Answer
As the null hypothesis is not rejected, the researcher concludes that there is not enough evidence to show that the sequence of random integers generated does not occur randomly.
Use Excel to randomly generate a sequence of 20 integer values of 0s or 1s.
Work Step by Step
Hence, the random numbers generated form a sequence as follows:
\[\underline{1\text{ 1 1}}\text{ }\underline{\text{0}}\text{ }\underline{\text{1 1 1 1 1}}\text{ }\underline{\text{0}}\text{ }\underline{\text{1}}\text{ }\underline{\text{0}}\text{ }\underline{\text{1 1}}\text{ }\underline{\text{0}}\text{ }\underline{\text{1}}\text{ }\underline{\text{0}}\text{ }\underline{\text{1}}\text{ }\underline{\text{0 0}}\]
Define the total number of random numbers being sampled as n. Define \[{{n}_{1}}\] to represent the number of 1s and \[{{n}_{2}}\] to represent the number of 0s. Define r to represent the number of runs.
The provided sequence shows that there are 20 random numbers. Out of 20 random numbers, 13 random numbers are 1s and 7 random numbers are 0s.
In the provided sequence, the first three random numbers are 1s and form a run of length 3 as they are of the same type. The next random number of 0 is of length 1 as it is of another type. The next run consists of five random numbers of 1s. In this way, the number of runs is 12.
Hence, the values are \[n=20,\ {{n}_{1}}=13,\ {{n}_{2}}=7,\ \text{and}\ r=12\].
To test the randomness of the data, verify whether the requirements for randomness are satisfied.
1. The provided sample is a sequence of observations according to their order of occurrence.
2. The provided observations consist of two mutually exclusive categories: random numbers of 0s and 1s. Therefore, the requirements are satisfied.
A hypothesis test is conducted through the following steps to test the randomness of the data.
Step 1: Assume that the provided data is random. Now, formulate the null and alternative hypotheses as follows:
\[\begin{align}
& {{H}_{0}}:\ \text{the sequence of the data is random}\text{.} \\
& {{H}_{1}}:\ \text{the sequence of the data is not random} \\
\end{align}\]
Step 2: State the level of significance, α.
The level of significance is provided as \[\alpha =0.05\].
Step 3: The number of runs r is used to compute the test statistic. As the sample size 20 is equal to 20, the provided problem is a small-sample case. For a small-sample case, the number of runs r is the test statistic. The number of runs is obtained as 12. Therefore, the test statistic is \[r=12\].
Step 4: The values of the number of observations of the two types are obtained as \[{{n}_{1}}=13\ \text{and}\ {{n}_{2}}=7\] in the previous steps.
Use the critical values table for the number of runs to determine the intersection of the row that corresponds to \[{{n}_{1}}=13\] and the column that corresponds to \[{{n}_{2}}=7\]. This gives the lower and the upper critical values.
Hence, the lower and the upper critical values are obtained as 5 and 15, respectively.
The test statistic \[r=12\] is not less than the lower critical value 5, and not greater than the upper critical value 15. Hence, the null hypothesis is accepted.
Step 5: Conclude that there is not enough evidence to show that the sequence of random integers generated does not occur randomly. The provided data supports the assumption of randomness.
