Question

A medical researcher says that less than ​85% of adults in a certain country think that healthy children should be required to be vaccinated. In a random sample of 600 adults in that​ country, 83​% think that healthy children should be required to be vaccinated. At a .05 significance level ​, is there enough evidence to support the​ researcher's claim?

Answer 1

To determine if there is enough evidence to support the researcher's claim, we conduct a hypothesis test comparing the sample proportion to the null hypothesis proportion. Using the given values, we find a test statistic of -2.45, which is less than the critical value at a significance level of 0.05. Therefore, we reject the null hypothesis and conclude that there is enough evidence to support the researcher's claim.

Step-by-step explanation:

To determine if there is enough evidence to support the researcher's claim that less than 85% of adults think that healthy children should be required to be vaccinated, we need to conduct a hypothesis test.

First, we state the null and alternative hypotheses:

1. Null Hypothesis (H0): p = 0.85 (proportion of adults who think children should be required to be vaccinated is 85%)
2. Alternative Hypothesis (H1): p < 0.85 (proportion of adults who think children should be required to be vaccinated is less than 85%)

Next, we calculate the test statistic using the sample proportion and the null hypothesis:

z = (p bar - p0) / sqrt(p0(1-p0)/n), where p bar is the sample proportion, p0 is the null hypothesis proportion, and n is the sample size.

Using the given values, we find that z = (0.83 - 0.85) / sqrt(0.85(1-0.85)/600) = -2.45.

Finally, we compare the test statistic to the critical value at a significance level of 0.05. Since the test statistic (-2.45) is less than the critical value (-1.96 for a one-tailed test), we reject the null hypothesis.

Therefore, there is enough evidence to support the researcher's claim that less than 85% of adults think that healthy children should be required to be vaccinated.