Question

twelve different video games showing alcohol use were observed. the duration times of alcohol use were recorded with the times (seconds) listed below. what requirements must be satisfied to test the claim that the sample is from a population with a mean greater than 85 sec? are the requirements all satisfied?
77 15 560 53 0 57 207 43 102 0 2 60
are the requirements all satisfied?
a. yes. a normal quartile plot suggests that the sample is from a normally distributed population, and there are observations above and below 85 sec.
b. no. the conditions for a binomial distribution are not satisfied, and there is not enough information given to determine whether the sample is a simple random sample.
c. no. the sample size is not greater than 30, the sample does not appear to be from a normally distributed population, and there is not enough information given to determine whether the sample is a simple random sample.
d. yes. a normal quartile plot suggests that the sample is from a normally distributed population, and there is enough information to determine that the sample is a simple random sample

Explanation:

Step1: Check sample size

The sample size \(n = 12\). For hypothesis testing about the population mean, if the population standard deviation \(\sigma\) is unknown (which is usually the case when not given), we use the \(t -\)test. The requirements for the \(t -\)test are:

1. The sample is a simple random sample.
2. Either the population is normally distributed or \(n>30\).

Step2: Analyze each option

• Option A: Just having observations above and below \(85\) sec is not a valid requirement for the hypothesis test. The key requirements are about the sample being a simple random sample and the population distribution (or sample size).
• Option B: The problem is about testing a population mean, not a binomial distribution. So, the binomial - distribution - related reasoning is incorrect.
• Option C: The sample size \(n = 12<30\). There is no information given in the problem statement (other than the times) to confirm that it is a simple random sample. Also, looking at the data values \(0,0,2,15,43,57,60,63,77,102,207,560\), a normal quantile plot (which is usually needed to check normality, but not provided in the problem statement text) is likely to show non - normality (due to the outlier \(560\)).
• Option D: There is no information in the problem statement (as presented in the text) to confirm that it is a simple random sample. Just a normal quantile plot suggestion (not confirmed by the problem's text description) is not enough when the sample size is small and no information about the sampling method.

Answer:

C. No. The sample size is not greater than 30, the sample does not appear to be from a normally distributed population, and there is not enough information given to determine whether the sample is a simple random sample.