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?

Short answer

Null hypothesis: The proportion of babies that are boys is equal to 51.2%.

Alternative hypothesis:The proportion of babies that are boys is not equal to 51.2%.

Test statistic: -0.977

Critical value: 1.96

P-value: 0.3286

The null hypothesis is failed to reject.

There is not enough evidence to reject the claim that the proportion of babies that are boys is equal to 51.2%.

Yes, the results support the belief that 51.2% of the babies born are boys.

Step-by-step solution

Given information

A sample of 860 births is selected, out of which 426 areboys. It is claimed that 51.2% of newborn babies are boys.

Hypotheses

The null hypothesis is written as follows.

The proportion of newborn babies that are boys is equal to 51.2%.

\({H_0}:p = 0.512\)

The alternative hypothesis is written as follows.

The proportion of newborn babies that are boys is not equal to 51.2%.

\({H_1}:p \ne 0.512\)

The test is two-tailed.

Sample size, sample proportion,and population proportion

The sample size is n=860.

The sample proportion of babies that are boys is computed below.

\[\begin{array}{c}\hat p = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{boys}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{babies}}\;}}\\ = \frac{{426}}{{860}}\\ = 0.495\end{array}\].

The population proportion of babies that are boys is equal to 0.512.

Test statistic

The value of the test statistic is computed below.

\(\begin{array}{c}z = \frac{{\hat p - p}}{{\sqrt {\frac{{pq}}{n}} }}\\ = \frac{{0.495 - 0.512}}{{\sqrt {\frac{{0.512\left( {1 - 0.512} \right)}}{{860}}} }}\\ = - 0.977\end{array}\).

Thus, z=-0.977.

Critical value and p-value

Referring to the standard normal table, the critical value of z at\(\alpha = 0.05\)for a two-tailed test is equal to 1.96.

Referring to the standard normal table, the p-value for the test statistic value of -0.977 is equal to 0.3286.

Asthe p-value is greater than 0.05, the null hypothesis is failed to reject.

Conclusion of the test

There is not enough evidence to reject the claim that the proportion of babies that are boys is equal to 51.2%.

The results support the belief that 51.2% of the babies born are boys.