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CHAPTER 7 Tests of Hypotheses and Significance
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7.85. The apparent decrease is significant at the 0.05 level but not at the 0.01 level. 7.86. We would conclude that the result is unusual at the 0.05 level but not at the 0.01 level. 7.87. We cannot conclude that the first variance is greater than the second at either level. 7.88. We can conclude that the first variance is greater than the second at both levels. 7.89. No. 7.92. (a) 8.64 7.94. f(x) 7.90. (a) 0.3112 (b) 0.0118 (c) 0 (d) 0 (e) 0.0118. 0.96 oz (b) 8.64
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0.83 oz (c) 8.64
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7.93. (a) 6 (b) 4
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(0.68)4 x; expected frequencies are 32, 60, 43, 13, and 2, respectively.
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7.95. Expected frequencies are 1.7, 5.5, 12.0, 15.9, 13.7, 7.6, 2.7, and 0.6, respectively. 7.96. Expected frequencies are 1.1, 4.0, 11.1, 23.9, 39.5, 50.2, 49.0, 36.6, 21.1, 9.4, 3.1, and 1.0, respectively. 7.97. Expected frequencies are 41.7, 53.4, 34.2, 14.6, and 4.7, respectively. (0.61)xe x!
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7.98. f (x)
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7.99. The hypothesis cannot be rejected at either level. 7.100. The conclusion is the same as before. 7.101. The new instructor is not following the grade pattern of the others. (The fact that the grades happen to be better than average may be due to better teaching ability or lower standards or both.) 7.102. There is no reason to reject the hypothesis that the coins are fair. 7.103. There is no reason to reject the hypothesis at either level. 7.104. (a) 10, 60, 50, respectively (b) The hypothesis that the results are the same as those expected cannot be rejected at a 0.05 level of significance. 7.105. The difference is significant at the 0.05 level. 7.106. (a) The fit is good. (b) No.
7.107. (a) The fit is too good. (b) The fit is poor at the 0.05 level. 7.108. (a) The fit is very poor at the 0.05 level. Since the binomial distribution gives a good fit of the data, this is consistent with Problem 7.109. (b) The fit is good but not too good. 7.109. The hypothesis can be rejected at the 0.05 level but not at the 0.01 level.
CHAPTER 7 Tests of Hypotheses and Significance
7.111. The hypothesis cannot be rejected at either level.
7.110. Same conclusion.
7.112. The hypothesis cannot be rejected at the 0.05 level. 7.113. The hypothesis can be rejected at both levels. 7.114. The hypothesis can be rejected at both levels. 7.115. The hypothesis cannot be rejected at either level. 7.116. (a) 0.3863, 0.3779 (with Yates correction) 7.117. (a) 0.2205, 0.1985 (corrected) (b) 0.0872, 0.0738 (corrected) 7.118. 0.4651
7.119. (a) A two-tailed test at a 0.05 level fails to reject the hypothesis of equal proportions. (b) A one-tailed test at a 0.05 level indicates that A has a greater proportion of red marbles than B. 7.120. (a) 9 (b) 10 (c) 10 (d) 8 7.121. (a) No. (b) Yes. (c) No.
7.122. Using a one-tailed test, the result is significant at the 0.05 level but is not significant at the 0.01 level. 7.123. We can conclude that brand A is better than brand B at the 0.05 level. 7.124. Not at the 0.05 level.
Curve Fitting, Regression, and Correlation
Curve Fitting
Very often in practice a relationship is found to exist between two (or more) variables, and one wishes to express this relationship in mathematical form by determining an equation connecting the variables. A first step is the collection of data showing corresponding values of the variables. For example, suppose x and y denote, respectively, the height and weight of an adult male. Then a sample of n individuals would reveal the heights x1, x2, . . . , xn and the corresponding weights y1, y2, . . . , yn. A next step is to plot the points (x1, y1), (x2, y2), . . . , (xn, yn) on a rectangular coordinate system. The resulting set of points is sometimes called a scatter diagram. From the scatter diagram it is often possible to visualize a smooth curve approximating the data. Such a curve is called an approximating curve. In Fig. 8-1, for example, the data appear to be approximated well by a straight line, and we say that a linear relationship exists between the variables. In Fig. 8-2, however, although a relationship exists between the variables, it is not a linear relationship and so we call it a nonlinear relationship. In Fig. 8-3 there appears to be no relationship between the variables. The general problem of finding equations of approximating curves that fit given sets of data is called curve fitting. In practice the type of equation is often suggested from the scatter diagram. For Fig. 8-1 we could use a straight line y a bx (1)
while for Fig. 8-2 we could try a parabola or quadratic curve: y a bx cx2 (2)
Sometimes it helps to plot scatter diagrams in terms of transformed variables. For example, if log y vs. x leads to a straight line, we would try log y a bx as an equation for the approximating curve.
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