Testing Claims About Proportions
In Exercises 9–32, test...

Question

Testing Claims About Proportions

In Exercises 9–32, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.

Births A random sample of 860 births in New York State included 426 boys. Use a 0.05 significance level to test the claim that 51.2% of newborn babies are boys. Do the results support the belief that 51.2% of newborn babies are boys?

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a hospital in Texas records 512 girls out of 10,020 births in a given year. At the 0.05 significance level, test the claim that 50% of newborns are girls. Do the results support the belief that 50% of newborns are girls? We're given 4 possible choices as our answers. For choice A, there is enough evidence to reject the claim. The proportion is significantly different from 50%. For choice B, the sample size is insufficient to draw any conclusion. For choice C, the test is invalid because the sample proportion is too close to 0.5, and for choice D, there's not enough evidence to reject the claim. The results support the belief that 50% of newborns are girls. So, our first step is to actually define our hypotheses. So, for our null hypothesis, which we'll label as H knot. We're going to have the true proportion of newborns being girls, we'll label that as P, and that's going to be equal to 0.50. And our alternative hypothesis, H1, is going to be that P is not equal to 0.50. And this is going to be a two-tailed test, because we are testing if the proportion is different from 50%, not just higher or lower. Next, we're going to need to calculate our proportion from our data. So we'll we'll label that proportion PAT, and that's going to be equal to the number of girls that were born in a given year, which in this case will be 512, divided by the total number of births, which is 1020. And when we calculate this, this is going to be equal to 0.50196. Next, we're going to use the Z test for proportions. And so recall that Z is going to be equal to the quantity of E hat minus Eno. Divided by the square root of the quantity of p knot, multiplied by the quantity of 1 minus p knot in quantity divided by N. Where Phat is the value that we calculated earlier, Pan knot. Is going to be equal to 0.5, and N is going to be the number of births, which is going to be equal to 1020. So, substituting in our values, Z is going to be equal to the quantity of 0.50196 minus 0.5 in quantity, divided by The square root Of the quantity of 0.5, multiplied by the quantity of 1 minus 0.5 in quantity, divided by 1020. And evaluating this expression, this is going to be approximately equal to 0.125. So here we have a Z score of 0.125, and when we use our tables, we see that we get a P value for a two-tailed test, that is about 0.900. And when we compare that to our significance level, we see that 0.900 is actually greater than the 0.05. And so that means That we fail to reject the no hypothesis. So, in conclusion, there is not enough evidence to conclude that the proportion of girls is different from 50%. And based on that conclusion, if we look at our answer choices, we see that the correct answer choice corresponds to answer choice D. There is not enough evidence to reject the claim. The results support the belief that 50% of newborns are girls. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Hypothesis Testing

P-value

Normal Approximation to the Binomial Distribution

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