Question

usa today reported that in 1992, 39% of all elementary school children claimed that when they grow up they want to do something to help other people. however, in 1995, 128 of a random sample of 317 of these some children claimed that when they grow up they want to do something to help other people. does this information indicate that the proportion has increased? suppose a = 0.05. choose the correct set of hypothesis ho: μ = .39 ha: μ > .39 ho: μ = .40 ha: μ > .40 ho: p = .40 ha: p > .40 ho: p = .39 ha: p > .39

Explanation:

Step1: Determine the parameter

We are dealing with a proportion. The claim is about whether the proportion of children who want to do something to help others has increased. So the parameter is \( p \) (proportion), not \( \mu \) (mean).

Step2: Set up the null and alternative hypotheses

The null hypothesis \( H_0 \) is the statement of no change. In 1992, the proportion was \( 0.39 \). But in 1995, we are testing if the proportion has increased. However, when setting up hypotheses for a significance test, we use the value that we are comparing against. Here, if we assume the proportion in 1992 is \( 0.39 \), but when calculating the sample proportion \( \hat{p}=\frac{128}{317}\approx0.404 \). But the correct way is to set up the null hypothesis as the value from the previous known proportion (if we consider the 1992 value as a baseline for comparison in a general hypothesis - testing framework for proportion increase). But if we consider the problem as a test for proportion (not mean), and if we assume that the original proportion (the one we are comparing the new sample to) is \( p = 0.39 \) (the 1992 value). But wait, no! Wait, the formula for the sample proportion is \( \hat{p}=\frac{x}{n}\), where \( x = 128\) and \( n=317\), \( \hat{p}\approx0.404\). But in hypothesis - testing for proportion, the null hypothesis is a statement about the population proportion. If we want to test if the proportion has increased from the 1992 value of \( 0.39\), the null hypothesis \( H_0:p = 0.39\) and the alternative hypothesis \( H_a:p>0.39\)

Answer:

\( H_0:p = 0.39\), \( H_a:p>0.39\) (the last option in the given list)