In a table of random digits, each digit is to occur with a probability of 0.1.
a) A student examines a list of 200 random digits and counts only eleven 4’s and thus claims that the table is not really random. Explain the error in the student’s thinking.
b) How many 7’s would you expect to find in a random digit table consisting of 1000 digits?
part A is think is that they assumed that if it was random then the numbers woud be distributed evenly.
part b is 100?
I would EXPECT 100 on the average, but I would expect the actual number to be within sthree standard deviations of that.
The students arror is understanding what expected value is versus samples.
a) The error in the student's thinking is assuming that in a table of random digits, each digit will occur with equal frequency. However, in a random sequence, the occurrence of each digit is independent of the previous or future digits. So, it is possible to have variations in the frequency of specific digits, even if the overall sequence is random. In this case, finding only eleven 4's in a list of 200 digits does not invalidate the randomness of the table. It is within the realm of possibility for the number 4 to occur less frequently in that specific sample.
b) Given that each digit has a probability of occurring with 0.1, we can expect that in a random digit table with 1000 digits, the number 7 will appear approximately 100 times. This calculation is done by multiplying the probability (0.1) by the total number of digits (1000): 0.1 x 1000 = 100. So, you would expect to find around 100 7's in a random digit table consisting of 1000 digits.
a) The error in the student's thinking lies in their assumption that a random distribution of digits would result in an equal number of each digit. In a truly random sequence, there is no guarantee of equal distribution. Each digit has a probability of occurring, but that does not mean they will appear in equal frequency. In the case of the given table with a probability of 0.1 for each digit, it is entirely possible to have more or fewer occurrences of a specific digit than others. So the fact that there are only eleven 4's in a list of 200 random digits does not indicate any issue with randomness. It is within the realm of probability for such an outcome to occur.
b) To determine the expected number of 7's in a random digit table consisting of 1000 digits, we need to consider the probability distribution. Since each digit occurs with a probability of 0.1, we can calculate the expected number of 7's by multiplying the probability by the total number of digits.
Expected number of 7's = Probability of occurrence * Total number of digits
In this case, the probability of occurrence for 7 is 0.1, and the total number of digits is 1000, so we have:
Expected number of 7's = 0.1 * 1000 = 100
Therefore, we would expect to find approximately 100 occurrences of the digit 7 in a random digit table consisting of 1000 digits.