Chapter 6
                      Answers to selected problems

6.1 .006, .018, .216, .135

6.3 (a) .12; (b) .16; (c) .19; (d) 0; (e) 6; (f) 5.85

6.5 3/25, 4/25, 9/25, 21/25

6.7 59280, 1/59280, 1000, 1/1235

6.9 (a) mu: 5, sigma-squared: 4; 
    (b) Means: 1, 3, 4, 5, 6, 7;    P: .04, .16, .16, .16, .32, .16 
        variances: 0, 2, 8, 18     P: .36, .32, .16, .16
    (c) 5, 4, Both of them coincide with the corresponding population
        parameter. 
    (d) 1.414 

6.11  (b) (mean, prob): (1, .36), (2, .24), (3, .04), (5, .24), 
                        (6, .08), (9, .04); 
      (c) (var, prob): (0, .44), (2, .24), (18, .08), (32, .24); 
          (sd, prob): (0,.44), (1.414, .24), (4.243, .08), (5.657, .24); 
      (d) 3, 9.6, 3.10; 
      (e) 3, 9.6, 2.04; Sigma is not equal to E(s).

6.13 (a) 64; (b) 7.5, 1.25; (d) .645

6.15 1/4, 1/4, 1/28, yes, No because the two variables are not independent.

6.17 (a) yes; (b) 1/324, 5/648; (c) 7/2; (d) 14, 7/2

6.19 (a) .0025; (b) .0149; (c) .0446; (d) .0025; (e) .0149; (f) .0446

6.21 3.5, .313

6.23 (a) 3, 12.667, 3.559; (b) 18, 202.67, 14.24; (c) 3, .4449; (d) 18, 2.848

6.25 .906

6.27 (a) 0,0   0,6   6,0   6,6; 
     (b) Each sample has probability 1/4. 
         sample means: 0,  3,  3, 6. 
         s^2:          0, 18, 18, 0. 
         s^2-hat:      0,  9,  9, 0.; 
     (c) s^2: (value, prob)     =      (0, .5), (18, .5). 
         s^2-hat: (value, prob) =      (0, .5),  (9, .5).; 
     (d) 9, 4.5. The population variance is equal to 9.

6.29 (a) .558; (b) .0099

6.31 (a) 1.068; (b) .826

6.33 (a) 1.273; (b) 1, 0, .4909

6.35 (a) 1.155; (b) .385, .0047, .0049

6.37 (a) 5.55 lb.; (b) .1364 lb.; (c) 0; (d) .514

6.39 mu = 7, sigma = 2.408; (a) .5; (b) .45; (c) .0043

6.41 (a) 3/216, (b) 13/216

6.43 Without cc: (a) .9808, (b) .0287; With cc: (a) .9826, (b) .0262

6.45 mu=5, var=5, P=.5000 without continuity correction; P = .4726 with cc

6.47 mu = 157.0 lb \pm 2.2 lb

6.49 (a) 52.3 +- 1.9; (b) 52.3 +- 2.3; (c) 52.3 +- 3.0; (d) the first

6.51 (a) Because the sample size is less than 30. 
     (b) mu = 105.6 +- 8.2
     (c) We can be 95\% confident that the interval [97.4, 113.8] captures 
         the population mean.

6.53 (a) [78.8, 87.6];  
     (b) This confidence interval is statistically significant. It proves, 
         in the statistical sense, that the population mean is greater 
         than 75.

6.55  A 95% confidence interval for the population mean would be 76 +- 2.8, 
      and the number 81 lies outside this interval. Therefore the evidence 
      is statistically significant. One or both of the researcher's 
      figures (81 and 5) must surely be incorrect, unless the sample is 
      not random or there was some mistake in taking the readings. 
      Anyway there is almost certainly something wrong here.

6.57  (-infty , 26.3]

6.59  6
6.61  Two different confidence intervals are 74.25 \pm .98
      and 74.25 \pm 1.18. They certainly cannot both be correct. 
      I have no opinion on which one is better, captain. No, I don't have 
      a feel for standard deviation either.

6.63  mu = 403.0 +- 10.4

6.65  (a) mu = 3.2 E(12) +- .27 E(12), 
      (b) We can be 95% confident that the interval 
          [2.9 times 10^{12}, 3.5 times 10^{12}] captures the population mean.

6.67 (a) mu = 6 +- 5.996 min. 
     (b) The evidence that the student is, on the average, more than a 
         minute late to class each day is not statistically significant
         at the 90% confidence level.

6.69 (a) mu = 26 +- 114, 
     (b) At a 95% confidence level, the data do not constitute 
         statistically significant evidence that the population 
         mean is less than 100.

6.71 (a) mu = (9.2 +- .18)E(5), 
     (b) The only assumption we must make is that the sample is random.

6.73  mu = 107.4 +- 1.353 degrees centigrade. It is assumed that the chemist 
      is not making any systematic errors and that the only errors are random 
      experimental errors. It is also assumed that the chemist's procedures 
      will generate a normally distributed population of measurements.

6.75  (a) mean: 46, s: 30.15, C.I.:  48 \pm 25.21, 
      (b) mu = 50,    (c) yes.

6.77  (a) True period = 125.8 +- 4.0 hours. This is the true period as 
          determined by the student's data. It will be a valid confidence 
          interval for the true period of the star if there is no bias in the 
          measurements. 
      (b) The accepted period of Delta Cephei is 5.366 days.  Since this 
          number is within the confidence interval determined from 
          the student's data, the student's data do not appear to be 
          biased.  Good job, student! 
      (c) 128 hours 39 +- 10 minutes

6.79  (a) mu = 40.5 +- 11.86 in.; 
      (b) The confidence interval is valid only under the assumption that 
          the target population, ``sturgeons that may be observed in that 
          location by people like the biologist'', is normally distributed.

6.81  (a) .176,   (b) .0068,   (c) 41.5 \pm 17.8

6.85 K = .000240 +- .000066

6.87  (a) .6; (b) .648; 
      (c) .1104 (without continuity correction), .1308 (with cc),
          .1303 (precise value)

6.89  1/15

6.91  From table: (a) .8554; (b) .28

6.93  mu = 45.3 +- .93