Question

3. Fred, like all normal people, loves to bowl. He never has an open frame. That is, each frame is either a strike (x) or a spare (/). Strikes are better. Here is his data a. What is the conditional probability of a strike in the frame following a strike? b. How many runs for Fred? c. The runs test for this data gives the following. Expected Number of Runs: 11.2857; s.d: 2.1853 z-value--1.0460 Explain the results of this test. Does it show evidence of the hot hand?

Answer 1

Answer:

giventhat

a. As per the data, the total number of strikes by Fred = 11-1 = 10. (We ignore one observation from the end because it cannot be followed by another observation)
Again, number of cases in which a frame is a strike, AND the preceding frame was also a strike = 7.
Therefore, the conditional probability of a strike in the frame following a strike
= No. of cases where a strike was preceded by a strike / No. of strikes
= 7/10
= 0.70

b. Each group of successive values of the two outcomes is called a run. For example, the 3 successive strikes at the beginning constitute a run. Similarly, the 5 successive spares in the middle constitute a run.
Therefore, the entire data set consists of 7 runs.

c. The z-statistics in a runs test tests the null hypothesis of randomness against an alternative hypothesis of non-randomness of the outcomes.
Here, z-value = -1.0460 p-value for the z-value = 0.147781
Therefore, assuming a two-sided alternative hypothesis, based on the sample evidence, we reject the null hypothesis at the 5% level of significance.
In other words, the sample shows evidence of a hot hand at the 5% level of significance (i.e. the outcome deviates from randomness).