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 α = 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}} let p be the population proportion of successes, where a success is an adult in the country who thinks that healthy children should be required to be vaccinated. state h₀ and hₐ. select the correct choice below and fill in the answer boxes to complete your choice. (round to two decimal places as needed.) a. h₀: p < \square hₐ: p ≥ \square b. h₀: p = \square hₐ: p ≠ \square c. h₀: p ≠ \square hₐ: p = \square d. h₀: p ≤ \square hₐ: p > \square e. h₀: p ≥ \square hₐ: p < \square f. h₀: p > \square hₐ: p ≤ \square

Explanation:

Step1: Identify the claim

The researcher claims that less than 83% of adults think healthy children should be vaccinated. So the alternative hypothesis \( H_a \) is \( p < 0.83 \). The null hypothesis \( H_0 \) is the complement of the alternative in terms of the claim's direction for a one - tailed test. For a claim that \( p < k \), the null hypothesis is \( H_0:p\geq k \), but wait, no - wait, actually, in hypothesis testing for a proportion, the null hypothesis is a statement of equality or a non - strict inequality that is being tested against. Wait, the standard approach: the claim is \( p < 0.83 \) (researcher's claim). So the null hypothesis \( H_0 \) should be \( p\geq0.83 \) and the alternative \( H_a:p < 0.83 \)? Wait, no, looking at the options, option E is \( H_0:p\geq\square \) and \( H_a:p < \square \). The value of \( k \) here is 0.83, since the claim is about less than 83% (0.83). So we need to match the options. Let's check the options:

Option E: \( H_0:p\geq\square \), \( H_a:p < \square \). The value to fill in the square is 0.83, because the claim is that \( p < 0.83 \), so the null hypothesis is the opposite (or the statement we are testing against), so \( H_0:p\geq0.83 \) and \( H_a:p < 0.83 \).

Step2: Match with the option

Looking at the options, option E has \( H_0:p\geq\) and \( H_a:p < \), and the value is 0.83. So we select option E and fill in 0.83 in both boxes.

Answer:

E. \( H_0: p\geq 0.83 \)
\( H_a: p < 0.83 \)