Respuesta :
a) The probability the home team wins is of: 0.7291 = 72.91%.
b) The conditional probability that the home team wins, given that is is behind 5 points by the end of halftime is of: 0.3409 = 34.09%.
c) The conditional probability that the home team wins, given that it is ahead by 5 points at the end of the first quarter is of: 0.9015 = 90.15%.
How to obtain probabilities using the normal distribution?
The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.
- Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.
The mean and the standard deviation for each quarter are given as follows:
- Mean: 1.5.
- Standard deviation: square root of 6 = 2.45.
Hence for each case, the mean and the standard deviation are given as follows:
- Entire game -> Four quarters -> [tex]\mu = 6, \sigma = 9.8[/tex]
- After halftime -> Two quarters -> [tex]\mu = 3, \sigma = 4.9[/tex]
- After the first quarter -> Three quarters -> [tex]\mu = 4.5, \sigma = 7.35[/tex]
The probability of the home team winning is the probability of X being greater than zero for four quarters, hence it is one subtracted by the p-value of Z when X = 0, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Z = (0 - 6)/9.8
Z = -0.61
Z = -0.61 has a p-value of 0.2709.
1 - 0.2709 = 0.7291 = 72.91%.
When the home team is down five points at the end of the halftime, the probability of winning is P(X > 5) for two quarters, hence one subtracted by the p-value of Z when X = 5 as follows:
Z = (5 - 3)/4.9
Z = 0.41
Z = 0.41 has a p-value of 0.6591.
Z = has a p-value of 0.6591.
1 - 0.6591 = 0.3409 = 34.09%.
When the home team is up five points at the end of the first quarter, the probability of winning is P(X > -5) for three quarters, hence one subtracted by the p-value of Z when X = -5.
Z = (-5 - 4.5)/7.35
Z = -1.29
Z = -1.29 has a p-value of 0.0985.
1 - 0.0985 = 0.9015 = 90.15%.
More can be learned about the normal distribution at https://brainly.com/question/25800303
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