According to the Entertainment Software Association, the average age of "gamers" (those that routinely play video games)is 30 years old. A random sample of 33 gamers people had an average age of 32.7 years old. Assume that the population standard deviation for the age of gamers is 4.6 years. Use a 90% confidence interval to test the validity of this report and choose the one statement that is correct.
A)Because this confidence interval does not include 30 years, the report by the Entertainment Software Association is not validated.
B)Because this confidence interval does include 30 years, the report by the Entertainment Software Association is not validated.
C)Because this confidence interval does not include 30 years, the report by the Entertainment Software Association is validated.
D)Because this confidence interval does include 30 years, the report by the Entertainment Software Association is validated.

Question

Quizplus · Accepted Answer

CTo calculate the 90% confidence interval for the mean age of gamers, we use the formula for a confidence interval when the population standard deviation is known: $\bar{x} \pm z \frac{\sigma}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $z$ is the z-score corresponding to the desired confidence level, $\sigma$ is the population standard deviation, and $n$ is the sample size.Given: $\bar{x} = 32.7$, $\sigma = 4.6$, $n = 33$, and for a 90% confidence interval, the z-score is approximately 1.645.The margin of error (ME) is calculated as: $ME = 1.645 \times \frac{4.6}{\sqrt{33}} \approx 1.32$.Thus, the 90% confidence interval is $32.7 \pm 1.32$, which is approximately $31.38$ to $34.02$.Since 30 is not within this interval, the report by the Entertainment Software Association stating the average age is 30 years old is not validated by this sample data. However, the correct interpretation of a confidence interval not including a certain value (in this case, 30) does not directly validate or invalidate the report; it suggests that, based on this sample, the average age might be different from 30. Therefore, the correct statement should reflect the evidence provided by the confidence interval regarding the report's claim, but due to the nature of the question and the options provided, the closest accurate choice is that the confidence interval does not include 30 years, which would lean towards the report not being validated based on this sample data.

According to the Entertainment Software Association, the Average Age of "Gamers