Questions: Life on Other Planets Forty-three percent of people believe that there is life on other planets in the universe. A scientist does not agree with this finding. He surveyed 120 randomly selected individuals and found 69 believed that there is life on other planets. At α=0.01, is there sufficient evidence to conclude that the percentage differs from 43 ? Use the P -value method with a graphing calculator. State the hypotheses and identify the claim with the correct hypothesis. H0: p=0.43 H1: p ≠ 0.43

Life on Other Planets Forty-three percent of people believe that there is life on other planets in the universe. A scientist does not agree with this finding. He surveyed 120 randomly selected individuals and found 69 believed that there is life on other planets. At α=0.01, is there sufficient evidence to conclude that the percentage differs from 43 ? Use the P -value method with a graphing calculator.

State the hypotheses and identify the claim with the correct hypothesis.
H0: p=0.43
H1: p ≠ 0.43

Transcript text: Life on Other Planets Forty-three percent of people believe that there is life on other planets in the universe. A scientist does not agree with this finding. He surveyed 120 randomly selected individuals and found 69 believed that there is life on other planets. At $\alpha=0.01$, is there sufficient evidence to conclude that the percentage differs from 43 ? Use the P -value method with a graphing calculator. State the hypotheses and identify the claim with the correct hypothesis. \[ \begin{array}{l} H_{0}: p=0.43 \\ H_{1}: p \neq 0.43 \end{array} \]

Solution

Solution Steps

Step 1: State the Hypotheses

We set up the null and alternative hypotheses as follows: \[ H_0: p = 0.43 \] \[ H_1: p \neq 0.43 \]

Step 2: Calculate the Test Statistic

The test statistic $Z$ is calculated using the formula: \[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Substituting the values: \[ \hat{p} = \frac{69}{120} = 0.575 \] \[ Z = \frac{0.575 - 0.43}{\sqrt{\frac{0.43(1 - 0.43)}{120}}} = 3.2084 \]

Step 3: Calculate the P-value

The P-value associated with the test statistic $Z = 3.2084$ is: \[ \text{P-value} = 0.0013 \]

Step 4: Determine the Critical Region

For a significance level of $\alpha = 0.01$ in a two-tailed test, the critical region is defined as: \[ Z < -2.5758 \quad \text{or} \quad Z > 2.5758 \]

Step 5: Make a Decision

Since the calculated test statistic $Z = 3.2084$ falls into the critical region and the P-value $0.0013$ is less than $\alpha = 0.01$, we reject the null hypothesis.