0.182\);     (c)  \(P\approx0.948\);     (d)  \(P\approx0.0515\)

22. A random variable is normally distributed with the mean \(\mu=70\) and the standard deviation \(\sigma=10\).
    

    (a)   Calculate \(P(65\leqslant X\leqslant 75)\).

    (b)   Calculate \(P(X\leqslant 50)\).

    (c)   Calculate \(P(X\geqslant 85)\).

    (d)   Find \(a\) where \(P(X\leqslant a)=0.75\).

    Solutions:    (a)  \(P(65\leqslant X\leqslant 75)\approx0.383\);     (b)  \(P(X\leqslant 50)\approx0.0228\);     (c)  \(P(X\geqslant 85)\approx0.0668\);     (d)  \(a\approx76.7\)
23. A random variable \(X\) is normally distributed: \(X\sim N(20,25)\).
    

    (a)   Calculate \(P(15\leqslant X\leqslant 20)\).

    (b)   Calculate \(P(X\geqslant 23)\).

    (c)   Find \(a\) where \(P(X\leqslant a)=\frac{2}{3}\).

    (d)   Find \(m\) where \(P(X\geqslant m)=0.45\).

    Hint:    The notation \(X\sim N(20,25)\) means that \(X\) is normally distributed with the mean \(\mu=20\) and variance \(\sigma^2=25\) (which means that the standard deviation \(\sigma=5\)).
    Solutions:    (a)  \(P(15\leqslant X\leqslant 20)\approx0.341\);     (b)  \(P(X\geqslant 23)\approx0.274\);     (c)  \(a\approx22.2\);     (d)  \(m\approx20.6\)
24. A random variable \(X\) is normally distributed: \(X\sim N(30,9)\).
    

    (a)   Write down \(\mu\) and \(\sigma\).

    (b)   Calculate \(P(X\leqslant \mu)\) and \(P(X\geqslant \mu)\).

    (c)   Calculate \(P(\mu\leqslant X\leqslant \mu+\sigma)\) and \(P(X\geqslant \mu+\sigma)\).

    (d)   Calculate \(P(\mu-\sigma\leqslant X\leqslant \mu+\sigma)\).

    Solutions:    (a)  \(\mu=30,~ \sigma=3\);     (b)  \(P(X\leqslant \mu)=P(X\geqslant \mu)=0.5=50\%\);     (c)  \(P(\mu\leqslant X\leqslant \mu+\sigma)\approx0.341\),  \(P(X\geqslant \mu+\sigma)\approx0.159\);     (d)   \(P(\mu-\sigma\leqslant X\leqslant \mu+\sigma)\approx0.683\)
25. A random variable \(Z\) is normally distributed: \(Z\sim N(0,1)\).
    

    (a)   Write down \(\mu\) and \(\sigma\).

    (b)   Calculate \(P(Z\leqslant \mu)\) and \(P(Z\geqslant \mu)\).

    (c)   Calculate \(P(\mu\leqslant Z\leqslant \mu+\sigma)\) and \(P(Z\geqslant \mu+\sigma)\).

    (d)   Calculate \(P(\mu-\sigma\leqslant Z\leqslant \mu+\sigma)\).

    Solutions:    (a)  \(\mu=0,~ \sigma=1\);     (b)  \(P(Z\leqslant \mu)=P(Z\geqslant \mu)=0.5=50\%\);     (c)  \(P(\mu\leqslant Z\leqslant \mu+\sigma)\approx0.341\),  \(P(Z\geqslant \mu+\sigma)\approx0.159\);     (d)   \(P(\mu-\sigma\leqslant Z\leqslant \mu+\sigma)\approx0.683\)
26. A random variable \(X\) is normally distributed: \(X\sim N(10,4)\). The values of the random variable \(Y\) are obtained by multiplying \(X\) by 3 and then adding 6.
    

    (a)   Calculate \(P(X\leqslant 11)\) and \(P(Y\leqslant 39)\).

    (b)   Calculate \(P(Y\geqslant 30)\) and find the appropriate interval for \(X\).

    (c)   Find \(a\) where \(P(Y\leqslant a)=\frac{1}{4}\).

    (d)   Find \(b\) where \(P(33\leqslant Y\leqslant b)=0.3\).

