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Two brothers, Michael and Alan, were in a bowling league that met once a
week. From past experiences, it is known that both brothers' scores are
approximately normally distributed where Michael has a mean score of 150
with a standard deviation of 30, and Alan has a mean score of 165 with a
standard deviation of 15. Assuming that their scores are independent, which
of the following values is closest to the probability that Michael will have a
greater core than Alan in a single game?
0.16
0.28
0.31
0.33
0.37

Respuesta :

Answer:

The correct option is;

0.28

Step-by-step explanation:

The given parameters are;

The mean score for Michael, [tex]\bar x _1[/tex] = 150

The standard deviation, σ₁ = 30

The mean score for Alan,  [tex]\bar x _2[/tex] = 165

The standard deviation, σ₂ = 15

Taking n₁ = n₂ = 1

[tex]z=\dfrac{(\bar{x}_{1}-\bar{x}_{2})-(\mu_{1}-\mu _{2} )}{\sqrt{\dfrac{\sigma_{1}^{2} }{n_{1}}-\dfrac{\sigma _{2}^{2}}{n_{2}}}}[/tex]

Taking μ₁ - μ₂ = 0

[tex]z=\dfrac{(150-165)}{\sqrt{\dfrac{30^{2} }{1}-\dfrac{15^{2}}{1}}} \approx -0.577[/tex]

The p-value for a z-score of -0.577 from the z-table is 0.28434

Therefore, the probability that Michael, with mean score, [tex]\bar x _1[/tex] = 150 will have a greater score than Alan, with a mean score of [tex]\bar x _2[/tex] = 165 is 0.28434 ≈ 0.28

Therefore, the correct option is 0.28

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