Question

a. Determine the sample proportion.

b. Decide whether using the one-proportion $z-$test is appropriate.

c. If appropriate, use the one-proportion $z-$test to perform the specified hypothesis test.

$x=3$

$n=100$

$H_0:p=0.04$

$H_a:p\ne 0.04$

$\alpha =0.10$

Short answer

(a) The sample proportion is $0.03$

(b) It is not appropriate to use one proportion $z-$test.

(c) The required hypothesis test as well as the one proportion $z-$test cannot be carried out.

Step-by-step solution

Part (a) Step 1: Given information

The given data is

$n=100$

$x=3$

$H_0:p=0.04$

$H_a:p\ne 0.04$

$\alpha =0.10$

Part (a) Step 2: Explanation

The sample proportion is expressed as

$\hat{p}=\frac{x}{n}$

The sample proportion is calculated as

$\hat{p}=\frac{3}{100}$

$=0.03$.

Part (b) Step 1: Given information

The given data is

$x=3$

$n=100$

$H_0:p=0.04$

$H_a:p\ne 0.04$

$\alpha =0.10$

Part (b) Step 2: Explanation

Test the hypothesis as

$H_0:p=0.04$

$H_0:p\ne 0.04$

Calculate the value of$n{p}_0$

$n{p}_0=\left(100\right)(0.04)$

$=4$

Calculate the value of$n\left(1-p_0\right)$

$n\left(1-p_0\right)=\left(100\right)(1-0.04)$

$=100(0.96)$

$=96$

Hence, $n{p}_0<5$

Therefore, it is not appropriate to use one proportion $z-$test.

Part (c) Step 1: Given information

The given data is

$x=3$

$n=100$

$H_0:p=0.04$

$H_a:p\ne 0.04$

$\alpha =0.10$

Part (c) Step 2: Explanation

Because $n{p}_0$is fewer than $5$, the required hypothesis cannot be tested, and only one proportion $z-$test can be done.

As a result, the required hypothesis test as well as the one proportion $z-$test cannot be carried out.