    Hint:    \(Y=3X+6\) means that the variable \(Y\) has the mean \(\mu=3\cdot 10+6=36\) and standard deviation \(\sigma=3\cdot 2=6\), so \(Y\sim N(36,36)\).
    Solutions:    (a)  \(P(X\leqslant 11)=P(Y\leqslant 39)\approx0.691\);     (b)  \(P(X\geqslant 8)=P(Y\geqslant 30)\approx0.841\);     (c)  \(a\approx32.0\);     (d)  \(b\approx37.7\)
27. A random variable \(X\) is normally distributed with \(\mu=50\) and \(\sigma=15\). Find the standard score (the \(Z\)-score) for values: \(X=55\),  \(X=60\) and \(X=20\). Hence calculate:
    

    (a)   \(P(55\leqslant X \leqslant 60)\)

    (b)   \(P(X\leqslant 20)\)

    Hint:    The random variable \(Z=\frac{\textstyle X-\mu}{\textstyle\sigma}\) is normally distributed: \(Z\sim N(0,1)\).
    Solutions:    (a)  \(P(55\leqslant X \leqslant 60)=P(\frac{1}{3}\leqslant Z \leqslant \frac{2}{3})\approx0.117\);     (b)  \(P(X\leqslant 20)=P(Z\leqslant -2)\approx0.0228\)
28. Results of a psychological test are normally distributed with the mean 85 and standard deviation 15.
    

    (a)   How many percent of the population have the score greater than 100?

    (b)   How many percent of the population have the score between 70 and 80?

    (c)   How many percent of the population have the score smaller than 60?

    (d)   Find the value \(a\), given that 10% of the population have the score smaller than \(a\).

    Solutions:    (a)  \(P(X \geqslant 100)\approx0.159=15.9\%\);     (b)  \(P(70\leqslant X\leqslant 80)\approx0.211=21.1\%\);     (c)  \(P(X\leqslant 60)\approx0.0478=4.78\%\);     (d)  \(a\approx65.8\)
29. Results of a psychological test are normally distributed with the mean \(\mu=90\).
    

    (a)   How many percent of the population have the score between \(\mu-\frac{1}{2}\sigma\) and \(\mu+\frac{1}{2}\sigma\)?

    (b)   Find \(\sigma\), given that only 5% of the population achieved score greater than 120.

    Hint:    Use \(Z\)-score in part (a). In part (b), first find the value \(a\) so that \(P(Z\geqslant a)=0.05\) and then use \(Z=\frac{\textstyle X-\mu}{\textstyle\sigma}\) to find \(\sigma\).
    Solutions:    (a)  \(P(-\frac{1}{2}\leqslant Z\leqslant \frac{1}{2})\approx0.383=38.3\%\);     (b)  \(\sigma\approx18.2\)

Bivariate statistics

30. The following table shows the weights and heights of 6 students.
    \( \begin{array}{|l|r|r|r|r|r|r|}\hline \mathrm{weight~(kg)} & 61 & 70 & 72 & 82 & 85 & 88 \cr \hline \mathrm{height~(cm)} & 170 & 178 & 175 & 181 & 179 & 185 \cr\hline \end{array} \)
    

    (a)   Find the mean and standard deviation for weight and for height.

    (b)   Find correlation coefficient \(r\).

    (c)   Draw scatter graph.

    (d)   Write the equation of the regression line (line of best fit).

    Solutions:    (a)  \(\overline{x}\approx76.3,~\sigma x\approx9.46;~~ \overline{y}\approx178,~ \sigma y\approx4.69\);     (b)  \(r\approx0.912\);     (d)  \(y=0.452x+143\)
31. The following table shows the grades of 8 students in mathematics and in physics.
    \( \begin{array}{|l|c|c|c|c|c|c|c|c|}\hline \mathrm{mathematics} & 7 & 4 & 6 & 5 & 3 & 6 & 1 & 5 \cr \hline \mathrm{physics} & 5 & 5 & 4 & 4 & 2 & 7 & 3 & 5 \cr\hline \end{array} \)
    

    (a)   Draw scatter graph.

    (b)   Find correlation coefficient \(r\).

    (c)   Write the equation of the regression line.

    Solutions:    (b)  \(r\approx0.648\);     (c)  \(y=0.507x+2.03\)
32. The following table shows monthly incomes and number of children for several families.
    \( \begin{array}{|l|c|c|c|c|c|c|c|}\hline \mathrm{incomes} & 1200 & 1800 & 2100 & 2500 & 2800 & 3500 & 3800 \cr \hline \mathrm{children} & 4 & 3 & 2 & 2 & 3 & 1 & 1 \cr\hline \end{array} \)
    

    (a)   Draw scatter graph.

    (b)   Find the correlation coefficient \(r\).

    (c)   Draw the trend line and write its equation.

    Solutions:    (b)  \(r\approx-0.869\);     (c)  \(y=-0.00105x+4.93\) (Hint: You can use a thousand as a unit. In this case the equation of the trend line is: \(y=-1.05x+4.93\).)

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