The Genetics & IVF Institute developed its YSORT method to...

The Genetics \& IVF Institute developed its YSORT method to increase the probability of conceiving a boy. Among 152 women using that method, 127 had baby boys. Assuming that the method has no effect so that boys and girls are equally likely, find the probability of getting at least 127 boys among 152 babies. Does the result suggest that the YSORT method is effective? Why or why not?

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. In this context, each baby represents a trial with the probability of being a boy (success) of 0.5 under the null hypothesis, and the distribution is used to determine the likelihood of obtaining a given number of boys out of a specified number of births.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. In this situation, the null hypothesis is that the YSORT method has no effect (i.e., the probability of a boy is 0.5), and the test examines whether the observed data (127 boys out of 152) is consistent with this hypothesis.

P-value

The p-value is the probability of obtaining results at least as extreme as those observed, assuming that the null hypothesis is true. A very small p-value suggests that such an extreme result would be highly unlikely under the null hypothesis, thus providing evidence against it. In the context of this problem, calculating the p-value helps to determine if the observed number of boys is statistically significant.

Normal Approximation

The normal approximation is a method used to approximate the binomial distribution with a normal distribution when the number of trials is sufficiently large. This is particularly useful for calculating probabilities when computing the exact binomial probabilities becomes cumbersome. The approximation involves using the mean and variance of the binomial distribution along with a continuity correction to estimate the probability of observing a given number of successes.

Transcript

00:01 In this question we have been given to calculate the value of probability based on the given information.

00:05 So here we have the sample size of babies as 152 and it is given that assuming the probability of having a boy as well as girl is equally likely.

00:15 So it will be 1 divided with 2 and hence the q that is probability of having a girl will be 1 minus p.

00:22 So that is also equals to 1 divided with 2 and 1 divided with 2 can be written as 0 .5 right? so now we can say that the population mean which is having the formula n times p will be here 152 multiplied with 1 divided with 2 and that is here equals to 76.

00:40 So the value of sigma, that is population standard deviation will be under root of n pq and that is here equals to under root of 152 and then here it is multiplied with we can say 0 .5 and then it will be here 0 .5.

00:55 So, if we simplify this very expression, then we will have the value of this very expression that is standard deviation as 6.

01:05 It will be here 164.

01:07 So now we can say that the standardized value is the value decreased by the mean and divided by the standard deviation.

01:15 So basically we are using the z score formula to calculate the value of z and that will be here like this and here z value will be 126 .5.

01:25 Minus it will be here 76 divided with it will be here 6 .164.

01:31 So it is here equals to, we can say that it is almost 8 .19.

01:38 Now, solving this very problem further, we can say that let us determine the value of probability, which is probability of x greater than equals to 127, and that is here equals to probability of x, and it can be here written as greater than 126...

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