Question

In: Statistics and Probability

The probability of a student playing basketball is known to be 0.53; and the probability of...

  1. The probability of a student playing basketball is known to be 0.53; and the probability of a student playing soccer is known to be 0.43. If the probability of playing both is known to be 0.35, calculate:
    1. the probability of playing soccer
    2. the probability of playing at least one of basketball and soccer
    3. the probability of a student playing soccer, given that they play basketball?
    4. Are playing soccer and basketball independent? Justify
    5. (harder) a group of 29 randomly selected students attend a special seminar on the health benefits of playing sport. Of these 29, only 6 play neither after the seminar. State a sensible null hypothesis, test it, and interpret.

Solutions

Expert Solution

Given,

a)the probability of playing soccer p(s)=0.43

b)the probability of playing at least one of basketball and soccer

P(B or S)=P(B)+P(S)-P(B and S)

=0.53+0.43-0.35=0.61

c)the probability of a student playing soccer, given that they play basketball

P(S|B)=P(B and S)/P(B)

P(S|B)=0.35/0.53=0.6604

d)Are playing soccer and basketball independent:

no, playing soccer and basketball are not indepedent.because P(B and S)P(B)*P(S)

e)

Probability of playing neither basketball or Soccer= 1 - probability of playing at least one of basketball and soccer

= 1 - 0.61 = 0.39

Null Hypothesis H0: p = 0.39

Alternative Hypothesis H0: p 0.39

given,

n=29, x=6

p̂=6/29=0.2069

Since it is observed that ∣z∣=2.022>zc​=1.96, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0432, and since p=0.0432<0.05, it is concluded that the null hypothesis is rejected.

Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population proportion p is different than p0​, at the α=0.05 significance level.


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