Goodness of Fit Uniform Distribution
if every value has the same probability l/N of occurring. To test the h hypothesis that a random variable has a uniform distribution we obtain random sample of size n and record the frequency of occurrence fo of ea of the N values. Then we compute the (equal) frequencies fee that would expected if,the variable has a uniform distribution. Since the probability any value Is l/N and the sample contains n values, we expect the frequency of each value in the sample will be
one of N = 5 values and we have a sample of n = 400, then’ frequency for each value is
The sample X2, computed as in previous sections from the fa and fe frequencies, is used to decide whether the uniform distribution hypothesis will be accepted or rejected. EXAMPLE A winning number in a Massachusetts lottery is a four-digit number such as 1083or 4416. Digits in the winning number are assumed to be drawn at random. It is assumed that the winnirrg digit population has a uniform distribution; that is, each of the N = 10integers 0, 1, 2, 3, . . . ,9 has the same probability (I/N = litO) of being selected for each place in a winning number. Lou Diamond plays the lottery regularly. She keeps a running tally of digits in past winning numbers, and plays (bets on) a four-digit number made up of digits that have occurred most frequently in the past. Lou’s system is based on the supposition that some digits (the ones that have occurred most frequently in the past) have higher probabilities of occurring than others; that is, the winning digit population does not have a uniform distribution. We will use a random sample of 400 winning digits to test, at the 5 percent level, the hypothesis that the population distribution is uniform. The hypotheses are 1. Ho: the distribution is uniform Ha: the distribution is not uniform The sample data are given in the second column of Table 14.10. There are 400 observations. If Ho is true, each of the 10 digits has a probability .
elation is uniform. Consequently, the test result does not support Lou’s supposition that some digits have a higher probability of occurring than other digits. Figure 14.4 shows the observed and expected frequencies of occurrence that led us to accept Ho. Notice that the observed sample frequencies are not exactly uniformly distributed; however, their differences from uniform frequencies are not large enough to lead us to reject the uniform distribution hypothesis at the 5 percent level. Incidentally, the fact that the sample X2 = 16.55 is only a little less than Xa.05.9 = 16.919 might dampen our enthusiasm for accepting Ho; but the smallness of the difference does not alter the conclusion. The point of importance is this: in correctly applying statistical inference, the analyst must choose a value for a that is consistent with his or her attitude toward the consequences of rejecting a true Ho. That choice should be made before the sample statistic is computed; and that choice establishes the decision rule the analyst should be willing to apply-no matter how close the value of the sample statistic is to the value in the decision rule.
Related Stats Assignments
The Uniform Distribution
Suppose we have a continuous random variable that can take on values on a line segment. If the probability for an interval of a given length is the same rio matter where the interval is between the endpoints of the line segment, the probability distribution is called a uniform distribution. Figure 8.28 shows a uniform distribution over the line segment with endpoints 0 and 1000. The base of the rectangle is 1000;so its height must be 0.001 to make the total area equal to 1;that IS, 1000 x 0.001 = l. The robability “curve” for the uniform distribution is a horizontal line at height 0.00l. The mean of a uniform distribution is halfway between its endpoints. The range of a uniform distribution is the difference between its endpoints. In Figure 8.28,
the mean and range are, respectively, 500 and 1000. In the figure, any interval between 0 and 1000 which has a width of 100 will ave 100/1000, or one-tenth, of the area. In general, for a uniform distribution, the probability for an interval is the width of the interval divided by the range Thus, for the interval from a to b, whose width is b – a, the uniform distribution probability formula is
In the formula, both a and b must be between, or at, the endpoints. Any part of an interval which is outside the endpoints has a probability of zero. For example, for Figure 8.28,
EXAMPLE Suppose that the weight of maple sugar obtained by boiling down a tank of maple-tree sap is uniformly distributed with a mean of 10 pounds and a range of 1.8 pounds. (a) What are the smallest and largest weights of sugar obtained from a tank of sap? (b) What is the probability that a tank of sap will boil down to between 9 and 10.5 pounds?
SOLUTION a. The mean, 10, is the center point of a line segment whose length is the range, 1.8 pounds. Hence, the line segment extends 112(1.8) = 0.9 pound to the left and to the right of 10, that is, from 9.1 to 10.9 pounds. b. Dividing the length of the interval by the range gives
as the probability that a tankful will boil down to between 9 and 10.5 pounds of sugar.
EXERCISE (a) See the last example. What is the probability that a tankful will boil down to more than 9.5 pounds of sugar? (b) What is the height of the rectangle representing the distribution of the example?
ANSWER (a) 0.778; (b) 1/1.8. or about 0.5556.
Uniform distributions are sometimes used in problem situations where we are almost totally without information about the probability distribution for a continuous random variable: Then we may choose to assume that all intervals of a given length are equally likely, that is, we assume the distribution is uniform. Also, as you will find out in the next chapter, uniform distributions are used to demonstrate some important conclusions about the behavior of samples drawn from a population.