MATH c UNIVERSITY OF LEEDS Examination for the Module MATH1725 (May-June 2009) INTRODUCTION TO STATISTICS. Time allowed: 2 hours

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1 01 This question paper consists of 11 printed pages, each of which is identified by the reference. Only approved basic scientific calculators may be used. Statistical tables are provided at the end of the exam paper. c UNIVERSITY OF LEEDS Examination for the Module (May-June 2009) INTRODUCTION TO STATISTICS Time allowed: 2 hours Attempt ALL questions in Section A and TWO questions from Section B. Questions A1 to A10 require you to write down a single letter answer. Questions A11 to A20 require you to write down a short explanation. Your answers to Section A questions and Section B questions may be written in the same answer book. Sections A and B are each worth 50% of the examination marks. Questions A1 to A20 carry equal weight. 1 CONTINUED...

2 SECTION A Attempt ALL questions in Section A. Questions A1 to A10 require you to write down a single letter answer. A1. The incomes (in units of 1000) of ten directors of Marks and Spencer plc during 2008 were: 453, 1375, 698, 293, 701, 73, 57, 68, 79, 73. What does the sample median equal? A 57, B 137.5, C 186, D 293, E 387. A2. If Z N(0, 1), what is the value of P(Z > 1.15). A , B , C , D , E A3. Suppose variables X 1, X 2,..., X n have common mean µ and variance σ 2. Their mean X is said to be an unbiased estimator of µ. What does this tell you? A X = 1 n n i=1 X i, B E[ X] = µ, C Var[ X] = σ2, D µ is a special measure of spread. n A4. A sample correlation coefficient r xy equals 1. What does this definitely tell you about the corresponding scatter plot of y against x? A Data points closely scattered about a straight line, B data points all lie on a straight line with slope 1, C data points all lie on a straight line with zero slope, D data points all lie on a straight line with negative slope. A5. A least squares regression problem has n pairs of data (x i, y i ), i = 1, 2,..., n. The fitted least squares regression line is y = ˆα + ˆβx. Which quantity is minimised to derive ˆα and ˆβ? A n y i α βx i, B i=1 n (y i +α βx i ) 2, C i=1 n (y i α βx i ) 2, D i=1 n (y i +α+βx i ) 2. i=1 2 CONTINUED...

3 A6. The boxplot below shows the heights in metres for 25 male students and 25 female students. Females Males Height (m) Which of the following statements are true? (i) The median height of males is less than the median height of females. (ii) The semi-interquartile range of female heights is about 0.115m. (iii) The variability of male and female heights is about the same. A: (ii) only, B: (i) and (ii), C: (ii) and (iii), D: (iii) only. A7. Random variables X and Y have correlation coefficient 0.5. If X has mean 2 and variance 4, and Y has mean 1 and variance 1, what is the mean of X 2Y? A 2, B 1, C 0, D 1, E 2. A8. In question A7 above, what is the variance of X 2Y? A 2, B 0, C 2, D 4, E 6, F 7. A9. If random variables X and Y each have variance equal 3 and X + Y has variance 8, what does the covariance between X and Y equal? A 0, B 1, C 2, D 3, E not enough information to say. A10. For the χ 2 -distribution with 5 degrees of freedom, what is the value of χ 2 5 (10%)? A 9.236, B 11.07, C 15.09, D 15.99, E CONTINUED...

4 Questions A11 to A20 require you to write down a short explanation. A11. For a set of n observations, what is a dot-plot? A12. Briefly describe the central limit theorem. A13. A sample of n = 16 values has sample mean x = 1.44 and sample variance s 2 = Is the sample mean significantly different from zero? A14. Values x i and y i, i = 1, 2,..., n, lie on a horizontal line y = c where c is a constant. What does the sample covariance s XY equal? A15. Random variables X and Y are both discrete with joint probability function p(x i, y j ). How would you calculate the marginal probability function of X, p X (x i )? A16. In question A15 above, how would you calculate E[XY ]? A17. A random sample of size n is taken from a population of size N with replacement. If the population consists of R individuals of type A and N R of type B, what is the probability that the sample contains r of type A? A18. In question A17 above, if n is large and R/N is close to 1, what continuous distribution 2 could be used as an approximation when calculating the required probability? (State also the mean and variance of this distribution.) A19. In a sample of 161 first year Leeds University students, 21 did more than 5 hours of paid work in a given week of term. Use these data to obtain an approximate 95% confidence interval for the proportion of Leeds University students who do more than 5 hours of paid work in a week of term. A20. In a chi-squared test with ten groups the observed value of the chi-squared test statistic under some null hypothesis H 0 is χ 2 obs. What would extremely small values of χ 2 obs suggest about the experimental data? 4 CONTINUED...

