A toothpaste company wants to declare in an ad that more than $80\%$ of dentists recommend their toothpaste. The company performs

a hypothesis test at the $a=0\mathrm{.}05$ significance level to test their claim.

The company samples $500$ dentists. Of these, $406$ recommend their toothpaste.

Let $p$ represent the population proportion of dentists that recommend their toothpaste.

a$.(2$pts$)$ Specify the null and alternative hypotheses for the company's claim.

$H_0$:$p=0\mathrm{.}80$;$H_A$:$p>0\mathrm{.}80$

$H_0$:$p=0\mathrm{.}80$;$H_A$:$p0\mathrm{.}80$

$H_0$:$p=0\mathrm{.}80$;$H_A$:$p<mn\; id="EditorElement\_5163512">0</mn><mn\; id="EditorElement\_5163513">.</mn><mn\; id="EditorElement\_5163514">8</mn><mn\; id="EditorElement\_5163515">0</mn>$

b$.(2$pts$)$ Calculate the sample proportion. $($Round final answer to $4$ decimal places$)$

Sample proportion:

b$.(3$pts$)$ Calculate the $z$ or $t$ statistic. $($Do not use continuity correction. Round final answer to $4$ decimal places$)$

Statistic:

c$.(3$pts$)$ Find the $p-$value. $($Do not use continuity correction. Round final answer to $4$ decimal places$)$

p$-$value: