According to an article in Newsweek, the natural ratio of...

According to an article in Newsweek, the natural ratio of girls to boys is $100: 105 .$ In China, the birth ratio is 100: 114 (46.7\% girls). Suppose you don't believe the reported figures of the percent of girls born in China. You conduct a study. In this study, you count the number of girls and boys born in 150 randomly chosen recent births. There are 60 girls and 90 boys born of the $150 .$ Based on your study, do you believe that the percent of girls born in China is $46.7 ?$

Key Concepts

Test Statistic and p-value

A test statistic quantifies the difference between the observed sample data and what is expected under the null hypothesis, usually in terms of standard errors. The corresponding p-value measures the probability of obtaining results at least as extreme as those observed, assuming that the null hypothesis is true. These tools help in deciding whether to reject the null hypothesis based on the evidence provided by the sample.

Type I and Type II Errors

In hypothesis testing, a Type I error occurs when the null hypothesis is wrongly rejected when it is true, while a Type II error occurs when the null hypothesis is not rejected even though the alternative hypothesis is true. Understanding these errors is essential for evaluating the risks associated with making decisions based on sample data, particularly when assessing the accuracy of reported birth ratios.

Hypothesis Testing

Hypothesis testing is the process of making statistical decisions using experimental data. It involves setting up a null hypothesis (in this case, that the birth ratio of girls in China is 46.7%) and an alternative hypothesis (that it is different), then using sample data to determine whether there is enough evidence to either reject the null hypothesis or fail to reject it.

Sampling Distribution of Proportions

The sampling distribution of proportions represents the distribution of sample proportions one would obtain if many samples were drawn from the same population. This concept is crucial for understanding how likely it is to observe a particular proportion of girls (such as 40% in the study) if the true proportion were actually 46.7%, allowing for the quantification of sampling variability.

Transcript

00:01 Oh i've got a little battery i can't stop hang on while i plug in sorry about this just got to get my computer plugged in so it doesn't die mid -solving i can't see what i'm doing shouldn't take too long almost done there that one's going get a spot where i can now maneuver mover okay so our natural ratio of girl births to boy births is 100 to 105 and in china we're told that it's 100 to 114 we conduct a study of boy births and girl births and we get 60 girls and boys in studying randomly chosen 150 recent births and do we believe this percentage so the null hypothesis this gonna i'm not gonna go ahead and explain the whole thing but we're going to be considering a null hypothesis here.

01:09 If this is true, then an alternate hypothesis will be the opposite.

01:26 So it's not the same as the null.

01:28 Ok, so according to this, the claim is this.

01:34 So our claim is that this is 0 .467.

01:43 This is our null.

01:45 And our alternative is not equal to 0 .467.

01:56 That's simple enough.

01:58 So first we need to know the population.

02:01 So we're going to have q equals 1 minus p equals 1 minus 0 .467 equals 0 .533.

02:15 So this is a binomial experimental design.

02:33 And we can check if it follows this.

02:37 150 times 0 .467 which is equal to 150 times 0 .467 is 70 .05 is greater than 5.