5 SECTION B Attempt TWO questions from Section B B1. The following data give the heights of 100 male students at a certain university. Height Number of (in inches) students (a) Calculate the sample mean and variance for these data. (b) A suitable normal distribution is fitted to these data and some expected frequencies have been determined as shown in the table below. Height Observed Expected (in inches) Frequency Frequency Determine the expected frequencies for the remaining class intervals. (c) Test whether your fitted normal distribution gives a good fit to these data. 5 CONTINUED...

6 B2. (a) Pairs of measurements (x i, y i ), i = 1, 2,..., n, are made on each of n individuals. The least squares regression line for y given x is y = α + βx. Derive the least squares estimates of α and β. (b) The anxiety level of subjects in a certain stress situation was assessed using two different procedures. (I) The stait-trait anxiety inventory (STAI) consisting of twenty questions. (II) The linear analogue (LA) score in which the subject is asked to indicate on a 100mm scale their perceived anxiety level with 0mm on the scale corresponding to the statement I do not feel anxious at all and 100mm on the scale corresponding to the statement I could not feel more anxious. For ten subjects the STAI and LA scores are given below. yi 2 = 16323, i Subject i STAI score y i LA score x i x 2 i = 22411, i x i y i = i (i) Fit a least squares regression line for predicting the STAI score given an LA score. (ii) Use your fitted regression line to predict the STAI score for an LA score x = 20. (c) Define the residuals r i for your fitted model and show that the residual for subject 5 equals CONTINUED...

7 B3. (a) A study of blood alcohol levels (in mg/litre) at post mortem examinations of road accident victims involved taking one blood sample from the leg (column A) and another from the heart (column B). The results are tabulated below. Case A B Do these results indicate that there is a significant difference in blood alcohol levels for the same individual in the leg compared with the heart? Why is it reasonable to suppose these data can be regarded as matched-pairs? (b) The following data give the length (in mm) of cuckoo (cuculus canorus) eggs found in nests belonging to wrens (A) and reed warblers (B). A: B: Is there any evidence at the 1% level to suggest that the egg size differs between the two host species? Why is it unreasonable to suppose these data can be regarded as matched-pairs? (c) What do you understand by the phrase matched-pairs? 7 CONTINUED...

8 B4. (a) The random variables X and Y have means µ X and µ Y respectively, variances σ 2 X and σ 2 Y respectively, and the correlation coefficient between them is ρ. Write down the mean and variance of ax + by, where a and b are constants. (b) An unbiased six-sided die is rolled n times. Let X 1 denote the total number of 1 s observed in the n rolls, and X 2 denote the total number of 2 s observed in the n rolls. Both X 1 and X 2 have binomial distributions. Explain briefly why this is so and state the parameters of the binomial distributions. (c) What are the variances of X 1 and X 2? (d) The random variable U = X 1 + X 2 gives the total number of 1 s and 2 s observed in the n rolls. By considering the distribution of the random variable U and hence obtaining its variance, or otherwise, deduce that the correlation coefficient between X 1 and X 2 is ρ = 1 5. (e) Obtain the variance of the difference V = X 1 X 2. (f) Determine the correlation coefficient between U and V. (g) Describe briefly how you could verify whether U and V are independent. (Explicit calculation is not required.) 8 CONTINUED...

9 Normal Distribution Function Tables The first table gives Φ(x) = 1 2π x e 1 2 t2 dt and this corresponds to the shaded area in the figure to the right. Φ(x) is the probability that a random variable, normally distributed with zero mean amd unit variance, will be less than or equal to x. When x < 0 use Φ(x) = 1 Φ( x), as the normal distribution with mean zero is symmetric about zero. To interpolate, use the formula Φ(x) Φ(x 1 ) + x x 1 x 2 x 1 (Φ(x 2 ) Φ(x 1 )) x Table 1 x Φ(x) x Φ(x) x Φ(x) x Φ(x) x Φ(x) x Φ(x) The inverse function Φ 1 (p) is tabulated below for various values of p. Table 2 p Φ 1 (p) CONTINUED...

10 Percentage Points of the t-distribution This table gives the percentage points t ν (P) for various values of P and degrees of freedom ν, as indicated by the figure to the right. The lower percentage points are given by symmetry as t ν (P), and the probability that t t ν (P) is 2P/100. The limiting distribution of t as ν is the normal distribution with zero mean and unit variance. 0 t ν (P) P/100 Percentage points P ν CONTINUED...

11 Percentage Points of the χ 2 -Distribution This table gives the percentage points χ 2 ν (P) for various values of P and degrees of freedom ν, as indicated by the figure to the right, plotted in the case ν = 3. If X is a variable distributed as χ 2 with ν degrees of freedom, P/100 is the probability that X χ 2 ν (P). For ν > 100, 2X is approximately normally distributed with mean 2ν 1 and unit variance. 0 χ 2 ν(p) P/100 Percentage points P ν END