The distribution of birth weight for children born in Country A is N(3462, 500)

Question

The distribution of birth weight for children born in Country A is N(3462, 500) in grams. (a) (1 mark) In Country A, which is more unusual: a baby who is born at 2962 grams or a baby who is born at 3962 grams? Explain. (b) (1 mark) What proportion of Country A's children are born with birth weight betweern 2462 and 2962 grams? (c) (2 marks) Suppose Joe and Lily both come from Country A. Joe's birth weight was 3520 ms and Lily's birth weight was 3330 grams.w weighed more at birth than Lily, but less than Joe? The distribution of birth weight for babies born in Country B is N(3112, 425) (d) (2 marks) Baby Ralph has a birth weight of 3200 grams. What proportion of babies born in Country A had lower birth weight than Ralph? What proportion of babies born in Country B had lower birth weight than Ralph? (e) (1 mark) In which country do you think Ralph was born? Explain. (f) (1 mark) Find the proportion of babies in Country B that have a birth weight greater than the lowest 15%.

Explanation / Answer

1)
mean = 3462 and std. dev. = 500

(A)
2962 - 3462 = -500
3962 - 3462 = 500

As both 2962 and 3962 are equidistant from the mean, hence probability probability of happening both is same. Therefore none is unusual.

(B)
P(2462 < X < 2962)
2962 lies 1 sigma to the left of the mean and 2462 lies 2 sigma to the right of the mean.

P(X < 2962) = P(z < (2962 - 3462)/500) = P(z < 1) = 0.1587

P(X < 2462) = P(z < (2462 - 3462)/500) = P(z < 2) = 0.0228

P(2462 < X < 2962) = 0.1587 - 0.0228 = 0.1359

Required probability = 0.1359

(C)
P(3330 < X < 3520)

P(X < 3520) = P(z < (3520 - 3462)/500) = P(z < 0.116) = 0.5462

P(X < 2462) = P(z < (2462 - 3462)/500) = P(z < -0.264) = 0.3959

P(2462 < X < 2962) = 0.5462 - 0.3959 = 0.1503

Required probability = 0.1503

(D)
Country B, mean = 3112 and std. dev. = 425

For county A, P(X < 3200) = P(z < (3200 - 3462)/500) = P(z < -0.524) = 0.3001

For county B, P(X < 3200) = P(z < (3200 - 3112)/425) = P(z < 0.2071) = 0.582

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