Question

#3

Sand and clay studies were conducted at a site in California. Twelve consecutive depths, each about 15 cm deep, were studied and the following percentages of sand in the soil were recorded.

29.6

30.3

26.2

30.0

26.9

28.9

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26.5

27.5

27.8

29.5

29.3

23.8

​

Convert this sequence of numbers to a sequence of symbols A and B, where A indicates a value above the median and B denotes a value below the median.  Test the sequence for randomness about the median with a 5% level of significance. What is the value of the sample test statistic R, the number of runs?

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R = 8

R = 9

R = 7

R = 4

R = 6

Answer 1

Sand and clay studies were conducted at a site in California, 12 convective depths, each about 15 cm deep, were studied and the following percentage of sand in the soil were recorded.

29.6, 30.3, 26.2, 30.0, 26.9, 28.9, 26.5, 27.5, 27.8, 29.5, 29.3, 23.8

Now we want to test randomness about the median at 5% level of significance.

Now to calculate number of runs i.e. value of sample test statistic R at first we convert this sequence of numbers to a sequence of symbol A and B, where A indicates value above the median and B indicates value below the median.

[ Median:- median is the middle most value of the data set. When sample size n is odd then ((n+1)/2)th observation is the median and when sample size n is even then any value in between (n/2)th observation and ((n/2)+1)th is median, for simplicity we take mean of these two values as a median. So calculate median of these 12 observation at first we arrange the data set in increasing order and we get, 23.8, 26.2, 26.5, 26.9, 27.5, 27.8, 28.9, 29.3, 29.5, 29.6, 30.0, 30.3. Here, sample size n=12 i.e. even. So median is the any value between (n/2)=12/2=6th observation and ((n/2)+1)=7th observation. 6th observatuon=27.8 and 7th observation=28.9,for simplicity we take mean of these two value, median=(27.8+28.9)/2=28.35 ]

So about the median the sequence is given by, AABABABBBAAB

Now, value of sample test statistic R = number of runs = number of changes of A and B in the sequence = 8

[ AA=1, B=1, A=1, B=1, A=1, BBB=1, AA=1, B=1 So total 8 changes i.e. number of runs is 8 ]

Answer:- The value of sample test statistic R = Number of runs = 8

So the correct answer is 1st option. R=8