Question

The home team won 1359 out of 2430 games in 2010. Is there any noticeable “home

field advantage” in MLB at α = .01? (Slide #9-24)

1. a) What is n and success rate ̂? n= 2430, ̂=1359/2430 = 0.559

2. b) Set up H0 and H1. H0: p = 0.5(=p0) vs. H1: p > 0.5

3. c) Which conditions should you check before applying CLT? Success/Failure, Independence condition

4. d) Decide an appropriate distribution from H0. N(0.5, √( p0q0/n) = 0.010) by CLT

5. e) Set the decision rule at α = 0.01. z.01 = 2.33If z > 2.33, we can reject H0.

6. f) Compute a test statistic and make a decision. z = ( ̂ –p0) / SE = 5.9 > z.01 We can reject H0.

7. g) Express the decision in terms of the problem.
   There is sufficient evidence to conclude that there is a home field advantage in MLB at α=.01

h) Find the SE (standard error), ME (Margin of error) and the 99% confidence interval. SE = . . =0.010; ME = z∙SE = 2.33 * 0.010 = 0.0233;

CI = .559 ± .0233 = [0.5357, 0.5823]
i) What would you say about home field advantage using the above CI?

Answer 1

a)

b)

The null and alternative hypotheses are defined as,

This is a one-tailed test

c)

The necessary conditions are,

1) The normality condition: np>=30

2) The independence condition: requires the independence of the observations in the sample and this condition holds if the sample size is 10% or less of the population (10% rule) when sampling is being done without replacement

3) Simple random sample: The samples are collected by a simple random sampling method.

d)

The mean and the standard deviation for the sampling distribution of the proportion is obtained using the following formula,

where,

e)

The critical values are obtained from the standard normal distribution table for the significance level = 0.01 and for a one-tailed test.

If the obtained z statistic is greater than the critical value, the null hypothesis will be rejected.

f) Compute a test statistic and make a decision. z = ( ̂ –p0) / SE = 5.9 > z.01 We can reject H0.

The z-statistic is obtained using the formula,

Since the z-statistic is greater than the critical value at a 1% significance level, the null hypothesis is rejected.

g)

There is sufficient evidence to conclude that the home team has home-field advantage in MLB at a 1% significance level.

h)

The standard error is obtained using the following formula,

The margin of error is obtained using the forllowing formula,

where the z critical value = 2.576 for 99% confidence interval.

The 99% confidence interval is obtained using the following formula,

i)

Since the hypothesized proportion, p0 doesn't lie within the 99% confidence interval and the 99% confidence is significantly greater than the proportion, p0 = 0.50 we can conclude that the success rate for the home team is significantly larger than 0.50.