Question

A​ gender-selection technique is designed to increase the likelihood that a baby will be a girl. In the results of the​ gender-selection technique, 850 births consisted of 440 baby girls and 410 baby boys. In analyzing these​ results, assume that boys and girls are equally likely.

a. Find the probability of getting exactly 440 girls in 850 births.
b. Find the probability of getting 440 or more girls in 850 births. If boys and girls are equally​ likely, is 440 girls in 850 births unusually​ high?
c. Which probability is relevant for trying to determine whether the technique is​ effective: the result from part​ (a) or the result from part​ (b)?
d. Based on the​ results, does it appear that the​ gender-selection technique is​ effective?

Answer 1

a) binomial probability formula:

P(X = 440) = (850 choose 440) * (0.5)^440 * (0.5)^(850-440) ≈ 0.066

b) The mean number of girls in 850 births is 0.5 * 850 = 425, and the standard deviation is sqrt(850 * 0.5 * 0.5) ≈ 18.38. To standardize the value of 440, we calculate the z-score: z = (440 - 425) / 18.38 ≈ 0.81. Using a standard normal table, we find that the probability of getting a z-score of 0.81 or higher is about 0.209. Therefore, the probability of getting 440 or more girls in 850 births is approximately 0.209.

If boys and girls are equally likely, getting 440 girls in 850 births is not necessarily unusually high. However, getting 440 or more girls in 850 births has a probability of only 0.209, which is relatively low.

c) For figuring out whether the strategy works, the likelihood from section (b) is more important. This is because it accounts for the possibility of obtaining more than 440 girls, which would be even stronger support for the effectiveness of the method.

d) The gender-selection method may be successful based on the results, but we can't say for sure. It's unlikely to have 440 or more girls in 850 births, but it's not impossible either (0.209). The efficiency of the method would require further research to be determined.