Question

a producer of gel pens claims that 96% of its pens can write more than 10,000 words without losing ink. a random sample of 500 pens is collected, and it is found that 470 of the pens can write more than 10,000 words without losing ink. let $hat{p} =$ the proportion of the sample of pens that can write more than 10,000 words.the probability that 94% or fewer of these gel pens can write more than 10,000 words is 0.0115. does this result provide convincing evidence against the producer of the gel pens?○ yes, the difference between the sample proportion and the parameter is 2%, which is less than 5%.○ yes, the probability of seeing the sample result is so far from what is expected that the probability of it occurring by chance alone is very unlikely ($0.0115 < 0.05$).○ no, it is expected that at least 470 gel pens from this producer will write more than 10,000 words.○ no, the difference between the sample result and what is expected is not extreme enough. the probability of it occurring by chance alone is not unlikely ($0.0115 < 0.05$).

Explanation:

Step1: Calculate sample proportion

$\hat{p} = \frac{470}{500} = 0.94$

Step2: Compare p-value to significance level

Given $p=0.0115 < 0.05$, so the result is statistically significant, meaning the observed outcome is unlikely to happen by chance if the producer's claim (96% success rate) is true.

Answer:

Yes, the probability of seeing the sample result is so far from what is expected that the probability of it occurring by chance alone is very unlikely (0.0115 < 0.05).