Question

a poll found that 82% of american adults believed that there was a conspiracy in the death of president kennedy. a later poll asked 1088 american adults if they believe there was a conspiracy in the assassination, and found that 881 believe there was a conspiracy. do the data show that the proportion of americans who believe in this conspiracy has decreased? test at the 10% level.
state the hypotheses.
$h_0: p$
$h_a: p$
calculate $hat{p}$. round to four decimal places.
$hat{p}=$
calculate the test statistic. round to three decimal places.
$z =$
find the p - value. round to four decimal places.
$p - value=$
state your decision.
since the p - value is less than.10, reject $h_0$.
since the p - value is less than.10, fail to reject $h_0$.
since the p - value is greater than.10, fail to reject $h_0$.
since the p - value is greater than.10, reject $h_0$.
interpret the results.
at the 10% level of significance, there is enough evidence to show that the proportion of american adults who believe that there is a conspiracy in the death of president kennedy is more than 82%, the value from the earlier poll.
at the 10% level of significance, there is not enough evidence to show that the proportion of american adults who believe that there is a conspiracy in the death of president kennedy is more than 82%, the value from the earlier poll.
at the 10% level of significance, there is enough evidence to show that the proportion of american adults who believe that there is a conspiracy in the death of president kennedy is less than 82%, the value from the earlier poll.
at the 10% level of significance, there is not enough evidence to show that the proportion of american adults who believe that there is a conspiracy in the death of president kennedy is less than 82%, the value from the earlier poll.

Explanation:

Step1: State the hypotheses

The null hypothesis $H_0$ is that the proportion has not decreased, so $H_0:p\geq0.82$. The alternative hypothesis $H_A$ is that the proportion has decreased, so $H_A:p < 0.82$.

Step2: Calculate $\hat{p}$

The sample - proportion $\hat{p}=\frac{x}{n}$, where $x = 881$ and $n = 1088$. So $\hat{p}=\frac{881}{1088}\approx0.8097$.

Step3: Calculate the test - statistic

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 $p_0 = 0.82$, $\hat{p}=0.8097$, and $n = 1088$.

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Step4: Find the p - value

Since $H_A:p < 0.82$, the p - value is the probability of getting a z - score less than the calculated z - value. Using a standard normal table or calculator, the p - value for $z=-0.885$ is $P(Z < - 0.885)\approx0.1884$.

Step5: Make a decision

The significance level is $\alpha = 0.10$. Since the p - value ($0.1884$) is greater than $\alpha=0.10$, we fail to reject $H_0$.

Step6: Interpret the results

At the 10% level of significance, there is not enough evidence to show that the proportion of American adults who believe that there is a conspiracy in the death of President Kennedy is less than 82%, the value from the earlier poll.

Answer:

$H_0:p\geq0.82$
$H_A:p < 0.82$
$\hat{p}\approx0.8097$
$z\approx - 0.885$
p - value $\approx0.1884$
Since the p - value is greater than $.10$, fail to reject $H_0$.
At the 10% level of significance, there is not enough evidence to show that the proportion of American adults who believe that there is a conspiracy in the death of President Kennedy is less than 82%, the value from the earlier poll.