Question

A poll surveyed 503 video gamers, and 106 of them said they prefer playing games on a console rather than a computer. An executive at a game console company claims that more than 25% of gamers prefer consoles. Does the poll provide convincing evidence that the claim is true? Use the level of significance.

Answer 1

Answer:

The claim has no evidence to be supported.

On the contrary, there is enough evidence to support the claim that less than 25% of gamers prefer consoles.

Step-by-step explanation:

This is a hypothesis test for a proportion.

The sample has a size n=503.

The sample proportion is p=0.211.

p=X/n=106/503=0.211

The sample proportion is less than 25%, so we will test the claim that less than 25% of gamers prefer consoles.  

Then, the null and alternative hypothesis are:

$H_0: \pi=0.25\\\\H_a:\pi$

The significance level is assumed to be 0.05.

 

The standard error of the proportion is:

$\sigma_p=\sqrt{\dfrac{\pi(1-\pi)}{n}}=\sqrt{\dfrac{0.25*0.75}{503}}\\\\\\ \sigma_p=\sqrt{0.000373}=0.019$

Then, we can calculate the z-statistic as:

$z=\dfrac{p-\pi+0.5/n}{\sigma_p}=\dfrac{0.211-0.25+0.5/503}{0.019}=\dfrac{-0.038}{0.019}=-1.9685$

This test is a left-tailed test, so the P-value for this test is calculated as:

$\text{P-value}=P(z$

As the P-value (0.0245) is smaller than the significance level (0.05), the effect is  significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that less than 25% of gamers prefer consoles.