Question

A researcher at a medical college conducted a study of 60 randomly selected male soccer players and concluded that frequently "heading" the ball in soccer lowers players' IQs. The soccer players were divided into two groups, based on whether they averaged 10 or more headers per game.

Mean IQs were reported in the article, but the sample sizes and standard deviations were not given. Suppose that these values were as given in the accompanying table.

	n	Sample Mean	Sample sd
Fewer Than 10 Headers	33	114	10
10 or More Headers	27	105	8

(a) Does the sample data support the conclusion that the mean IQ of soccer players who frequently head the ball is lower than that of those who do not frequently head the ball? Test the relevant hypotheses using α = 0.05. (Use a statistical computer package to calculate the P-value. Use μ_{fewer than 10 headers} − μ_{10 headers}. Round your test statistic to two decimal places, your df down to the nearest whole number, and your P-value to three decimal places.)

t	=
df	=
P-value	=

State your conclusion.

Reject H₀. We have convincing evidence that the mean IQ for those who average fewer than 10 headers per game is greater than the mean IQ for those who average 10 or more headers per game.

Do not reject H₀. We do not have convincing evidence that the mean IQ for those who average fewer than 10 headers per game is greater than the mean IQ for those who average 10 or more headers per game.    

Reject H₀. We do not have convincing evidence that the mean IQ for those who average fewer than 10 headers per game is greater than the mean IQ for those who average 10 or more headers per game.

Do not reject H₀. We have convincing evidence that the mean IQ for those who average fewer than 10 headers per game is greater than the mean IQ for those who average 10 or more headers per game.

(b) Can you conclude that heading the ball causes lower IQ?

No, since this was an experiment we cannot conclude that heading the ball causes lower IQ.

Yes, since this was an observational study we can conclude that heading the ball causes lower IQ.    

Yes, since this was an experiment we can conclude that heading the ball causes lower IQ.

No, since this was an observational study we cannot conclude that heading the ball causes lower IQ.

No, since the sample evidence does not support that conclusion we cannot conclude that heading the ball causes lower IQ.

Answer 1

(a)

H0: μ_{fewer than 10 headers} - μ_{10 headers} = 0

H1: μ_{fewer than 10 headers} - μ_{10 headers}  > 0

Assuming the population sd to be equal among the two groups, pooled variance is,

sd² = [(n1 - 1) s1² + (n2 - 1) s2² ] / (n1 + n2 - 2)

= [(33-1) * 10² + (27-1) * 8²) ] / (33 + 27 - 2)

= 83.86207

Standard error of difference =

= 2.3764

Test statistic = μ_{fewer than 10 headers} − μ_{10 headers}  / Std Error

= (114 - 105) / 2.3764

t = 3.79

df = n1 + n2 - 2 = 33 + 27 - 2 = 58

P-value = P(t > 3.79, df = 58) = 0.0002

Since p-value is less than the 0.05 significance level,

Reject H₀. We have convincing evidence that the mean IQ for those who average fewer than 10 headers per game is greater than the mean IQ for those who average 10 or more headers per game.

(b)

Yes, since this was an observational study we can conclude that heading the ball causes lower IQ.