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.

Stem Cell Survey Adults were randomly selected for a Newsweek poll. They were asked if they “favor or oppose using federal tax dollars to fund medical research using stem cells obtained from human embryos.” Of those polled, 481 were in favor, 401 were opposed, and 120 were unsure. A politician claims that people don’t really understand the stem cell issue and their responses to such questions are random responses equivalent to a coin toss. Exclude the 120 subjects who said that they were unsure, and use a 0.01 significance level to test the claim that the proportion of subjects who respond in favor is equal to 0.5. What does the result suggest about the politician’s claim?

Short answer

Nullhypothesis: The proportion of adults who responded in favor is equal to 0.5.

Alternativehypothesis: The proportion of adults who responded in favor is not equal to 0.5.

Teststatistic: 2.694

Criticalvalue: 1.96

P-value: 0.0071

The null hypothesis is rejected.

There isenough evidence to reject the claim that the proportion of adults who responded in favor is equal to 0.5.

Because it appears that more than half of individuals support utilizing federal tax dollars to fund medical research using stem cells derived from human embryos, the politician's claim that people's reactions are random and have no significant meaning is completely false.

Step-by-step solution

Given information

The number of adults who responded in favor is equal to 481, and the number of adults opposed is equal to 401.

Hypotheses

The null hypothesis is written as follows:

The proportion of adults who responded in favor is equal to 0.5.

\({H_0}:p = 0.5\)

The alternative hypothesis is written as follows:

The proportion of adults who responded in favor is not equal to 0.5.

\({H_1}:p \ne 0.5\)

The test is two-tailed.

Sample size, sample proportion, and population proportion

The sample size is computed below:

\[\begin{array}{c}n = 481 + 401\\ = 882\end{array}\]

The sample proportion of adults who responded in favor isas follows:

\[\begin{array}{c}\hat p = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{adults}}\;{\rm{who}}\;{\rm{responded}}\;{\rm{in}}\;{\rm{favour}}}}{{{\rm{Sample}}\;{\rm{Size}}}}\\ = \frac{{481}}{{882}}\\ = 0.545\end{array}\]

The population proportion of adults who responded in favor is equal to 0.5.

Test statistic

The value of the test statistic is computed below:

\(\begin{array}{c}z = \frac{{\hat p - p}}{{\sqrt {\frac{{pq}}{n}} }}\\ = \frac{{0.545 - 0.5}}{{\sqrt {\frac{{0.5\left( {1 - 0.5} \right)}}{{882}}} }}\\ = 2.694\end{array}\)

Thus, z=2.694.

Critical value and P-value

Referring to the standard normal distribution 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 distribution table, the p-value for the test statistic value of 2.694 is equal to 0.0071.

Since the p-value is less than 0.05, the null hypothesis is rejected.

Conclusion of the test

There is enough evidence to reject the claim that the proportion of adults who responded in favor is equal to 0.5.

Since it appears that more than 50% of the adults favor using federal tax dollars to fund medical research using stem cells obtained from human embryos, the politicians claim that the responses of the people are random and do not hold any substantial meaning incorrect.