Question

a poll found that 80% of american adults believed that there was a conspiracy in the death of president kennedy. a later poll asked 1026 american adults if they believe there was a conspiracy in the assassination, and found that 811 believe there was a conspiracy. do the data show that the proportion of americans who believe in this conspiracy has decreased? test at the 5% level.
state the hypotheses.
$h_0: p = 0.8$
$h_a: p < 0.8$
calculate $hat{p}$. round to four decimal places.
$hat{p}=$
calculate the test statistic. round to three decimal places.
$z = 0.768$
find the p - value. round to four decimal places.
$p - value = 0.7208$
state your decision.
since the p - value is less than.05, reject $h_0$.
since the p - value is greater than.05, fail to reject $h_0$.
since the p - value is less than.05, fail to reject $h_0$.
since the p - value is greater than.05, reject $h_0$.

Explanation:

Step1: Calculate sample proportion $\hat{p}$

The sample proportion $\hat{p}$ is calculated as the number of successes (people who believe in the conspiracy) divided by the sample size. Here, the number of successes $x = 811$ and the sample size $n=1026$. So, $\hat{p}=\frac{x}{n}=\frac{811}{1026}\approx 0.7904$.

Step2: Calculate test - statistic (already given as $z = 0.768$)

The formula for the $z$-test statistic for a proportion is $z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$, where $p_0$ is the hypothesized proportion, $\hat{p}$ is the sample proportion, and $n$ is the sample size. Here $p_0 = 0.8$, $\hat{p}\approx0.7904$, and $n = 1026$. But since $z = 0.768$ is already given, we use it for finding the p - value.

Step3: Find the p - value

For a left - tailed test (since $H_a:p<0.8$), the p - value is $P(Z0.05$.

Step4: Make a decision

We compare the p - value with the significance level $\alpha=0.05$. If $p - value<\alpha$, we reject $H_0$, and if $p - value>\alpha$, we fail to reject $H_0$. Since the p - value is greater than $0.05$, we fail to reject $H_0$.

Answer:

$\hat{p}\approx0.7904$
Since the p - value is greater than $.05$, fail to reject $H_0$.