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?
85 15 597 40 0 56 202 45 157 0 2 57
a. the sample observations must be a simple random sample
b. the conditions for a binomial distribution must be satisfied
c. at least one observation must be above or below 85 sec
d. either the population is normally distributed, or n>30, or both
are the requirements all satisfied?
a. yes. the conditions for a binomial distribution are satisfied, and there is enough information to determine that the sample is a simple random sample
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. 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
d. yes. a normal quartile plot suggests that the sample is from a normally distributed population, and there are observations above and below 85 sec
e. 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
f. no. the sample size is not greater than 30, the sample does not appear to be from a normally distributed population, and there are no observations above 85 sec

Explanation:

When conducting a hypothesis test for the mean of a population, if the population standard deviation \(\sigma\) is unknown (which is likely the case here as it is not given), we use the \(t -\)test. The requirements for the \(t -\)test are:

• The sample must be a simple random sample.
• Either the population is normally distributed, or \(n>30\) (where \(n\) is the sample size), or both.

In this case, \(n = 12\) (since there are 12 data points). For a small - sample \(t -\)test (\(n<30\)), we need to check the normality of the population. A normal quantile plot can be used to check if the sample comes from a normally distributed population. Also, the sample should be a simple random sample.

Answer:

A. The sample observations must be a simple random sample; D. Either the population is normally distributed, or \(n > 30\), or both.
For the second part:
If we assume that a normal quantile plot (which is a common way to check normality for small samples) suggests that the sample is from a normally distributed population and if we can assume (or it is given, even though the problem statement is a bit unclear in terms of full information) that the sample is a simple random sample (since the problem is set up as a hypothesis - testing problem, we often assume the basic sampling requirements like simple random sampling unless stated otherwise), then the requirements are satisfied.
D. Yes. A normal quantile 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.