Question

a medical researcher says that less than 83% of adults in a certain country think that healthy children should be required to be vaccinated. in a random sample of 400 adults in that country, 80% think that healthy children should be required to be vaccinated. at \\(\alpha = 0.05\\), is there enough evidence to support the researcher’s claim? complete parts (a) through (d) below.
when you know the number of successes \\(x\\), the sample size \\(n\\), and the population proportion \\(p\\), it can be easier to use the formula shown below to find the standardized test statistic when using a \\(z\\)-test for a population proportion \\(p\\).
\\ z = \frac{x - np}{\sqrt{npq}} \\
\\(\dots\\)
(c) find the standardized test statistic \\(z\\).
\\(z = \boxed{}\\)
(round to two decimal places as needed.)

Explanation:

Step1: Identify values

We have \( n = 400 \), \( p = 0.83 \), sample proportion \( \hat{p}=0.80 \), so \( x = n\hat{p}=400\times0.80 = 320 \), \( q = 1 - p=1 - 0.83 = 0.17 \).

Step2: Substitute into formula

Use the formula \( z=\frac{x - np}{\sqrt{npq}} \). Substitute \( x = 320 \), \( n = 400 \), \( p = 0.83 \), \( q = 0.17 \):

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Answer:

\( z\approx - 1.60 \)