Answer :
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:
[tex]H_0: \pi=0.25\\\\H_a:\pi<0.25[/tex]
The significance level is assumed to be 0.05.
The standard error of the proportion is:
[tex]\sigma_p=\sqrt{\dfrac{\pi(1-\pi)}{n}}=\sqrt{\dfrac{0.25*0.75}{503}}\\\\\\ \sigma_p=\sqrt{0.000373}=0.019[/tex]
Then, we can calculate the z-statistic as:
[tex]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[/tex]
This test is a left-tailed test, so the P-value for this test is calculated as:
[tex]\text{P-value}=P(z<-1.9685)=0.0245[/tex]
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.