Runs Test with Quantitative Data In Exercises 21–23, use the f...

Question

Runs Test with Quantitative Data In Exercises 21–23, use the following information to perform a runs test. You can also use the runs test for randomness with quantitative data. First, calculate the median. Then assign a + sign to those values above the median and a - sign to those values below the median. Ignore any values that are equal to the median. Use α = 0.05
Use technology to generate a sequence of 30 numbers from 1 to 99, inclusive. Test the claim that the sequence of numbers is not random.

Use technology to generate a sequence of 30 numbers from 1 to 99, inclusive. Test the claim that the sequence of numbers is not random.

Accepted Answer

Hello there. Today we want to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A scientist records the following 26 temperature readings in units of degrees Celsius over several days. 21, 23, 25, 28, 30, 33 35 38, 40, 42 45 47 49 22 24 27 29 31. 34 36 39 41 43 46 4850. Using a runs test at alpha equals 0.05, determine if the sequence is random. Awesome. So it appears for this particular problem we're asked to take the information that is provided to us by the problem itself, and we're asked to use a runs test at a level of significance alpha of 0.05 in order to determine if the sequence is random for this particular problem. So once again, since we ultimately know that we're Trying to figure out whether or not this sequence is considered random. Let's read off our multiple choice answers to see what our final answer might be. A is the sequence is random. B is the sequence is not random. C is the test is inconclusive, and D is the sample size is too small to conclude. Awesome. So, with that in mind, our first step now is we need to order our numbers, so we need to put it from smallest value to highest number, so we're putting in a numerical order, which we will get 21. 22 23 24. 25, 27, 28, 29. 30, 31, 33, 34, 35, 36. 38 39, 40, 41, 42, 43. 45, 46, 47. 48, 49, and 50. That's our first step putting all of our data points from smallest value to largest value, so in numerical order. Our second step now is we need to determine the median, which once again, as we should recall, the median is the average of the 13th and 14th values. So we need to take parentheses 35 + 36 divided by 2, which is equal to 35.5 when we plug that expression into our calculator. Our 3rd step now is we need to assign positive. We need to assign, this simple positive to values that are above, so I'll put an arrow for above 35.5, and we need to apply a negative symbol to values that are below. I'll have an arrow pointing down to indicate below 35.5. And we need to ignore values that are equal to 35.5, which in this case, there are none. So that will mean for our fourth step when we add Positive signs to 36. 38 39 40 41 42 43 45 46 47 4849, and 50. So this is, once again, when we count. All the values that are assigned a positive symbol will be 13, let me count them up, they'll be 13 values that are considered positive. And on the other hand, all the, and we'll make a note that these are for positive, I'll put positive in quotation marks. So, all of our values that are considered negative. will be as follows. 21 22 23 24 25 27 28 29 30 31 33 34, and 35. Which is also, when we count these values up, they will be equal to 13. So that will mean that N subscript 1 or N1 will be equal to 13 and N2 will be equal to 13. Fantastic. So, our 5th step now is we need to determine the sequence of signs, which when we determine the sequence of signs, it will Look like the following, so we'll have negative, negative, negative, negative, negative, negative, negative, positive, positive, positive, positive, positive, positive, negative, negative, negative, negative, negative, negative, positive, positive, positive, positive, positive, positive, and positive. So that will mean that the 1st 7 are negative, the next 6 are positive, and the next 6 are negative, and the last 13 values are positive. So that will mean, as a result, there will be 4 runs, which I'll call R for short for runs. So there are 4 runs. So that will mean, let me make a note that the number of runs, which we'll call G, is equal to 4. Fantastic. Our sixth step now is we need to recall and note that the number of positive values N1 is equal to 13, and the positive of negative values N2 is equal to 13. And referring to our Larsson textbook, where we have a level of significance, alpha is equal to 0.05. And when N1 is equal to 13 and N2 is equal to 13, we can safely state that the lower critical value, which we'll call G L GL for short, will be equal to 8, and we will say that the upper critical value, which we'll call G subscript U, will be equal to 20. Fantastic. And our 7th and final step is we need to observe the number of runs, which once again G is equal to 4, and comparing, we will see that 4 is less than 8, that is G is less than G subscript L. and because G is less than G subscript L, we can safely We can safely say that we can reject the null hypothesis, so I'll put an X next to H0 because H subscript 0 denotes the null hypothesis. So because G is less than G subscript L, we can reject the null hypothesis, and because we can reject the null hypothesis, we can therefore safely. that this particular sequence is not random. So looking at our multiple choice answers, the correct answer, without a doubt, will have to be multiple choice answer letter B, which states that the sequence is not random. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Runs Test for Randomness

Median and Data Categorization

Significance Level (α) and Hypothesis Testing

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