Question

A test of $H_0:p=0.65$against $H_a:p<0.65$has test statistic z = -1.78.

(a) What conclusion would you draw at the 5% significance level? At the 1% level?

(b) If the alternative hypothesis were $H_0:p=0.65$, what conclusion would you draw at the 5% significance level? At the 1% level?

Short answer

a. At $5\%$the null hypothesis gets rejected and at $1\%$it is not rejected.

b. At the $5\%$significance level and at the $1\%$level the null hypothesis doesn't get rejected.

Step-by-step solution

Introduction

The null hypothesis recommends that no measurable relationship and importance exists in a bunch of given single noticed variables, between two arrangements of noticed information and estimated peculiarities.

Explanation Part (a)

The null and alternative hypotheses are,

$\begin{array}{r}{H}_0:p=0.65\\ H_a:p<0.65\end{array}$

Test statistic z = $-1.78$

Calculating the p-value,

$$\begin{gathered} =P(Z>|-1.78\left|\right) \\ =0.0375 \end{gathered}$$

Thus At $5\%$the null hypothesis gets rejected as the p-value is less and at $1\%$it is not rejected as p-value is greater.

Explanation Part (b)

The null and alternative hypotheses are,

$\begin{array}{r}{H}_0:p=0.65\\ H_a:p\ne 0.65\end{array}$

Calculating the p-value,

$=2\times P(Z>|Z\left|\right)$

$=2\times 0.0375=0.0750$

Hence at the $5\%$significance level and at the $1\%$level the null hypothesis doesn't get rejected as the p-value is greater.