In Exercises 5–8, use the runs test with a significance level ...

Question

In Exercises 5–8, use the runs test with a significance level of α = 0.05. (All data are listed in order by row.)

Odd and Even Digits in Pi A New York Times article about the calculation of decimal places of π noted that “mathematicians are pretty sure that the digits of π are indistinguishable from any random sequence.” Given below are the first 25 decimal places of π. Test for randomness in the way that odd (O) and even (E) digits occur in the sequence. Based on the result, does the statement from the New York Times appear to be accurate?

Accepted Answer

Hello, everyone. Let's take a look at this question together. A researcher flips a coin 25 times and records the outcomes using H for heads and T for tails. The sequence of outcomes is as follows, where we have the outcomes of the coin being flipped 25 times with H representing heads and T for tails. Using a significant level of 0.05, conduct a runs test to determine whether the sequence appears random with respect to heads and tails. Based on your conclusion, does the coin flipping process appear to be random? Is it answer choice A, yes, because the number of runs falls within the critical limits? Answer choice B, yes, because the number of runs falls below the lower critical limit. Answer choice C yes, because the number of runs falls above the higher critical limit, or answer choice D no because the outcomes are not evenly split between heads and tails. So in order to solve this question and conduct a runs test, we must first organize our data into a set of runs, and we do so by first counting the number of H's and T's in our data, which is the number of heads represented by H. will be represented as N1, which equals 13, and the number of tails, which is the letter T is represented as N2, which is equal to 12. And then we can count the number of runs which are sequence changes from H to T or T to H. and then we group the identical consecutive outcomes to count the runs, which gives us the following table of the different run numbers and the outcomes for each run. And based on the table, we can identify the number of runs is equal to 15. So now we can get the critical values for the runs test. Which we locate the critical values corresponding to our N1 equaling 13, our N2 equaling 12, and an alpha of 0.05. And from table A 10, the critical values are the lower limit is 8, and the upper limit is 19, and then the next step is to compare the observed runs to the critical limits. Since if the observed number of runs is less than the lower limit or greater than the upper limit, we reject the null hypothesis of randomness. And if the observed number of runs is between the lower limit and the upper limit, meaning it is inclusive, we fail to reject the null hypothesis. And based on our data, the observed. Runs is equal to 15, and since the number of observed runs is between the lower limit and the upper limit, the number of runs falls within the acceptable range. And so the observed number of runs falls within the critical limits. Therefore, we fail to reject the null hypothesis, and at the 0.0. 5 significance level, the sequence appears to be random, indicating that the coin flipping behaves like a random process. So looking at our answer choices, we can identify that answer choice A is the correct answer, as the coin flipping process does appear to be random because the number of runs falls within the critical limits, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Runs Test

Significance Level (α)

Randomness in Sequences

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