Mathematical Statistics

Solutions and Hints for Exercises

Chapter 1

Exercise 1.1

The sample is not random because inhabitants without entry in the telephone book cannot be selected.

Exercise 1.2

It is recommended to use SPSS avoiding long‐winded calculations by hand. Write the 81 different quadruples of the numbers 1, 2, 3 due to the random sampling with replacement into the columns y₁, y₂, y₃, y₄ of a SPSS data sheet (Statistics Data Editor). In the command sequence ‘Transform – Compute Variable’, denote the effect variable by ‘Mean’ and form (y₁ + y₂ + y₃ + y₄)/4 using the command MEAN = MEAN(y1,y2,y3,y4). See also the SPSS syntax below. Now the mean values occur in column 5 of the data sheet. Analogously create the variable s2 = VARIANCE(y1,y2,y3,y4) and in column 6 of the data sheet s2 is given.

After performing of the command sequence ‘Analyze – Descriptive Statistics – Descriptive’, the mean value and the variance of the population are calculated from the means (set under options) of MEAN and s2. The value of the VARIANCE of the variable MEAN must be multiplied by to get the population variance of the sample mean (2/3)/4, because from the population of N = 81 samples of size 4 SPSS calculates a sample variance with denominator . The corresponding graphical representations are obtained via ...