This question: 1 point(s) possible A research center claims that $28 \%$ of adults in a certain country would travel into space on a commercial flight if they could afford it. In a random sample of 1200 adults in that country, $31 \%$ say that they would travel into space on a commercial flight if they could afford it. At $\alpha=0.10$, is there enough evidence to reject the research center's claim? Complete parts (a) through (d) below. (b) Use technology to find the P-value. Identify the standardized test statistic. \[ z=\square \] (Round to two decimal places as needed.) Identify the P-value. \[ P=\square \] (Round to three decimal places as needed.) (c) Decide whether to reject or fail to reject the null hypothesis and (d) interpret the decision in the context of the original claim.
Solution
Step 1 ：State the null hypothesis as \(H_0: p = 0.28\) and the alternative hypothesis as \(H_1: p \neq 0.28\).
Step 2 ：Calculate the standardized test statistic using the formula \(z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\).
Step 3 ：Substitute \(\hat{p} = 0.31\), \(p_0 = 0.28\), and \(n = 1200\) into the formula to get \(z = \frac{0.31 - 0.28}{\sqrt{\frac{0.28(1 - 0.28)}{1200}}} \approx 2.04\).
Step 4 ：Find the P-value by looking up the z-score in a standard normal distribution table or using technology. The P-value is approximately 0.041.
Step 5 ：Compare the P-value with the significance level \(\alpha = 0.10\). Since the P-value (0.041) is less than \(\alpha\), reject the null hypothesis.
Step 6 ：\(\boxed{\text{There is enough evidence at the 10% level of significance to reject the research center's claim that 28% of adults in the country would travel into space on a commercial flight if they could afford it. The sample data suggest that the proportion is different from 28%.}}\)
