Twenty years ago, $56 \%$ of parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. A recent survey found that 280 of 750 parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. Do parents feel differently today than they did twenty yoars ago? Use the $\alpha=0.1$ level of significance.
Because $n p_{0}\left(1-p_{0}\right)=184.8> 10$, the sample size is less than $5 \%$ of the population size, and the sample can be reasonably assumed to be random, the requirements for testing the hypothesis are satisfied.
(Round to one decimal place as needed.)
What are the null and altemative hypotheses?
$H_{0}: P=0.56$ versus $H_{1}: P \neq 0.56$
(Type integers or decimals. Do not round.)
Find the test statistic.
$z_{0}=\square$ (Round to two decimal places as needed.)
Answer
So, the test statistic z0 is approximately \(\boxed{-10.32}\) (rounded to two decimal places)
Step 1 ：State the null hypothesis (H0) and the alternative hypothesis (H1). H0: P = 0.56, H1: P ≠ 0.56
Step 2 ：Calculate the sample proportion (p̂) by dividing the number of successful outcomes by the sample size. In this case, p̂ = 280 / 750 = 0.3733
Step 3 ：Calculate the test statistic (z0) using the formula: z0 = (p̂ - P0) / sqrt[(P0(1 - P0)) / n]. Here, P0 is the proportion in the null hypothesis and n is the sample size
Step 4 ：Substitute the values into the formula: z0 = (0.3733 - 0.56) / sqrt[(0.56(1 - 0.56)) / 750]
Step 5 ：Simplify the expression inside the square root: z0 = -0.1867 / sqrt[0.2464 / 750]
Step 6 ：Calculate the denominator: z0 = -0.1867 / 0.0181
Step 7 ：Finally, calculate the test statistic: z0 = -10.32
Step 8 ：So, the test statistic z0 is approximately \(\boxed{-10.32}\) (rounded to two decimal places)
