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.
$h_0: p$
$h_a: p$
calculate $\beta$. round to four decimal places.
$\beta =$
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.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 $h_0$.
interpret the results.
at the 5% 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 80%, the value from the earlier poll.
at the 5% 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 80%, the value from the earlier poll.
at the 5% 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 80%, the value from the earlier poll.
at the 5% 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 80%, the value from the earlier poll.

Explanation:

Step1: State the hypotheses

The null hypothesis $H_0$ is that the proportion $p$ has not decreased, so $H_0:p = 0.8$. The alternative hypothesis $H_1$ is that the proportion has decreased, so $H_1:p<0.8$.

Step2: Calculate the sample proportion $\hat{p}$

The sample size $n = 1026$ and the number of successes $x = 811$. So $\hat{p}=\frac{x}{n}=\frac{811}{1026}\approx0.7904$.

Step3: Calculate the test - statistic $z$

The formula for the test - statistic in a one - sample proportion test is $z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$, where $p_0 = 0.8$, $n = 1026$ and $\hat{p}\approx0.7904$.

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Step4: Calculate the p - value

Since this is a left - tailed test, the p - value is $P(Z < z)$. Looking up $z=-0.768$ in the standard normal table, the p - value is $P(Z < - 0.768)\approx0.2206$.

Step5: Make a decision

Since the p - value ($0.2206$) is greater than the significance level $\alpha = 0.05$, we fail to reject $H_0$.

Step6: Interpret the results

At the 5% 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 80%, the value from the earlier poll.

Answer:

$H_0:p = 0.8$
$H_1:p<0.8$
$\beta$ is not applicable in the steps above (it is the probability of Type - II error and not directly calculated in this test procedure as given in the problem structure)
$z\approx - 0.768$
$p - value\approx0.2206$
Since the p - value is greater than.05, fail to reject $H_0$.
At the 5% 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 80%, the value from the earlier poll.