Question

After game 1 of the World Series (of baseball, a best-of-seven series), the announcers announced that over the previous 20 years, it had happened 12 times that the team that won the first game went on to win the series. They seemed to be suggesting that winning a series 60% of the time was surprisingly high. Is it? In other words, assuming that the two teams are equally likely to win a game and that the games are independent events, what is the probability that the team that won the first game wins the series?

Answer 1

Solution

We will address the problem with hypothesis testing.

Now, to work out the solution,

Given,

‘assuming that the two teams are equally likely to win a game and that the games are independent events', the probability that the team that won the first game wins the series should be ½. So, our approach would be to test if the given data, i.e., 12 times happening out of 20 times is consistent with this probability.

Let p = the probability that the team that won the first game wins the series

Claim :

Winning a series 60% of the time was surprisingly high.

Hypotheses:

Null H₀ : p = p₀ = Vs Alternative H_A : p > ½ [claim]

Test Statistic:

Z = (p_nat - p₀)/√{p₀(1 - p₀)/n}

where

p_nat = sample proportion and

n = sample size.

Calculations:

p0	0.5
n	20
x	12
phat	0.6
Zcal	0.8944272
α(assumed)	0.05
Zcrit	1.6448536
p-value	0.1855467

Distribution, Significance Level, α, Critical Value and p-value:

Under H₀, distribution of Z can be approximated by Standard Normal Distribution, provided

np₀ and np₀(1 - p₀) are both greater than 10.

So, given a level of significance of α%, Critical Value = upper α% of N(0, 1), and

p-value = P(Z > Zcal)

Using Excel Functions: Statistical NORMSINV and NORMDIST, critical value and p-value are found to be as shown in the above table.

Decision:

Since Zcal < Zcrit, or equivalently since p-value > α, H₀ is accepted.

Conclusion :

There is not enough evidence to suggest that the claim is valid and hence we conclude that

it is not necessary that winning the first game would imply winning the series .Answer

DONE